Critical Half-Wave Problems
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Lenzmann, Enno
In this talk, I will a give a survey of recent results about the mass-critical nonlinear half-wave equation on the line. Furthermore, I will discuss work in progress on the energy-critical half-wave maps equation, which posesses some intriguing connections to completely integrabe spin chains and the theory of minimal surfaces
 Thank you so much and thanks to the organizers for giving me the opportunity to give a talk. And actually my original plan was as said in the title to talk about critical half-wave problems which would mean two the L2 critical problem that Patrick Girard was talking about on Monday and another new problem which I will now try to focus on because Patrick did such a great job, it's hard to really add something to this. So let me explain what I want to discuss is what I phrase is a half-wave maps problem. Okay so what's that? So it's an equation that is extremely easy to write down. So let's see and I will say a few words in a moment. So the wedge means the cross product in R3 and U is our unknown function depending on time on an interval say from 0 to capital T and say it's defined on Rd with values in S2 which we always of course think being embedded in R3. Yeah so this is a quite I mean simple minded at first glance generalization of the Schrodinger maps problem. So you just take the say Landau-Lichtschitz equation. This would be the Laplacian or it would be minus the Laplacian but minus signs can be easily so to say absorbed by redefining direction of time. This is in Hamiltonian equation by the way so I will show to you soon. So I might get some various sign mistakes in my formulas but they don't matter so that's only where it matters I think I have them right. So let's see. Okay so this is the half-wave maps equation and so there are two reasons why I recently started to work on that. First of all I've seen this equation before but so only recently I realized that this has a really somewhat very nice physical interpretation that's the one bonus thing about it which I tried to make somewhat clear to you here and the second thing is which I was always also afraid about is that you cannot do much with that that is rather quite unexplicit what you can do compared to what like Schrodinger maps or wave maps equation where you have a nice explicit structure say of a covariant or wave maps and so on it studies stability and problems like that but it turns out to be wrong this statement that I will show to you that you can do a lot of it explicitly because I will as my main result today discuss that in the one dimensional case which is the energy critical case I will show to you how to completely classify all traveling solitary wave solutions in an explicit form in terms of so-called Blaschke products. Okay so this is something I recently found and something that amazed me and gives me hope to continue this study of this energy critical problem. Okay so as I said this is an Hamiltonian equation so there's an energy functional attached to it which of course looks like this so I guess what this is going to be so maybe there's a half here if you like that and for later purposes I will just do this here for record that this is of course as expressible as a double integral which will as you see make a connection to a very interesting physical application of this model and the Poisson bracket attached to that so yet you can see that you can write this equation of as a Hamiltonian equation of motion is the one that is given by functions with values in s2 say the components at x and y satisfy a Poisson bracket structure which is given by this expression so this is the antisymmetric Levy-Civita symbol and the delta function so if you if you use this structure at least formally you see that the half-wave maps equation is formally equivalent to this evolution equation given by the Poisson bracket given with h okay so in particular h is at least a formally conserved quantity so let me now start and discuss the motivation behind this equation so first of all it is clear how that you get to each other equation? oh yeah if you are clear I mean you have to plug in what this Poisson structure is and you work out what this is by plugging this in and then you finally see that this is the equation that you get I mean if you take for instance here the Dirichlet integral you get the Landau-Liffchitz equation with that Poisson structure the Poisson structure is some universal thing it's just the energy function of it determines whether you have the half-wave maps equation as I call it or the Landau-Liffchitz equation. What is the Landau-Liffchitz equation? The Landau-Liffchitz is the Schrödinger maps equation. I don't have to write it down yeah Schrödinger map or Landau-Liffchitz coming from physics like the continuous version of the Heisenberg equation for a ferromagnetic system okay yeah so d equal to 1 by the way would correspond to the energy critical case everything above d equal to 2 and higher is of course an energy sub-critical a supercritical sorry by the way and the energy critical case it's also physically most relevant as I will explain in a second and there's also another thing attached to it it has a conformal invariance property okay so this is the case I will focus on okay so the motivation so one motivation comes from differential geometry I mean recently there was some some growing interest in what is called a half harmonic map with target s2 okay so that would correspond of course obviously to static solutions to my flow equation huh so in particular these static solutions will play also an important role for this time dependent equation of course you might also think about something like a half harmonic map heat flow which would be a parabolic analog to this equation which I don't discuss here so this is something that was recently introduced Riviera and Daliou so in particular the regularity theory for these equations like say if you're in the critical dimension that an h1 half map is actually a smooth map like an analog of Elan's regularity theory for harmonic maps in two dimensions yeah but it's a different thing because Riviera used some kind of commutator type arguments to get this regularity result and also from a more geometric point of view in a very recent work by Fraser and Shane it is seen that this equation corresponds to a free boundary minimal disk problem I will