in this paper , we consider the cauchy problem for the cubic - quintic nonlinear schrdinger equation ( nls ) on : which arises as the natural hamiltonian evolution associated to the energy here is a complex - valued function of .this model appears in numerous problems in physics , including field theory , nonlinear optics , the mean - field theory of superconductivity , langmuir waves in plasma physics , and the motion of bose einstein condensates ; see , for example , .this ubiquity is strongly connected to the particular signs appearing in and . as we hope to convince the reader, the combination of focusing cubic nonlinearity and defocusing quintic nonlinearity is physically very natural and leads to interesting mathematics .  for systems at low densities ,nonlinear effects should be weak and hence it is natural to perform a taylor expansion on the associated part of the energy and keep only the lowest nonlinear term .this produces the cubic term in , at least in a gauge invariant setting . such a term with a positive coefficient ( i.e. , the defocusing case ) represents an inherent repulsion of the constituents of the system and leads to global solutions that disperse .a mathematically rigorous proof of this assertion has been available for some time ; see , for example , .a cubic term with a negative coefficient ( the focusing case ) is physically a very natural scenario .atoms / molecules do experience an attractive ( van der waals ) force at moderate densities ; indeed , such attraction underlies the condensation of gases into liquids at low temperatures . similarly , the focusing cubic nls is also commonly used to model the self - focusing of laser beams in certain nonlinear materials .as might be expected , the three - dimensional focusing cubic nls has also received considerable mathematical attention .it was proved in the 1970 s that blowup occurs for an open set of large initial data ; see . for a more up - to - date view of this phenomenon , see and the references therein .  in the physical systemsalluded to above , no true singularity occurs .the very concentration that mathematics predicts degrades the accuracy of the model ; new physics comes into play , preventing further collapse . from a mathematical point of view , the simplest way to incorporate such phenomenology is to introduce a defocusing nonlinearity of the next higher power .a deeper understanding of the underlying physics is , of course , needed to predict the appropriate coupling constant .  by making an appropriate choice of units for space , time , and the solution values , it is possible to scale away any coupling constants in front of the nonlinearities ; only the signs remain .unlike the case of a single power nonlinearity , this exhausts all scaling symmetries .indeed , the dynamics of solutions living at disparate length scales are inherently different ; the amplitude and spatial scale of a solution affects the relative strengths of the linear dispersion and each of the two nonlinearities .we will exploit two conservation laws associated to the flow in addition to the energy , namely , the mass and momentum ; these are given by respectively . through the rescalingused to normalize the equation , information about the relative strength of the two nonlinearities is transfered to the mass and energy .this behooves us to consider initial data for as broad a mass / energy region as we are able . in section [ sec : me ] , we determine precisely which mass / energy pairs are actually possible ; figure [ f : me ] summarizes our results .ultimately , we would like to produce a ` phase diagram ' for , indicating which dynamical behaviours are possible in which regions of the mass / energy plane .this is the long - term dream that guides the investigations in this paper .there is no reason to introduce momentum as a third axis in this phase diagram .the galilei symmetry can be exploited to normalize the momentum to zero , while leaving the mass unchanged and modifying the energy in the obvious manner ( see the proof of proposition [ prop : p(u)=0 ] ) .indeed , this transformation simply amounts to passing to the rest frame of the centre of mass .the physical intuition espoused earlier suggests that should have global solutions .this has been proved rigorously :  [ t : matador ] the initial value problem admits a unique global solution in the class and the solution depends continuously ( in ) on the initial data .furthermore , the solution obeys conservation of mass , energy , and momentum .the cited paper presents a proof of uniqueness only within the subclass of solutions that also belong to .however , the ideas needed to upgrade this to the ` unconditional uniqueness ' formulated above appear already in ; see also for an adaptation of the argument to an equation very similar to.  the global well - posedness of can not be proved by the same simple direct arguments that apply to the defocusing cubic nls .this stems from the fact that the quintic nonlinearity is energy - critical in three spatial dimensions . to better explain this point and its effects on the analysis ,let us first redirect our discussion to the defocusing quintic nls in , whose energy functional is given by   the rescaling is a symmetry of solutions to .this scaling also preserves the energy functional , which is why is termed energy - critical .  for defocusing nonlinearities with power smaller than five , energy conservation prevents concentration .this combined with simple contraction mapping arguments yields a proof of global well - posedness . for the energy - critical problem , this is not the case ; indeed , the scaling symmetry shows that concentration is perfectly consistent with energy conservation .note that we can not use other nls conservation laws to prevent concentration ; energy has the highest regularity among all known conservation laws .the development of methods to tackle this problem of criticality constitutes one of the major breakthroughs in the study of nonlinear dispersive equations . in the case of nls , it was precisely for the equation that this breakthrough was first made :  [ t : quintic ] equation admits a unique global solution for every initial data .this solution obeys   the spacetime bound provides an explicit expression that solutions disperse , something that we previously intuited must hold in the defocusing case .in fact , this bound leads to the following conclusion : there exist asymptotic states so that this says that the solution scatters ( to the linear flow ) as .more formally , this is the statement of asymptotic completeness of wave operators .the question of existence of wave operators is easily settled by contraction mapping arguments , even in this scaling - critical case .the proof of theorem [ t : quintic ] is long and subtle .this remains true , even after incorporating the extensive developments spawned by this breakthrough over the intervening decade ( cf . ) .  as will be evident from the arguments in section [ sec:6 ] ,any solution to can be embedded as a solution to in a certain scaling regime .correspondingly , any reasonably quantitative proof of theorem [ t : matador ] must automatically imply theorem [ t : quintic ] . the key theme underlying the proof of theorem [ t : matador ] in is to argue conversely , namely , to start with theorem [ t : quintic ] and treat the cubic term as a perturbation .( this idea is further expanded upon in ) .moreover , it is shown that if one considers the cubic - quintic nls with _ both _ nonlinearities defocusing , then scattering holds ( in ) even for large initial data .the papers also prove a scattering result for our equation . specifically , they show that if the mass of the initial data is sufficiently small , depending on the norm of the initial data , then scattering holds .to be more precise , sufficiently small mass means smaller than the reciprocal of a tower of exponentials in the norm .this is the best one can achieve via these perturbative methods , because of the best known quantitative bounds in .one of the main results of this paper is a proof of scattering for initial data whose mass and energy belong to the larger region in the mass / energy plane :  [ thm : main ] let be such that belongs to the region defined in section [ sec : virial ] .then , the unique global solution to the cubic - quintic nls satisfies in particular , the solution scatters in both forward and backward in time .the region is depicted in figure [ f : r preview ] .we will give a precise description of later in the introduction , once we have covered the necessary prerequisites . for now , we note the following consequence of theorem [ thm : main ] : there is a mass threshold so that scattering holds for all initial data with , irrespective of the energy ( or -norm ) .  theorem [ thm : main ] is not a perturbative result .we will be able to show that scattering fails for certain solutions whose mass and energy lie on the boundary of .moreover , the exact value of the mass threshold noted above is dictated by the optimizers of a certain gagliardo nirenberg - type inequality ( see section [ sec : gnh ] ) .we show that these optimizers are radially symmetric solutions to a certain elliptic pde .this reduces matters to an ode problem that is readily susceptible to numerical investigation via the shooting method . in this way, we are lead to the assertion that scattering holds for .scattering does not hold for all initial data ; our equation admits solitons . in this paper , we will use the word ` soliton ' to refer to solutions to of the form .naturally , the galilei symmetry can be exploited to introduce ( or remove ) translational motion .  