come to this later that is to say it is also naturally connected that you look for something which is a minimal surface say inside the ball the unit ball with a certain condition how the boundary of this minimal surface hits the boundary of the ball which is s2 and in a in this setting it has to hit perpendicular and you can actually see that this problem is equivalent to this to study the free boundary minimal just yes this is a boundary equation I will come to this point where you see that it's a completely natural connection to the theory of minimal surfaces okay I come to this and B this is the physical motivation behind that of course I will later look for traveling solitary waves which will be generalizing these half harmonic maps in the in the sense that there's another non-zero right hand side okay and I will classify also all these solutions and B is physics and this is a typical thing you have to know the right names to look for because physicists like these names you see weak ordering Gibbs state if you don't know what who weak was who Gibbs was it's hard to find out in the literature what's this about once you have the right code words it works so the first two names that somewhat play a role in these business are Haldane and Chastry two physicists who wrote down a Hamiltonian for a spin chain and I will be a little bit formal here of course which is some object like that okay these are spin operators I won't go into any details and there is a so to say interaction among spins this is on a discrete lattice say in one dimension so you have fixed lattice sides and xk xk plus one and fourth are the lattice sides and to each lattice sides you attach a quantum spin I don't want it to get go into details what this is about and they interact with certain what's called long-range potentials that might be a sine squared of the difference of the lattice sides or there's also the so to say algebraic model where this is a square of this and so on there's a whole zoo of this and the point about this these are exactly solvable models in the sense you can calculate the spectrum and so forth and you can also find lax pairs and so on these are long-range quantum chains which have a very nice particular structure from the physical point of view so it's hard to draw what these quantum operators do actually but I installed Planck's constant here just to indicate that in a certain limiting regime which you can think of the semi-classical limit or also a large spin limit you are at least formally led to what is called a system of classical spins so then this Hamiltonian becomes something which you could phrase as a calogero mosa type Hamiltonian with spin which would look like this that formally at least the spin operators get replaced by something of this form where the sk's are now unit vectors in three space okay so in the classical regime the spins are given by an arrow which lies on the unit's phi in three-dimensional space that's a at least for a physicist a typical procedure to go from a quantum spin system to a classical spin system so if you just had just nearest neighbor interactions so this was in this function but just only nearest neighbors couple you get the Heisenberg chain and the continuum limit of the Heisenberg chain in the sense that this parameter a shrinks to zero would be the classical Landau-Lischitz Schrodinger maps equation however here you can of course somewhat anticipate that in the next step of taking this limit you get that this sum leads to a double integral of that form that you have a continuous spin variable which I call now s which is of this form and we are here on one space dimension making this rigorous makes it of course one step is to study actually that the dynamics converge in a certain sense and of course in higher dimensions might be even more difficult in the critical case I think it's still doable but it's it's it's not so simple and on account of the fact that this is of unit lengths you can of course rewrite it as this which is the Hamiltonian I wrote above in one space dimension okay again this Kaluger-Romosa type Hamiltonian also shares the magic property that you have lax pairs you have a very explicit structure here and you might wonder if this survives actually for this limiting PDE there's a strong indication for that because I will show to you an explicit classification of all solitary waves it might be an hint that this is a completely integrable system which is yet an energy critical so it shows a criticality which in principle can also say that you have a finite time blow up and so on so let's see so now I come to the more yes exactly in dimension two and larger it's super critical and in one dimension it's energy critical and in principle you can think like for energy critical Schrodinger maps you have also blob solutions that in this case you might have because of an integrable structure too many conservation laws which forbid such a thing but it could also be like for the cubic chigo equation that they only live up to certain level of regularity and you still see some kind of solo left type growth phenomena which are a little bit counter to it the first if you think of a completely into if I think of a completely integrable system so what I'm here interested in is to consider the traveling solitary waves for this model and this is interesting in one respect first that for the Landau-Liffchitz equation such a thing does not exist and you mentioned they are only static finite energy solutions you don't cannot create a traveling solitary waves for the Landau-Liffchitz equation you might think you can apply a Galilean boost but this is not working but of course for the wave maps equation you can do a Lorentz boost to get a traveling wave map if you wish now but here this is not a Lorentz invariant system so we cannot apply something but you will see that we get a rather explicit way of constructing these things and I'll show to you that this is the only way to get them okay so you make an ansatz of course that you say I have a velocity given a real number v and you say I look for solutions of the form utx is equal to say some profile uv of x minus vt and this is simple to see that this now satisfies an equation of this form okay there's a minus here but it's inessential okay and I have to be a little bit more precise what I mean here I mean a solution which is an h dot one-half going from the real line to the unit sphere okay and there's a certain boundary condition at infinity I fixed say this south pole if you wish maybe huh okay this is always understood implicitly one thing which I will completely skip is that you might ask for a regularity theory for this equation of course we want to ultimately that any solution is a smooth solution because the arguments I will present you will in some way need this kind of property but I will completely skip this and just mention that joint work with my former postdoc Armin Cicorah you get this smooth okay naturally of course it's not obvious at all because the perturbation so to say that you add is also first order operator and it's a massive so to say change of the equation we know already smoothness for the v is equal to zero case okay but let's leave this issue aside for a moment so what you can do first is of course you can try to cook up a solution okay how could you maybe try to do that first there's one way of course you can think you attack this variationally this is critical points of the at least formally of the energy and then you have to find another functional which is a side condition maybe it's such that the left hand side comes from that side condition in v is a Lagrange multiplier okay call this functional p of u v constant which in a sense would correspond to a linear momentum but this is a tricky business in this setting I will not write down you can in fact find a functional at least formally so that you get this the problem is as you will see is that both functionals show a conformal invariance property and hence discussing minimizing say sequences is not so simple it's a little bit like the plateau problem so it might be a little bit painful to attack this problem variationally but in principle it should be doable okay one thing so we won't follow this road second road is across an implicit function theorem at least for small v you might be able to construct in the neighborhood of these whatever are the the half harmonic maps at least slow moving traveling waves okay but we won't do that so there's a lucky punch here which does the job okay at least you get solutions you make an ansatz so you start with a half harmonic map which corresponds to a static solution and you can find such things like just living say in the equator plane so the third component is zero of course then I have to make sure that this point also lies in that plate but it's fine okay and now you make an ansatz that for uv and that's sort of a magic thing here but you will see a geometric reason for that later of course I made a much more general ansatz first because I was thought I was thinking about kind of mimicking Lorentz boosts however alpha and beta are just constants and if and only if alpha and that it's not true there's a there's a sign so if alpha is 1 minus v squared so this of course assumes that v is less than unity which means the solitary waves cannot propagate faster than the speed one in my units okay what let's assume that for a moment and b is equal to v you get a solution that's quite fascinating that you get an explicit cheap trick to construct so to say boosted solutions so it's it's even is there any variance in the equations that leads to this? I don't think so I don't think so there's no Lorentz invariance there's no Galilean invariance and the funny thing is also that this transform which comes from the static to a traveling one is just acting so to say from the outside on the target I mean you don't have to do anything in the arguments of x what if it's just one verb yeah yeah but still I mean you could I mean okay let me draw a picture the picture is this don't you have to adjust the boundary conditions? huh okay I changed the boundary conditions but I can rotate back so I can always that's everything I say is modular rotation on the target that's true I violate the boundary condition but I could rotate back and then I would get the same boundary condition but I'm gentle about that so so what I did is that this half harmonic map and of course f and g are certain special functions but it's a certain kind of parametrization of the great of a great circle gets in a sense boosted to some other circle which lives here and in the extreme case when v say tends to 1 you just on the get the constant which just lives on the tip of the sphere okay okay so the theorem that I'm going to talk about now says this is the only way to get the solutions okay the boosted solutions and gives an explicit characterization of all the possible functions f and g you can have here so theorem so let you let v be a real number and uv so this equation makes perfect sense if it's an h dot one-half function and distribution of sense solve that then you have two cases if v is bigger and modulus than one or equal to one the only way is that you have a trivial solution it's a constant so it's nothing else can happen and if v satisfies this then what rotations of course are on the sphere on the target you have to have that uv of x is of that form which I wrote down and now what is f and g v sorry yes where f f is the real part say of a function capital f and g is the imaginary part it's the real part of a holomorphic function in fact on the boundary living on the upper half plane and on the boundary you get the little f and g is the imaginary part of that function and f is of the form f of z is equal to what is called a finite Blush product so there's a k running from one to say d and there's maybe a lambda k z minus ak there's a shift of course that is maybe possible plus i and these lambda k's are real numbers not zero and the ak's are arbitrary points in r and d is a natural number of course okay yes for v equal to zero you can characterize that but the point is and this is the point that you can really reduce it to the case v is equal to zero so I will explain that the proof is really an advancement of understanding the case when v is equal to zero okay but so can this be understood directly on the stationary solution of the half harmonic map or unique okay when I look at the stationary solution the half harmonic maps this is not completely easy to see but doable okay with what's known in the literature in the recent literature say the point is that you can so to say also undo the boost transformation I will explain that okay yeah yeah it gives you in particular characterization of all half harmonic maps because it's a case v is equal to zero okay and there's of course a saying there is