as well as limiting the region where scattering can occur, solitons also constitute an essential ingredient in any purported phase diagram of the dynamical behaviour of . section [ sec : solitons ] is devoted to the study of solitons for our equation .pohozaev identities show that solitons can only exist when ; see lemma [ l : poho ] .that solitons do indeed exist for all such frequencies is then deduced from the general results of .note that the soliton profile is necessarily different for differing ; recall that in the case of a single focusing nonlinearity there is a scaling symmetry , which means that solitons have the same profile for all .the paper actually shows that admits infinitely many radially symmetric solitons for each .we focus our attention on what are known as ground state solitons , that is , solitons for which is non - negative .as discussed in section [ sec : solitons ] , these exist for all ; indeed , there is a unique ground state soliton for each such .we curtail our investigation in this particular way because the ground state solitons are the smallest solitons in a certain mass / energy sense ; see theorem [ t : solitons](i ) .in particular , it is reasonable to predict that it is these solitons which mark the boundary of the region in the mass / energy plane where only scattering occurs .note that pure scattering behaviour will not hold in any region where both mass and energy exceed the mass and energy of a soliton .this is easily seen by constructing solutions containing a soliton and a radiation term .as part of our analysis of the ground state solitons , we are able to determine the kernel of the linearized operator about : specifically , we prove that it is spanned by the components of .this spectral condition is an invaluable stepping stone for any subsequent rigourous analysis of the stability or asymptotic stability of soliton solutions , at least by the usual methods .in particular , our verification of the spectral condition allows one to apply the arguments in to see that solitons are stable wherever the mass / energy curve is concave and unstable where it is convex .figures [ f : pw ] , [ f : numerics1 ] and [ f : numerics2 ] depict the mass / energy curve for the solitons , based on numerics .we note that there is an upper branch of solitons where the curve is convex ( signaling instability ) and a lower branch where it is concave ( indicating stability ) .note that from our result on the linearized operator , we know that is a real - analytic -valued function ; the explanation for the cusp is that and both vanish at the same point .indeed , by , both vanish to the same order .our numerics show a single cusp , corresponding to the vanishing of at a single point , which in turn is the global minimum of and the global maximum of .unfortunately , we have not been able to prove this ; it constitutes our conjecture [ conj : mass ] .the solitons have been the subject of several numerical studies in the physics literature , in their role as light bullets " or 3d spatiotemporal optical solitons " .these numerics confirm our independent investigations and should be consulted by readers interested in plots of mass or energy against the frequency parameter .  the behavior of unstable ( upper branch ) soliton solutions to has been investigated numerically in lower dimensions .these authors observe that solutions beginning near the unstable branch of solitons approach a soliton on the lower branch , shedding their excess mass / energy in the form of radiation .  for nls with a single focusing nonlinearity, ground state solitons can be characterized as optimizers of certain gagliardo nirenberg inequalities .this has played an important role in many investigations , beginning with the seminal work . section [ sec : gnh ] is devoted to the discussion of an analogue for our problem . because there is a one - parameter family of ground state soliton profiles ( unlike the scale - invariant case of a single power nonlinearity ), we will need a one - parameter family of inequalities .our candidates are the following : where .the veracity of is easily deduced by interpolating between the classical gagliardo nirenberg inequality ( which corresponds to ) and the hlder inequality ( which corresponds to ) .in proposition [ prop : gn ] , we show that optimizers of are ground state solitons , up to scaling and translation .moreover , optimizing solitons have the property that , where is defined via in particular , no soliton occurs as the optimizer in for more than one value of  .however , we must acknowledge one short - comming of our results in this direction .numerics show that is strictly increasing , which would imply that each soliton occurs as the unique optimizer for the corresponding value of .we have been unable to prove this ; see conjecture [ conj : beta ] .  as mentioned earlier ,section [ sec : me ] is devoted to determining all feasible mass / energy pairs ; see figure [ f : me ] .a transition occurs at mass , where is a ground state soliton optimizing in the case . for masses below ,the energy is necessarily positive ; the infimal mass is zero , but this is not achieved . at the mass ,zero energy is still the infimal energy , but is now achieved ; moreover , it is achieved only by certain solitons . for masses strictly greater than , the infimalenergy is now negative and is achieved precisely by some soliton .our region is wholly contained in the mass strip where . indeed , in the region it is impossible , using only mass and energy variables , to segregate solutions that scatter from those that converge to a soliton plus radiation .beginning with section [ sec : virial ] we focus more tightly toward the proof of theorem [ thm : main ] . naturally , to obtain a non - perturbative proof of scattering, one needs some intrinsically nonlinear information about the equation .this is true even in the defocusing case , where this role is invariably fulfilled by ( traditional or interaction ) morawetz identities . for focusing equations , sharp thresholdsare typically determined via the virial identity ( suitably truncated ) ; see , for example , .one exception is the work on the mass - critical nls , which uses a hybrid of the virial and interaction morawetz identities .dodson - style variants are not advantageous for our problem. we will use the virial identity .the virial identity stems from the behaviour of a system under scaling ; however , it is not essential that the system has a scaling symmetry . as is easily verified , the operator is the generator of unitary dilations on .an elementary computation shows that if is a solution to , then this is the virial identity for our equation ; correspondingly , we will refer to the functional as the virial .  for soliton solutions , we have and correspondingly , .on the other hand , for solutions of the linear schrdinger equation , we have in particular , the virial of linear solutions is positive for large times .the way that the virial identity is employed in the proof of scattering is rather subtle .one of the key properties of the region is that for any solution to . nevertheless , it is _ not _ true that solutions whose virial is positive for all times must necessarily scatter .imagine , for example , a solution which consists of a soliton together with radiation . by suitable construction, one may decouple the contributions of the two parts of the solution to the virial ; indeed , decoupling will be automatic in the limits .as solitons have zero virial and radiation has positive virial , such a solution would have positive virial for all times , but definitely does not scatter .our example highlights a central problem that arises when applying conservation laws and similar identities : one only observes the aggregate of all the parts of the solution .unwanted behaviour of one part of the solution can not be ruled out if it can be compensated for by unreasonable behaviour of another part of the solution . while sometimes space and/or frequency localization techniques can be applied , this is untenable for generic large data due to the resulting combinatorial complexity .  in , bourgain introduced the induction on energy technique to overcome an obstruction of precisely this type .his solution to the combinatorial morass is to inductively exclude unreasonable behaviour at ever increasing energies .( the energy is a coercive conserved quantity for the problem treated in . )the guiding principle , advanced significantly in , is the following : assume all solutions with energy have been shown to scatter ; this is the inductive hypothesis .we wish to show that this remains true for all solutions with , where is a very small parameter .suppose that at some time , such a solution can be written as two well - separated parts , each with energy at least .then , we may apply the inductive hypothesis to approximate by the sum of two solutions , each of which scatters . that the parts are well - separated is essential .our equation is nonlinear ; it is only in such a regime that one may obtain an approximate solution by summing two solutions . for the models studied in and in this paper , two solutionsare well - separated if they live in different spatial locations or at differing length scales .  as consequence of the preceding , we see that the inductive step is reduced to proving scattering for solutions with that consist of a single piece ( up to errors of size ) , with a clearly - defined location in space and spatial / frequency scale .this substantially enhances the efficacy of conservation laws and monotonicity formulae .more recent applications of the induction on energy technique employ a contradiction argument , closer in spirit to the well - ordering principle . some of these ideas are already hinted at by the language used in , for example , the phrase `` minimal energy blowup solution '' .however , the true mathematical realization of this belongs to keraani and to kenig merle .this is the variant we will describe below .it leads to proofs that are more modular and simpler to understand ; it has also fueled an explosion of applications of the underlying paradigm .next we will explain the construction of a minimal blowup solution , which captures the essence of the induction on energy argument .after that , we will return to our discussion of section [ sec : virial ] and the role of the virial identity .  until now, the size of solutions has been determined by two variables : mass and energy . for a workable notion of minimality , we need to combine them into a single quantity . to this end, we introduce a continuous map in subsection [ ss : exhaustion of r ] .we will also regard as a function of solutions via . for expositional clarity, we will arrange that if and only if and define when .proposition [ p : all about d ] describes further properties of this function .given , we define in this way , theorem [ thm : main ] becomes the statement that for all .suppose now that theorem [ thm : main ] were to fail and let be the supremum of all values of for which is finite .failure of theorem [ thm : main ] is the assertion that . note that , so .in fact , proposition [ thm : small data scattering ] shows that . this proposition is a refinement of the small - data theory developed in .the positivity of corresponds to the ` base step ' of the induction .  under our contradiction hypothesis, there must be a sequence of solutions so that if we could conclude that a subsequence of converges in , perturbation theory ( see proposition [ perturbation of 3 - 5 ] ) would guarantee that the limit is a minimal blowup solution. it would be minimal because and a blowup solution because .the assertion that bounded sequences of solutions converge ( subsequentially ) to a solution is a narrow form of the well - known palais smale condition in the calculus of variations .( our sequence is bounded in because is bounded . )the usual palais smale condition is immediately broken by the presence of non - compact symmetries . for our problem ,the pertinent symmetries are spatial and temporal translations , as well as a vestigial / broken scaling symmetry .the first step in handling this issue is to prove an appropriate concentration - compactness principle for the _ linear _ equation .this is the task of proving a linear profile decomposition ; it is discharged in section [ sec:5 ] .our concentration - compactness principle has strong similarities to earlier work on the energy - critical case ; however , our proofs follow the slightly different path laid out in . for our problem, we must work in rather than the homogeneous space and prove decoupling of -norms as well as -norms .the techniques needed to adapt the argument have appeared before in the context of other problems with broken symmetries , . nevertheless , as a service to the reader we provide full details .  the linear profile decomposition, theorem [ thm : profile decomposition ] , decomposes a subsequence of initial data into linear combinations of asymptotically orthogonal ( as ) bubbles of concentration .each bubble appears at an -dependent location in space - time at some characteristic length scale .as noted earlier , for solutions at very small length scales , the quintic nonlinearity is dominant they behave as solutions to the quintic nls .the role of section [ sec:6 ] is to treat bubbles living at such small length scales by exploiting theorem [ t : quintic ] .  in section [ sec:7 ]we combine the results of sections [ sec:5 ] and [ sec:6 ] to prove that a version of the palais smale condition holds for our problem ( see proposition [ prop : palais - smale ] ) and then that failure of theorem [ thm : main ] would imply the existence of a minimal blowup solution ( see theorem [ prop : existence min blowup sol ] ) .it is here that we perform the ` inductive step ' : the sequence of initial data can not split into more than one bubble of concentration , because this would violate the minimality of . in the language of , _ dichotomy_ can not occur .this argument requires that strictly decreases if either the mass or the energy is decreased ( cf .proposition [ p : all about d](v ) and ( vi ) ) .  because they are exact symmetries of our equation ,space and time translations can be employed directly to tackle these sources of non - compactness . however , the problem of scaling seems to have simply disappeared in the statements of proposition [ prop : palais - smale ] and theorem [ prop : existence min blowup sol ] .let us explain .both nonlinearities are mass supercritical ; correspondingly , solutions with bounded mass but living at very large length scales behave essentially linearly . by the strichartz inequality ,linear solutions obey the spacetime bound .thus solutions living at very large length scales can not arise as minimal blowup solutions .on the other hand , by the analysis of section [ sec:6 ] , solutions living at very small length scales can not blow up either and so can not occur as minimal blowup solutions .naturally , if the sequence of solutions all live at intermediate length scales , there is no need to rescale in order to achieve subsequential convergence .theorem [ prop : existence min blowup sol ] gives more than just the existence of a minimal blowup solution ( under the assumption that theorem [ thm : main ] fails ) .it shows that any such minimal counter - example consists of a single well - localized bump .more precisely , it shows that such a solution is almost periodic modulo translations , that is , there is an -valued function of time so that in fact , this very strong property of such minimal blowup solutions is a rather trivial consequence of our strong palais smale condition .it is now that we finally see the true power of the induction on energy paradigm .it tells us where to center our virial identity and at what radius we can safely truncate ; the latter is dictated by compactness .there is one additional subtlety , however : we need to control how much moves . in proposition[ prop : p(u)=0 ] we observe that minimal blowup solutions have zero momentum .otherwise , one could apply a galilei boost , which preserves the mass and the blowup property , but reduces the energy . in proposition[ prop : control x(t ) ] we use this to constrain the motion of ; specifically , we show that as .this is sufficient to exploit a truncated version of the virial identity to show that is inconsistent with the region admitting an almost periodic solution .this is done in section [ sec:8 ] . as our previous arguments showthat failure of theorem [ thm : main ] guarantees the existence of such a solution , this completes the proof of theorem [ thm : main ] .  by realizing the arguments laid out above , we will be able to show that scattering occurs on any region of the mass / energy plane with the following property : note that non - vanishing of the virial on such a larger set is needed to perform the induction argument we described .  in section [ sec : virial ]we determine the largest region obeying ; this is the region appearing in theorem [ thm : main ] .as we will see , one may determine this region by finding for each mass the least energy at which the virial vanishes .this is formalized in definition [ d : rr defn ] .  from prior investigations ,it is reasonable to imagine that the boundary of the region is marked by solitons ; that is , the functions that achieve minimal energy among those with fixed mass and zero virial are solitons .we will prove that this is _ not _ the case for our model .we find this a startling new observation .it places a formidable limitation on existing technology and raises curious questions for further investigation ; most notably , what is the true extent of the mass / energy region where only scattering holds ?we are able to prove that for some values of the mass , functions with zero virial that achieve energy are precisely solitons , but for other values of the mass such functions are non - trivial rescalings of soliton profiles .these rescalings are _ not _ solitons in their own right !such non - soliton virial obstructions do not appear in earlier work .see theorem [ t : rbdry ] for further information .theorem [ t : mmr ] gives general information on the shape of the curve .let us now discuss the relation to the works , which considered the case where the highest nonlinearity is energy - critical and focusing .in that setting , the scattering threshold is dictated by the radial soliton associated to the purely energy - critical problem , which is unstable to finite - time blowup .in particular , the lower order nonlinearity does not alter the threshold determined earlier in .the authors of the papers consider variational problems based on minimizing a free energy of the form subject to vanishing of the virial , expanding on the methodology introduced by payne and sattinger .( in , . )it is not difficult to see from the results of this paper that our problem has the following properties : ( a ) minimizers may exist ( depending on ) , but no soliton ever occurs as such a minimizer and ( b ) the region can not be exhausted by sub - level sets of such functions .more specifically , we draw the reader s attention to the concavity proved in theorem [ t : feasible ] and the shape of the curve of rescaled solitons shown in figure [ f : numerics1 ] .( the facts about this curve proved in lemma [ l : rw ] are sufficient to verify ( a ) and ( b ) , but the convexity shown by the numerics makes it instantly apparent . )  in closing this introduction , we wish to express two thoughts about future directions :  \1 .we believe that the discovery of the new non - soliton obstructions to the usual tools used to prove scattering is almost as important as the positive results that we prove .it breaks the long - standing tradition that minimization subject to zero virial produces solitons ( or wave collapse with a soliton profile ) . in doing so , it highlights an unexpected inadequacy of prior methods and hopefully will stimulate the investigation of substitutes for the virial identity in the treatment of large data scattering .we hope that this paper may provide some preliminary guidelines for the problem of large - data scattering for the gross pitaevskii and cubic - quintic problems with non - zero boundary conditions at infinity .these problems are much more subtle .the gauge and galilei symmetries are both broken and the mass ( more accurately , the mass defect relative to the constant background ) is no longer coercive . on the positive side ,these problems are known to be well - posed for large data and to scatter for small data .there have also been substantial advances in understanding the structure of solitons for these equations .nonetheless , considerable obstacles ( both variational and dispersive in character ) currently stand in the way of a proof of scattering below the soliton threshold .we write to indicate that there exists some constant so that . if the constant depends on some parameter , we write .we write if .a pair of exponents is _ admissible _ if and . given a spacetime slab , we define analogously , denotes the corresponding dual strichartz spaces .these notations ( introduced in ) allow a compact expression of the strichartz estimates for the schrdinger propagator on :  [ lem : stri ] let be an interval in and let be a solution to then ,   we will often apply this estimate to the derivative of solutions to ; for example , combining with sobolev embedding , we observe that   let us now describe our notations for littlewood paley projectors .fix that is non - negative , radial , and such that if and if .we then define fourier multipliers as follows : \hat{f}(\xi ) , \\\text{and } \quad \widehat{p _ { > n } f}(\xi):=\bigl[1- \psi\big(\tfrac{\xi}{n}\big)\bigr]\hat{f}(\xi).\end{gathered}\ ] ]  for , we have   the brezis - lieb lemma is a refinement of fatou s lemma that has proven invaluable in the calculus of variations :  [ lem : b - l ] fix and suppose is bounded in and converges almost everywhere to .then   as one last preliminary , we remind the reader of a particular consequence of local smoothing ; see ( * ? ? ?* lemma 3.7 ) , ( * ? ? ?* lemma 2.5 ) , or ( * ? ? ?* corollary 4.15 ) .[ l : keraani3.7 ] given , \times\{|x|\leq r\})}^3 \lesssim    t^{\frac25 } r^{\frac{11}5 } \| e^{it\delta } \phi \|_{l^{10}_{t , x}}\| \nabla \phi \|_{l^2_x}^2.\ ] ]the purpose of this section is to discuss soliton solutions of the cubic - quintic nls , that is , solutions of the form with and .evidently , this corresponds to the analysis of the elliptic equation ( elliptic regularity guarantees that distributional solutions in are actually classical solutions , so we need not quibble about the appropriate notion of solution . )  in this paper , we are primarily interested in understanding thresholds for scattering . as we will see , this allows us to focus our attention on positive radial solutions to ; these are typically known as ground states . incidentally , by the well - known results of gidas , ni , and nirenberg , positive solutions to are automatically radial ( about some point ) .  before stating our main results about ground state solitons , we pause to recall some fundamental identities valid for all solutions :  [ l : poho ]let obey for some .then in particular , if then .pairing the equation with yields , from which it also follows that .the second identity , , follows from pairing the equation with .as it is not given that , a little extra care is needed here ; complete details can be found in .  for ,the polynomial appearing in is non - negative ; this then forces .lastly , taking a linear combination of and yields which shows that unless .[ t : solitons ] for each , there is a unique non - negative radially symmetric solution to in fact , is strictly positive and a decreasing function of .moreover ,   is a non - degenerate saddle point of , when viewed as a functional on .the morse index is equal to one .furthermore among all non - zero solutions to , achieves the minimal value of .the map is indeed , it is real analytic .the norm of is strictly increasing ; indeed , using the notation , we also have   as we have , and where is the unique positive radially symmetric solution to .  as we have , , and ; indeed ,   the remainder of this section is devoted to the proof of this theorem .existence of solitons for follows from the main theorem in .we remind the reader of the construction , since we will use it in verifying other parts of the theorem . defining , one first shows that the variational problem admits an optimizer that is non - negative , radially symmetric , and non - increasing ( as a function of radius ) .this is a standard argument , using rearrangement inequalities and the strauss lemma ( compactness of the embedding for radial functions ) .as an optimizer , obeys the euler lagrange equation = 0 { \quad\text{with}\quad } \lambda=\tfrac16\int |\nabla \tilde p_\omega|^2\,dx,\ ] ] the value of the lagrange multiplier being determined via the pohozaev identities for this equation .lastly , observe that is the sought - after soliton .the variational argument gives a bound on the supremum of the solution , which we will use later : {\tfrac12}+\smash[b]{\tfrac12}\smash[t]{\sqrt{1 - 4\omega } } } \quad\text{for all and .}\ ] ] to see this , we consider a generic trial function for and the competitor , where is chosen so that obeys the constraint . as whenever , it is not difficult to verify that if exceeds at some point , then and so will be a better trial function .this proves .uniqueness of non - negative radial solutions to the equation follows from ( * ? ? ?* theorem  ) .as discussed in that paper , there is a lengthy history of proving uniqueness of such ground state solutions ; however , is the earliest paper we have been able to find which covers the cubic - quintic equation .  that is strictly positive , rather than merely being non - negative , follows from the fact that it solves a second - order ordinary differential equation ( when viewed as a function of radius )specifically , at any point where one must also have to avoid a sign change , but then uniqueness for the ode forces .a similar argument also shows that is a _strictly _ decreasing function of .the significance of this is that the strict rearrangement inequalities of brothers and ziemer , then guarantee that any optimizer of must automatically be spherically symmetric ( about some point ) and thus ( by the uniqueness of radially symmetric solutions ) a translate of .this proves that is uniquely characterized by the variational problem.  incidentally , ode methods also allow one to see that is a real - analytic function of and that as , . for these assertions ,see theorems 1.8.1 and 3.8.1 in .we have now verified all the assertions at the beginning of theorem [ t : solitons ] . to aid the reader in navigating the proofs of the numbered parts of the theorem , each is given its own subsection below .theorem [ t : solitons ] gives considerable insight into the nature of the mass / energy curve for the system of solitons , which is depicted in figure [ f : pw ] .some further results are included in subsection [ ss : ii ] ; however , at least one conspicuous question remains open :  [ conj : mass ] there is an so that is strictly decreasing for and strictly increasing for .this conjecture is strongly supported by numerics . by, we see that the map would have the opposite monotonicities , so giving rise to the cusp seen in figure [ f : pw ] . bywe see that this conjecture would also demonstrate the convexity and concavity of the mass / energy curve when and , respectively , that is depicted in figure [ f : pw ] .   recalling the definitions of mass and energy shows that thus , critical points of this functional are precisely solutions to .a proof that the solution constructed by the variational argument minimizes over the class of all non - zero solutions to can be found in .the hessian of at is given by the operator note that this operator is also important as the linearization of around .thirdly , this operator appears when seeking ground states via the shooting method ; specifically , it gives the equation for the derivative of the solution with respect to the shooting parameter , which is the value of the solution at the origin .as is radially symmetric , the operator decomposes as a direct sum of operators , one in each angular momentum eigenspace .moreover , separation of variables associates to each such restricted operator an ode that can be studied by sturm liouville methods . with this in mind , we regard as a function of , rather than , whenever this is more convenient .note also that is a relatively compact perturbation of and so has only discrete spectrum on .the statement of theorem [ t : solitons ] focuses attention on the zero angular momentum component of because standard simple arguments of general applicability ( see below ) yield the structure at higher angular momentum .indeed , the central subtle question is whether zero is an eigenvalue of the restriction of to the space of radially symmetric functions .this question arises , for example , in understanding the stability / instability of the ground state solitons ; see .it is also a key step in the influential coffman kolodner approach to uniqueness of the ground state ( cf . ) .  while a number of general uniqueness theorems have been proven by the coffman kolodner method , we have not found an instance that is applicable to the cubic - quintic problem of interest to us .our proof of theorem [ t : solitons](i ) is strongly influenced by these papers , particularly , and our success here perhaps leads to further generalization this method .we do not pursue this since the competing approach used in provides the uniqueness statement we need .proposition [ p : ell ] in this subsection reduces the study to the key question identified above ; proposition [ p:2.5 ] settles it .note that the presence of in the null space of is not surprising : from a pde point of view it corresponds to the translation invariance of , while from a variational perspective it represents the same invariance for .[ p : ell ] fix and consider the restriction of to functions of the form where is a spherical harmonic of degree .when the operator has exactly one negative eigenvalue ; it is simple .  when there are no negative eigenvalues .zero is an eigenvalue and its eigenspace is spanned by the three components of .  when the operator is positive definite .we treat the parts in order , beginning with .first observe that belongs to and is radially symmetric . using this as a trial vector shows that has negative spectrum; indeed , a little computation shows   next we show that has at most one negative eigenvalue ( counting multiplicity ) .this is to be expected from the variational characterization of : it is defined by a singly constrained minimization problem and one constraint can only counteract one concave direction . to make this heuristic rigorous we argue by contradiction ; specifically , we suppose that there is a two - dimensional subspace for which next , we apply the implicit function theorem to construct a two - dimensional surface of functions obeying the constraint to be precise , having chosen a real - valued with w\,dx = 1 , ] for which means that rhs is strictly positive .( recall that changes sign at and remains negative there - after . )this yields a contradiction because and making lhs negative .  verifying the upper bound on in is the new input .suppose and so on . comparing the equations for and we see that this implies that oscillates faster than on the interval  ] . to see this note that and while remains positive , it oscillates faster than . thus has at least one zero in  ] .we now compare and on .first note that .if has no zeros on , then it oscillates faster than because on this interval .this is of course self - contradictory , since vanishes at both ends of this interval ( this was our contradiction hypothesis ) .  to reach a contradiction, we will show that does not have a zero in . to this end , suppose that did vanish there and let denote the smallest such zero .a repetition of the second derivative argument above ( now with reversed signs ) shows that must change sign at . however , by we also have for sufficiently large .thus there is a point at which also vanishes and such that on .now , on this smaller interval , oscillates more slowly than .this yields a contradiction by forcing to have a zero in .fix .first , we observe that belongs to a real - analytic branch of solutions to and subsequently identify this branch as consisting of ground state solitons .  as is a relatively compact perturbation of , theorem [ t : solitons](i ) together with the fredholm theorem guarantees that it is an isomorphism of onto .( recall that the subscript indicates radially symmetric functions . )thus , applying the implicit function theorem to the mapping , we see that there is a real analytic curve of solutions to defined in a neighbourhood of and obeying .note that the implicit function theorem ( or the contraction mapping argument used to prove it ) also guarantees that is smooth with values in and so ( by sobolev embedding ) also smooth when viewed as a map into .  to conclude that , it suffices to show that is non - negative , which we will do via a spectral theory argument . for each , the schrdinger operator a zero eigenvalue , as witnessed by itself .moreover , when this is an isolated simple eigenvalue at the bottom of the spectrum ( as follows , for example , from the weyl criterion and sturm comparison ) .thus , by eigenvalue / vector perturbation theory , the zero eigenvalue of must also be at the bottom of the spectrum for small .this in turn guarantees that must be positive ( cf .* theorem xiii.46 ) ) .this completes the proof of part ( ii ) of theorem [ t : solitons ] .   using just and the definitions of mass and energy of , we obtain where and .thus at any point where , = - \frac{1}{2 } \bigl(\frac{dm}{d\omega}\bigr)^{-1}.\end{aligned}\ ] ]  for much of what follows , it is convenient to introduce the notation which generalizes the notion introduced in theorem [ t : solitons ] , namely , .  by exploiting the pohozaev identities from lemma [ l : poho ] , we obtain very compact expressions for the key quantities associated to our solitons : }{3}\int |\nabla p_{\omega}|^2\ , dx , \label{eq : w24 } \\\int p_{\omega}^6\ , dx&= \beta(\omega ) \int |\nabla p_{\omega}|^2\,dx ,   { \quad\text{and}\quad } & e(p_{\omega } ) & = \tfrac{1-\beta(\omega)}{6 } \int |\nabla p_{\omega}|^2\,dx .\label{eq : w6e}\end{aligned}\ ] ]  numerical investigations give compelling evidence for the following :  [ conj : beta ] the mapping is injective ; that is , ground state solitons are uniquely identified by the ratio .several properties of solitons are singled out by their ratio .for example , a soliton has zero energy if and only if it has . in subsequent sections ,several results have cumbersome formulations because we have not yet been able to verify this conjecture .  by theorem [ t : solitons](iv , v ) , injectivity of equivalent to its strict monotonicity .there is another equivalent formulation of conjecture [ conj : beta ] , relating it directly to the variational characterization of , namely , = \tfrac23\log\bigl[\tfrac16\!\int |\nabla p_\omega|^2\,dx \bigr ]   \ \ \text{is a strictly convex function.}\ ] ] this equivalence is easily verified using and .the identity stated here follows immediately from .we turn next to the proof of .it is very simple : by and we have and so yields   the proof of is rather more complicated .we begin by computing the matrix elements of given in table [ table : lw ] . differentiating with respect to yields which we immediately deduce the top row of matrix elements in table [ table : lw ] . for the second row, we use the relation which is a direct consequence of , followed by and ..key matrix elements of . here[ cols="^,^,^,^",options="header " , ]   the minimum of the mass of as varies is represented by the cusp point in figure [ f : pw ] .note that combined with the analyticity of and guarantees rigorously that this will indeed be a cusp .  in section [ sec : virial ] we will find that has a further significance : the curves of solitons and rescaled solitons meet at this point .from we see that the mass / energy curve ( see figure [ f : pw ] ) steepens as grows .our last result for this subsection captures this phenomenon in a manner that will be used twice in the proof of theorem [ t : rbdry ] .[ l : cuspy ] fix with . if for all , then . if on the other hand , for all , then .  from and integration by parts, we have thus using we deduce \,\tfrac{2d\omega}{\omega^2}.\end{aligned}\ ] ] this immediately proves the lemma .note that we obtain strict inequalities because is a non - constant analytic function ; in particular , it is not constant on the interval .the key idea in our analysis of this case is to change variables to this new unknown obeys which we regard as a perturbation of the well - studied problem with . to this end , let denote the unique non - negative radially symmetric solution to the function is strictly positive , smooth , and exponentially decaying .most important for us is the fact that the linearized operator has been shown to be an isomorphism of onto ; see .thus , by the implicit function theorem , there is a real - analytic family of solutions to defined in a neighbourhood of that obeys (x ) + o(\omega^2 ) \qquad\text{in sense.}\ ] ]  that is non - negative and so must equal follows by the ground state argument presented in subsection [ ss : c1 ] .we now move on to studying asymptotics of the mass and energy .the leading terms are almost immediate from the above ; see below .the second order terms require the observation that which follows from , , and direct computation . indeed , using this , , and we have \\ & = \omega^{-1/2 }\| g \|_{l^2_x}^2  + \omega^{1/2 } \langle g + x\cdot \nabla g , g^5 \rangle + o(\omega^{3/2 } ) \\ & = \omega^{-1/2 } \| g \|_{l^2_x}^2  + \tfrac12 \omega^{1/2 } \| g \|_{l^6_x}^6  + o(\omega^{3/2 } ) .\end{aligned}\ ] ] note that this also gives asymptotics for the -norm of via the identity .  using the pohozaev identities and the asymptotics for the mass we deduce we also used here the leading term asymptotic for the -norm of , which is evident from .  a similar analysis yields .   we will exploit the variational characterization of solitons . as there , we write .first we consider an explicit trial function defined by when , when , and is the interpolating linear function of for intermediate values of .the parameters and are chosen so that as we have ; a slightly longer computation reveals and therefore .  by the variational characterization of , via the rescaled function , we then deduce that   the remainder of the argument is based on a direct analysis of the optimizer , which is necessarily positive , radial , and decreasing .we define new lengths and so that as throughout and when , the constraint guarantees that and that in particular , , which is to say , .  we can now get a lower bound on by comparing it to the least value taken by any function that equals on the sphere of radius and equals on the sphere of radius .this minimum is achieved by a harmonic function , specifically , by .thus , from this and our earlier upper bound on , we deduce that from the size of , , and , we also obtain .rescaling we obtain and then by employing the pohozaev identities ,   this completes the proof of the final part of theorem [ t : solitons ] .the appearance of solitons as optimizers in sobolev and gagliardo nirenberg inequalities has played a key role in the analysis of the focusing nls with a single power nonlinearity . in particular , this is an essential ingredient in the determination of thresholds for well - posedness and for scattering in these cases ; see .  in this sectionwe discuss an analogous characterization of solitons for the cubic - quintic problem in terms of a one - parameter family of inequalities we dub gagliardo nirenberg hlder inequalities ; see below .we suggest this name because they interpolate between the gagliardo nirenberg and hlder inequalities as the parameter varies .while the veracity of the inequality below for some constant is easily seen by taking a ( weighted ) geometric mean of these classical inequalities , this argument tells us little about the optimal constant or any optimizing functions .[ prop : gn ] fix and let denote the optimal ( i.e. infimal ) constant so that then and the inequality admits optimizers ; moreover , every optimizer is of the form where , , , and is a ground state soliton with .theorem [ t : solitons ] shows that is real - analytic with as and as .thus for any , there are only finitely many so that .one shortcoming in our result is that we are unable ( at this time ) to guarantee that all solitons appear as optimizers of nor are we able to preclude that two different solitons are both optimizers for the same inequality . as will be apparent from what follows , a positive resolution of conjecture [ conj : beta ]would obviate these concerns .that follows from as noted above .it can also be deduced in a similar manner from sobolev embedding ( specifically , ) .next , we prove that the optimal constant is achieved . to this end , let and let be a sequence realizing the infimum ( i.e. ) . by the usual rearrangement inequalities , we may assume that are non - negative , radially symmetric , and non - increasing . by exploiting the fact that is invariant under the rescaling , we may also arrange that and for all .passing to a subsequence , if necessary , we may assume that converges to some in both the weak topology on and the norm topology on .the latter assertion relies on the fact that are radially symmetric and the well - known compactness of the embedding ; see , for example , .sobolev embedding further guarantees that weakly in .next we verify that . by the normalizations and hlder s inequality , we have which shows that indeed .recall that the norm on any banach space is weakly lower semicontinuous .applying this in the numerator of and using that in the denominator yields thus equality holds throughout this line and is an optimizer for .  it remains to demonstrate that all optimizers are ground state solitons up to the obvious symmetries .we begin by treating the case that is non - negative and radially symmetric ; at the end of the proof we will reduce to this case via rearrangement inequalities .  as is a minimizer of , it must satisfy the corresponding euler - lagrange equation : for all .thus , direct computation shows that is a distributional solution to the following equation :   a little further computation shows that if we define where are given by then , satisfies   as is non - negative and radially symmetric , theorem [ t : solitons ] shows that uniquely identifies as being for this value of ; moreover ,   to complete the discussion of optimizers for we now need to consider the case that is not non - negative and radial .standard rearrangement inequalities guarantee that the symmetric decreasing rearrangement of will also be an optimizer . by what we have just proven , then be the rescaling of a ground state soliton .consequently , all level sets of have zero measure .thus by the strict rearrangement inequalities of , it follows that agrees with up to translation and multiplication by a unimodular complex number .this shows that for some , , , and a ground state soliton with , thereby completing the proof of proposition [ prop : gn ] .[ lemma : varitions of m , e ] fix and let denote a ground state soliton that optimizes the -gagliardo nirenberg hlder inequality .then , the optimal constant in that inequality is given by moreover , if  and is a ground state soliton that optimizes the -gagliardo - nirenberg - hlder inequality , then   as is a ground state soliton with , it obeys the relations and ; in particular , and . substituting these into leads immediately to .as , proposition [ prop : gn ] guarantees that is not an optimizer for the -gagliardo - nirenberg - hlder inequality . using and the fact that obeys the relations and , this yields   reversing the roles of and , we also obtain   combining and gives from which we will derive the remaining assertions of the lemma .  skipping over the middle term in and rearranging gives which then implies because .  from and the first inequality in we deduce while follows from and the second inequality in .note that when or , and follow directly from .finally , follows from and .  in view ofwe may say colloquially that however , we have not proved that uniquely determines a ground state soliton .correspondingly , the proper formulation is that holds for _ any system _ of optimizing solitons .  as a counterpoint ,recall from that is strictly increasing . combining this with yields the following :if and are optimizers for the gagliardo nirenberg hlder inequalities with parameters , respectively , then .this does not resolve conjecture [ conj : beta ] because we do not know that every occurs as an optimizer in for some .as is increasing for and decreasing for , the inequalities and imply  [ 1,\infty) and decreasing on .}\ ] ] note that the energy is negative for ; see .the purpose of this section is to give an essentially complete description of the possible pairs for .recall that our results are summarized in figure [ f : me ] .the symbol appearing in this figure represents a ground state soliton that optimizes the gagliardo - nirenberg - hlder inequality with parameter , as in lemma [ lemma : varitions of m , e ] .we will continue to use this notation throughout the section .numerically , we find that is unique ; its vital statistics can be found in table [ table : num ] .although we do not have a proof that is unique , the masses of all such minimizers are identical . indeed , shows that the mass of can be expressed in terms of the optimal constant in via .we will describe the possible mass / energy pairs by characterizing the possible energies for each fixed value of the mass . to this end , it is convenient to introduce the notations this infimium is always finite ; indeed , below shows that .[ t : feasible ] is continuous , concave , non - increasing , and non - positive .furthermore ,  if , then and .if , then . in particular , .if , then . note that is now achieved .if , then and .  when we see that is achieved .all such minimizers are ground state solitons with , up to translation and multiplication by a unimodular constant .[ r : strictly ] in the proof below , we also see that is strictly decreasing on the interval , with negative slope at , as depicted in figure [ f : me ] .this figure also shows to be smooth on this interval .we do not currently have a proof of this ; it would , however , follow from conjectures [ conj : mass ] and [ conj : beta ] .as enforces , item ( i ) follows immediately .given a non - zero and , consider .clearly ; moreover , .this shows that for , the set of possible energies is a semi - infinite interval .it remains only to investigate the curve and to determine when this infimium is achieved .the analysis below will show that minimizing sequences ( and minima , when they occur ) can be made radially symmetric .we chose to use the multiplier above , rather than , because it preserves this radial symmetry .as a consequence , we see that all feasible mass / energy pairs can be realized by radial functions and hence , by functions with zero total momentum .  given and ,. then in particular , as , which shows that for all .the gagliardo nirenberg hlder inequality with parameter gives when the optimal constant is written in the form .consequently , using young s inequality ( with powers and  ) , \bigl [ \tfrac12 \|\nabla u\|_{l^2_x}^2 + \tfrac{1}{6 } \|u\|_{l^6_x}^{6 } \bigr ] .\end{aligned}\ ] ]  from we see that whenever .notice that combined with the preceding analysis , this completes the proof of part ( ii ) of the theorem .  by the same analysiswe see that when .moreover , , as can be seen from .this settles part ( iii ) of the theorem .next we show that whenever .observe that if and , then choosing , , and using that , we deduce that .  applying the same rescaling argument to a generic trial function shows that is non - increasing , as stated in the theorem .incidentally , if the infimum is achieved for some , then applying the rescaling argument to an optimizer shows that the right derivative of at is strictly negative .( this derivative exists by virtue of the concavity proved below . )when combined with the results on the existence of extremizers proved below , this justifies the claims made in remark [ r : strictly ] .having shown that for , we will now be able to prove that the minimum is achieved for these values of .to this end , let be a minimizing sequence for this variational problem ( i.e. , and ) .using rearrangement inequalities , we see that may be taken radially symmetric , while the identity shows that is bounded in .thus , passing to a subsequence , we may guarantee that converges to some weakly in and as well as strongly in . from weak lower semicontinuity of norms we deducethat note that this guarantees and hence .actually , equality must hold in both parts of .if , then by rescaling as in we could produce a function with and , which is clearly in violation of the definition of .the same violation would arise if but .in conclusion , this demonstrates that when the infimal energy value is actually achieved .this completes the verification of part ( iv ) of the theorem .having finished the proof of the numbered parts of the theorem , it remains to show that optimizers are solitons ( when ) and to demonstrate the concavity and continuity of .( we have already proved that is non - positive and non - increasing . )  that optimizers for aresolitons follows immediately from the euler lagrange equation , . indeed ,having chosen to write the lagrange multiplier as , we precisely recover . by the sharp rearrangement inequalities of see that optimizers must be non - negative ( after multiplication by a unimodular constant ) and spherically symmetric ( about some point ) .thus , by lemma [ l : poho ] and theorem [ t : solitons ] , optimizers must be ground state solitons , up to symmetries .that these solitons have follows from and the fact that .  as we have shown that is monotone , continuity follows from mid - point concavity , which then implies concavity .thus it remains only to show that this will be effected by a moving hyperplane argument .given with , choose a plane that partitions the mass into parts and .reflecting one side or the other through the chosen plane yields two functions and each of which is even with respect to reflection in the plane and which obey and .  directly from the construction we have . taking the infimum over all with yields .the key new player in this section is the _ virial _ , to be regarded as a functional on . while not in perfect accord with the historical meaning ( see ) , our use of the term ` virial ' is consistent with the modern usage as can be seen from the following _ virial identity _ : for any solution to .taking a linear combination of the pohozaev identities and shows that whenever is a soliton solution to .looking back to the manner in which the pohozaev identities were derived yields an alternate expression of the vanishing of the virial for solitons , namely , for any , as solitons are precisely the points where , we deduce that solitons have zero virial .although the early sections of this paper focus on variational and elliptic problems , it is important to remember that theorem [ thm : main ] is a dynamical statement ; we wish to describe the long - time behaviour of solutions to for an open set of initial data .this focuses our attention on as a dynamical substitute for the pohozaev identities .  for solutions that remain well - localized around a fixed point , such as non - moving solitons, lhs will be the time derivative of a bounded function ; thus , by integrating over long time intervals , we deduce that vanishes in an averaged sense .we will show below that in a certain region in the mass - energy plane , which then precludes this sort of soliton - like behaviour .it is far from immediate to see why solutions in the region that merely fail to scatter should remain both in one place and sufficiently well localized that one may apply the argument just sketched .indeed , much of the second half of the paper is devoted to showing that the existence of a non - scattering solution in the region would guarantee the existence of _ another _ solution in the region to which this argument can be applied .this other solution will be constructed as a minimal blowup solution .it will take considerable effort to shoehorn our problem into an application of .however , we have little choice .this is not a small - data problem ; the nonlinear effects are as strong as the linear effects .indeed , there are soliton solutions on the boundary of the region ; see theorem [ t : rbdry ] . as a consequence, we must exploit some fundamentally nonlinear information .the virial identity is the only truly pertinent tool that we know .with this motivating discussion complete , we now turn to the analysis of the region in the mass / energy plane where the virial is strictly positive , beginning with some definitions .[ d : rr defn ] for , we define with the understanding that if no function obeys both and .we then define where is the common mass of all soliton optimizers of the -gagliardo nirenberg hlder inequality , as in section [ sec : me ] .this definition does not make it immediately apparent that functions with mass / energy belonging to the region have positive virial ; this will be shown in theorem [ t : mmr ] .theorem [ t : feasible ] shows that that the restrictions and appearing in are purely for expository clarity ; the set of with would be the same if they were omitted .indeed , if , then the minimal feasible energy with mass is achieved by a soliton , which has zero virial .theorem [ t : feasible ] also shows that whenever .the general features of are shown in figure [ f : r ] . in subsection[ ss : ev ] , we study the variational problem and verify some of the features depicted in figure [ f : r ] .subsection [ ss : exhaustion of r ] is devoted to constructing an exhaustion of that will allow us to prove theorem [ thm : main ] by performing induction on a _ single _ variable .some of the basic structural properties of the region are rigorously justified in theorem [ t : mmr ] below . later in this subsection, we will discuss the boundary of more thoroughly ; see theorem [ t : rbdry ] .  [t : mmr ] if for some , then .  for , the infimum is achieved and is both strictly decreasing and lower semicontinuous as a function of .recall that denotes the infimal energy that is possible for functions with mass . from theorem [ t : feasible ]we know that for and for .  before beginning the proof of theorem [ t : mmr ] ,we first give two lemmas .both are based on straight - forward , but rather cumbersome , computations of the effect of rescaling on the mass , energy , and virial of a function .[ l : energy+virial rescaling ] suppose is not identically zero and that either + to 1.5em or + to 1.5em and  .+ then there exists so that obeys , , and .note that and .direct computation shows that   suppose . fromwe see that .thus , there is at least one so that .let denote the smallest such , which allows us to infer that for all .it then follows from that is decreasing on the interval , and so .  by construction , at . combining this with yields  .this proves part ( i ) of the lemma .we turn now to part ( ii ) and so suppose that and .then \int |\nabla u|^2\,dx ] .similar computations give \int |\nabla u(x)|^2\,dx\end{aligned}\ ] ] the energy bound stated in now follows by combining this estimate with the second part of .suppose and .we wish to show that .that follows from the definition of . if , on the other hand , , then by lemma [ l : energy+virial rescaling](i ) there is a rescaling of with the same mass , and implies , so contradicting the original assumption .next we show that when . to do this ,we define later we will introduce a class of rescaled solitons that include this example ; see lemma [ l : rw ] . direct computation ( cf .the proof of lemma [ l : rw ] ) shows that   as is an optimizer for the ( )-gagliardo nirenberg hlder inequality , so too is .this allows us to express the optimal constant as where the second equality follows by exploiting .thus and so young s inequality ( with powers and ) yields this inequality immediately shows that whenever .  we have now completed the discussion of . when all assertions follow from theorem [ t : feasible ] .this theorem shows that for such , the minimal energy under the constraint is achieved by a soliton , and hence by a function with zero virial .thus .moreover , the proof of this theorem shows that is continuous and strictly decreasing for .we are now left to consider the middle interval .applying lemma [ l : rescaling2 ] with shows that for all such  .in particular , it is possible to achieve with .  using the classical gagliardo nirenberg inequality from , we observe that whenever .combining this with we deduce \tfrac{1}{m(u ) } { \quad\text{whenever}\quad }  v(u)=0 { \quad\text{and}\quad } m(u ) < m(q_1).\ ] ] this proves that when .next we show that is strictly decreasing when . given ,let be a sequence with , , and .as shown above , guarantees that , for an absolute constant .thus by lemma [ l : rescaling2 ] , there is a corresponding sequence with , , and . sending , we deduce that .next we show that the infimum is actually achieved for . to this end , let be a minimizing sequence at mass .replacing by their symmetric decreasing rearrangement does not affect the mass constraint , does not increase the energy , but may result in .however , by applying lemma [ l : energy+virial rescaling ] we may still conclude that there is a minimizing sequence comprising radially symmetric functions .  by, our optimizing sequence is bounded in .passing to a subsequence ( if necessary ) we may ensure weak convergence in and , as well as norm convergence in , due to radial symmetry .let us denote the limit by .  from these modes of convergenceit follows that , , and .we will show that equality holds in all three cases ; consequently , is the sought - after optimizer .if , we could conclude that , which is inconsistent with the fact that is a strictly decreasing function , as proved above .( here we also invoke lemma [ l : energy+virial rescaling ] if it happens that . )thus .it follows then from lemma [l : energy+virial rescaling ] and the definition of as an infimium that and .the only assertion of theorem [ t : mmr ] that remains unproven is that of lower semicontinuity .this will follow readily from the ideas already presented .as is decreasing , lower semicontinuity is equivalent to right continuity . in this way , we see that failure of lower semicontinuity at a point would guarantee the existence of a sequence decreasing to with .now let be radially symmetric functions with , , and .( we just proved the existence of such optimizers above . ) passing to a subsequence we may assume that weakly in and strongly in . using the lower semicontinuity of , , and under such convergence together with the strict monotonicity of , we then deduce that , , and , just as in the proof of the existence of optimizers .this then falsifies the assertion that .the appearance of , a particular rescaling of , in the proof of theorem [ t : mmr ] suggests that we will need to consider functions other than solitons if we wish to understand the boundary of .this is indeed the case , as will be borne out by theorem [ t : rbdry ] .indeed , it will be shown that an optimizer in the variational description of is either a soliton or a special type of rescaled soliton ; moreover both cases do occur .the statement and proof of theorem [ t : rbdry ] are simpler if we present the basic properties of these special rescaled solitons first .this is the purpose of the next lemma .[ l : rw ] fix . among all rescalings of with and is exactly one that obeys and , namely , }{4\sqrt{3\smash[b]{\beta(\omega ) } } } \ , x\bigr ) .\ ] ] moreover , {\beta(\omega)}}}\int |\nabla{p_\omega}(x)|^2\,dx ,   \quad m({r_\omega } ) = \frac{16\sqrt{3\smash[b]{\beta(\omega)}}}{9[1+\beta(\omega)]^2 } m({p_\omega } ) , \label{er : me}\\[0.5ex ] \text { with equality iff } , \label{er : m}\\ \text { with equality iff } , \label{er : e}\\ \tfrac{d\ } { d\omega } e({r_\omega } ) > 0 , \qquad [ \beta(\omega)-1 ] \tfrac{d\ } { d\omega } m({r_\omega } ) \geq 0 \ \text{with equality iff , } \label{er : de>0 } \\\lim_{\omega\searrow 0 } e({r_\omega } ) = \tfrac{1}{3\sqrt{3\beta(g ) } } m(g ) , { \quad\text{and}\quad } \lim_{\omega\searrow 0 } m({r_\omega } ) = \tfrac{16 \sqrt{3\beta(g)}}{9 } m(g ) , \label{er : w0}\end{gathered}\ ] ] where is the unique positive radial solution to , as in theorem [ t : solitons ] .recall that was defined in and that .after a few manipulations , this reveals that obeys proceeding similarly , but also exploiting and , yields \bigr\ } \int |\nabla{p_\omega}(x)|^2\,dx.\ ] ] it is now an elementary matter to verify that gives the only solution to the system and .  the formula for given in follows directly from , while the formula for is most easily found by mimicking .equality in and when is immediate as when .that strict inequality holds in all other cases then follows from {\beta}}}{9(1+\beta)^2 } = \frac{8(1 - 3\beta)}{3\sqrt{3\smash[b]{\beta}}(1+\beta)^3 }   { \quad\text{and}\quad } \frac{d\ } { d\beta } \frac{2}{3(1-\beta)\sqrt{3\smash[b]{\beta } } } = \frac{(3\beta-1)}{(3\beta)^{3/2}(\beta-1)^2}.\ ] ] note that there is no need to use the second formula when , for if , then shows that , while shows that .it remains only to verify and .differentiating the first relation in with respect to and then using gives \!\int |\nabla { p_\omega}|^2\,dx = - \tfrac{\beta-1}2 m({p_\omega } ) + \omega\tfrac{d\;}{d\omega } m({p_\omega}).\ ] ] from this , , , and finally , we then deduce that {\beta}}}\bigl[\tfrac{3\beta-1}2 m({p_\omega } ) - \omega \tfrac{d\;}{d\omega } m({p_\omega } ) \bigr ] > 0.\ ] ]  using we may rewrite as follows : .\ ] ] proceeding as above , we deduce that {\beta}}}{9(1+\beta)^2 } \tfrac{d\;}{d\omega } m({p_\omega } )   + \tfrac{8(1 - 3\beta)}{3\sqrt{3\smash[b]{\beta}}(1+\beta)^2 } \bigl[- \tfrac{\beta-1}{2\omega } m({p_\omega } ) + \tfrac{d\;}{d\omega } m({p_\omega})\bigr ] \\ & = \tfrac{8(1-\beta)}{3\sqrt{3\smash[b]{\beta}}(1+\beta)^2 } \bigl[- \tfrac{3\beta-1}{2\omega } m({p_\omega } ) + \tfrac{d\;}{d\omega } m({p_\omega})\bigr].\end{aligned}\ ] ] that \tfrac{d\ } { d\omega } m({r_\omega } ) \geq 0 ] , while can not be a rescaled soliton when .before we begin the proof of this theorem , we discuss the fuller picture provided by numerics .see figures [ f : numerics1 ] and [ f : numerics2 ] .we find that there is an intermediate mass so that is achieved solely by rescaled solitons when , and solely by solitons when .( numerics give . ) the transition in the type of minimizer is marked by a discontinuity in the derivative of and so is depicted as a kink in figure [ f : r ] .note that this kink is essentially invisible in figure [ f : numerics1 ] ; this is why we have included figure [ f : numerics2 ] .the mass / energy curve of crosses the mass / energy curve of whenever ; indeed , whenever .our numerics show that the mass / energy curves intersect exactly twice , once at and once at the ( unique ) point where .the intersection at is transverse . at the pointwhere , the two curves cross ; however , the formulas in lemma [ l : rw ] can be used to show that both have the same tangent and curvature .  excepting the point , we observe to be a smooth curve .let us briefly describe what is needed to achieve a rigorous proof of this .it follows from theorem [ t : solitons ] that both and are analytic -valued functions of ; thus is smooth except where or vanish or where the two mass / energy curves touch .  from the proof of lemma [ l : rw ] and conjecture [ conj : beta ] , it would follow that vanishes only once , namely , when , which corresponds to .recall that conjecture [ conj : beta ] implies uniqueness of the optimizer .  on the other hand conjecture [ conj : mass ]implies that vanishes only once , namely , at the minimal mass soliton .this corresponds to the cusp in figures [ f : pw ] , [ f : numerics1 ] and [ f : numerics2 ] . in this way, we see that the claimed shape of follows from our two conjectures along with the following assertion , which is supported by numerics : the mass / energy curve of intersects the _ lower _ branch of solitons at exactly one point ( which then serves to define ) and there is no intersection with the upper branch of solitons with .  to  toif obeys , then it follows that the radially symmetric rearrangement of also obeys . to see this , we first observe that , and . by lemma [ l : energy+virial rescaling ], we see that would be inconsistent with the definition of as an infimum .similarly , we must have .below we will show that non - negative radially symmetric functions obeying must agree with some or .in particular , this can be applied to . as all level sets of and zero ( volume ) measure , combining the strict rearrangement inequalities of with the arguments in the previous paragraph , we may conclude that any obeying must agree with some or some up to a translation and a phase rotation .  for the remainder of the proof we consider only that are non - negative and radially symmetric .as is a minimizer of the energy with constrained virial and mass , must satisfy the corresponding euler lagrange equation , provided and are linearly independent at the point .we first consider the possibility that they are linearly dependent .as it turns out , this is not an illusory scenario ; a little thought shows that this must occur at , which is the minimal mass at which it is possible to achieve zero virial .  as , so .thus linear dependence of and at guarantees for some . pairing this equation with yields by hypothesis . combining these immediately shows .  making the change of variables in , we find that is a non - negative radially symmetric solution to .thus by theorem [ t : solitons ] , it follows that for some .as is a rescaling of a soliton with and , it follows from lemma [ l : rw ] that , with the same value of . from the rescaling used above we deduce from that .in particular , .we now turn to the case that and are linearly independent . in this case , satisfies the euler lagrange equation , that is ,   if the lagrange multiplier , then is precisely the defining equation for a soliton ( cf . ) with .as is non - negative and radially symmetric , theorem [ t : solitons ] guarantees that for some .that follows from lemma [ l : energy+virial rescaling ] and the fact that minimizes the energy subject to and .  to show that actually , we will use lemma [ l : cuspy ]suppose instead that .combining and we see that is increasing at ; however , this function converges to as .thus we may choose so that and on the intervening interval .by lemma [ l : cuspy ] it then follows that .this contradicts the minimality of because is strictly decreasing , as shown in theorem [ t : mmr ] .this leaves the case .we first show that , arguing by contradiction .  as and linearly independent at , there exists so that and . without loss of generality, we may assume that .if then implies . from these values of the differentials , we see that obeys , , and for small . in view of lemma[ l : energy+virial rescaling ] , this contradicts the minimality of .thus , either , which was treated above , or , which we address next .  under the change of variables with the euler lagrange equation becomes .theorem [ t : solitons ] then allows us to conclude that for some . by lemmarefl : rw , to see that we need to show that .this follows by pairing with and using , as in the treatment of .lastly , a short computation shows   finally , we turn to the assertions of the second paragraph in the theorem .choose so that has the least mass possible among solitons ( theorem [ t : solitons ] guarantees that this is possible ) .by we know that and consequently , by .this shows that and consequently , there exists so that no soliton has mass smaller than .thus minimizers in this regime must be rescaled solitons .next we exploit the fact that is positive ( see ) to show that the possibility does indeed occur .let denote an optimizer for in the case .thus and .we claim that there exists so that for . taking this for granted for a moment , we see from the continuity of that as . in particular, there is some so that for , which guarantees that optimizers of must be solitons ( rather than rescaled solitons ) on this interval .it remains to verify the claim for and some .note that since is real analytic , failure of this claim would actually imply that on some interval of this type . by, this in turn implies that is increasing ( and so negative ) on this interval . recall that from above as . in this way , the assumption that our claim fails leads to the existence of an interval  ] whose sublevel sets exhaust .there are many ways to do this ; we choose one that is convenient for the subsequent analysis .in particular , to ensure that it is conserved by the flow , we will choose .note that property ( v ) below relies inherently on the shape of , specifically , the monotonicity of the function proved in theorem [ t : mmr ] .this property can not be relaxed ; it is precisely how our induction on manifests a simultaneous induction on the quantities and that are conserved by the flow .[ p : all about d ] there is a continuous function  ] such that where is a small constant to be chosen later .then by repeating the computation in the proof of proposition [ thm : small data scattering ] , we have where all spacetime norms are over . choosing sufficiently small , we obtain combining the subintervals yields the claim .our last result in this section is a stability result for .this is essential for any induction on mass / energy argument .[ perturbation of 3 - 5 ] let be a solution to the perturbed cubic - quintic nls for some function .suppose that for some .let with for some and let .there exists such that if for , then the unique global solution to with satisfies where is a non - decreasing function of , and .  without loss of generality, we may assume .by the usual combinatorial argument ( see , for example , ) and lemma [ lem : small2 ] , it suffices to prove the proposition under the additional hypothesis where is a small constant to be chosen later . from theorem [ t : matador ]we know that exists globally ; the point is to prove the estimate .let and let )} ] . to obtain the last inequality above we also used and the conservation of mass for to estimate therefore ,taking sufficiently small depending on and and then using the standard continuity argument to remove the restriction to $ ] , we derive .the goal of this section is to derive a concentration - compactness principle for the linear schrdinger propagator for bounded sequences of initial data in the _ inhomogeneous _ sobolev space .specifically , we prove that any bounded sequence in can be written as an asymptotically decoupling superposition of translations and rescalings of linear evolutions of fixed profiles plus a remainder whose linear evolution converges to zero in the -norm .the fact that is not a scale - invariant space makes the argument more complicated ; in particular , limiting profiles may not belong to . on the other hand , working in guarantees that the profiles can not live at arbitrarily large length scales ; cf . in proposition [ propo : inverse strichartz ] .a linear profile decomposition for the linear schrdinger propagator for bounded sequences in the _ homogeneous _ sobolev space was first obtained by keraani , relying on an improved sobolev inequality proved by grard , meyer , and oru .we should also note the influential precursor which treated the wave equation .a linear profile decomposition for the schrdinger propagator for bounded sequences in can be found in .the approach we follow here mirrors that in , rather than .in particular , we will build our linear profile decomposition based on an improved strichartz inequality , which states that if the linear evolution is large in the -norm , then at least one frequency annulus contributes non - trivially .the improved strichartz inequality is used to find the correct scaling parameters in the linear profile decomposition .[ propo : inverse strichartz ] let be a sequence such that passing to a subsequence if necessary , there exist such that all of the following hold : first , and if then additionally .secondly , furthermore , letting , & { \quad\text{if}\quad}{\lambda}_{\infty}>0,\\ { \lambda}_n^{-\frac12}e^{-it_n\delta }  \big[\big(p_{\geq \lambda_n^\theta}\phi\big)\big(\tfrac{x - x_n}{{\lambda}_n}\big)\big ] , & { \quad\text{if}\quad } { \lambda}_{\infty}=0 , \end{cases } \label{zinv0a}\end{aligned}\ ] ] with fixed , the following decoupling statements hold : = \|\phi\|_{\dot{h}^1_x}^2\gtrsim a^2\big(\tfrac{{\varepsilon}}{a}\big)^{\frac{15}{4 } } , \\ & \lim_{n\to\infty}\bigl[\|f_n\|_{l^2_x}^2-\|f_n-\phi_n\|_{l^2_x}^2 -\|\phi_n\|_{l^2_x}^2\bigr]=0 .\label{eq : decoupling l2}\end{aligned}\ ] ]  passing to a subsequence , we may assume that for all .by lemma [ lemma : improved strichartz ] , we have for all .thus , for each there exists such that on the other hand , by hlder , strichartz , and , combining and , we get therefore , there exist and such that (x_n)\bigr|\gtrsim { \varepsilon}^{\frac{15}{8}}a^{-\frac{7}{8}}.\end{aligned}\ ] ]  we choose the spatial scales to be .we note that is a bounded sequence ; indeed , by , bernstein s inequality , and , thus , passing to a subsequence we may assume that , where either or .it remains to find the profile . to this end , let as in lhs. by , we have thus , passing to a subsequence , we can find such that in . moreover , if , then for all large enough .thus by the uniqueness of weak limits and weak lower semicontinuity of the norm , we get that in this case .this proves .as is a hilbert space , the weak convergence of to in implies that combining this with the fact that and performing a change of variables , we obtain the equality in .  to complete the proof of , we must prove the lower bound on . to this end , let denote the kernel of the operator . as and , we have . thus (x_n).\end{aligned}\ ] ] invoking , we derive latexmath:[\[\label{eq : bound < phi , k1 > }  on the other hand , latexmath:[\ ] ] by the fundamental theorem of calculus combined with , , and , we get choosing sufficiently small and letting , we derive a contradiction .r. k. was partially supported by nsf grants dms-1001531 and dms-1265868 .t. o. acknowledges support from an ams simons travel grant .o. p. was supported by nsf grant under agreement dms-1128155 .m. v. was partially supported by the sloan foundation and nsf grants dms-0901166 and dms-1161396 .the authors are grateful to the hausdorff institute of mathematics and to the organizers of the program in harmonic analysis and partial differential equations ; they provided the perfect environment for us to meet and bring this project to completion .o. p. would like to thank prof .m. mari for edifying discussions during her visit to toulouse .any opinions , findings , and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the national science foundation .t. akahori , s. ibrahim , h. kikuchi , and h. nawa , _ existence of a ground state and blow - 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Abstract: we consider the cubic - quintic nonlinear schrdinger equation :   in the first part of the paper , we analyze the one - parameter family of ground - state solitons associated to this equation with particular attention to the shape of the associated mass / energy curve . additionally , we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one - parameter family of inequalities of gagliardo nirenberg type . building on this work , in the latter part of the paper we prove that scattering holds for solutions belonging to the region of the mass / energy plane where the virial is positive . we show this region is partially bounded by solitons but also by rescalings of solitons ( which are _ not _ soliton solutions in their own right ) . the discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial - based methodology .