no traveling wave with speed larger than one okay this is also one point okay and the interesting fact is that the energy of these guys is of this form there's a quantization of course involved like for harmonic maps the d the index of the blusher product however by tuning the v you can make this arbitrarily small so it's like the ill-to-critical half wave equation with patrick girard where you have l2 criticality but you have solitary waves which have arbitrarily small l2 mass and now you have solitary waves with arbitrary small energy energy is now the critical thing so it's completely different say from the land of life shits where you have don't have that okay okay okay all right so i have roughly 20 minutes what's the total okay 17 it counts okay let me explain to you about the proof okay okay this is actually two statements a and b okay i will along say something about a and but i will of course focus on the more interesting part b so what's the initial situation so think first that we are given a solution uv and think it's smooth okay to avoid any issues here so that the image of course it raises us a closed curve on the unit sphere okay and actually what we posteriori see that these are just great circles or in the case when v is equal to zero or when v is not equal to zero it's it's a circle on the sphere okay but first we don't know anything about this more than this so if you take so rewrite for simplicity little u is equal to uv i want to skip that index little v because writing is a bit difficult okay first thing of course that we do is also to see that you have a conformal invariance here is of course you consider this as a two-dimensional problem in a sense because this arises as a boundary of something yeah so you take the u to be the harmonic extension okay of little u that is capital u is a map from r2 plus which you can also identify with the complex alpha half plane into r3 it's a component wise harmonic function yeah and the boundary condition star is a boundary condition then because as you know it's a classical fact that operator just becomes the normal derivative with respect to this extension variable minus the normal derivative so effectively you study the problem minus v of dx capital u and there's a minus so it will change here but it's inessential of dy this is my new variable that adds to my space like this on the boundary okay fine so far so good so by the maximum principle actually there's a strong maximum principle that inside is strictly less than one well you see out that you the image of capital u is something inside the sphere of course because of this and the boundary is so to say given by this little u the boundary curve the profile of the solitary wave okay so far so good now comes the first step let's say okay capital u is a parametrization of a surface modulo there might be branching points and these things it's maybe not embedded but we just say it's a minimal surface provides a minimal surface that's the first step so how to see that the first idea is to use something which is called a hopf differential so it's a function that you cook up up depending on z say so z is just a notation of course for this and dz is the wilting derivative that's that's a fairly standard step step here what i do but i just want to explain it so i remember there should be maybe a fault to have everything nice here so now you take the derivative of capital u with respect to the z so it's actually just this you combine them so that's a c3 valued function it's a three vector with complex entries and you take the scalar product not the Hermitian product so take this product this is a standard thing in minimal surface theory so what you get is this Jesus it's a worm here nothing to do really with the questions it's the climate here okay okay so the the upshot is that this is a because u is a harmonic function this is a holomorphic function okay so you either really check that dz bar the anti- wilting derivative of this is zero hence it's a holomorphic function it's easy to see because dz bar times dz is just times the constant the Laplace operator okay so far so good and now comes the point that if you look at that function and you consider the imaginary part of this hopf differential on the boundary okay so then you use the boundary equation okay and now you have to explicitly work out what this means actually it means in terms of real parts that you have this minus maybe two i i'm not 100 sure whether there's a plus or minus but it doesn't matter here this okay so the imaginary part on the real axis of that holomorphic function in the upper half plane is the scalar product of d x u times scalar product y u but now look at the equation here this immediately tells you that at least when v is not equal to zero this has to be perpendicular to that vector and you see that the imaginary part of that holomorphic function on the real line is zero so you can extend it to the whole complex plane plane the imaginary part by odd reflection however the imaginary part of phi because on the boundary it's h1 half so this is actually an h dot one function so this is this times this is an l1 function it's integrable okay so then you can easily conclude that the imaginary part is identically zero okay so because it's a holomorphic function the real part has to be a constant a real constant however it's also an l1 so it's also identically zero so this hopf differential is actually identically zero fine so that tells you this is always equal to this and this is always zero so it's right angle so u capital u is a conformal map and this is just saying it's an harmonic function with which is conformal hence it's a minimal surface which you get now as a next step yes as a next step but there could be lots of minimal surfaces inside the unit ball of course they are very funny ones but they have to meet the boundary in a certain specific way they have to respect of course this boundary condition here yeah and when v is equal to zero this is something which I first referred as a free minimal disc which was studied by Fraser and Shane and they classified that this can only be a plain disc and then you are in the situation that the boundary is a great circle and then you are in good shape to get to get your classification theorem however they approve I could not see how to make it really work in the case when v is not equal to zero because you have a more complicated boundary condition okay but I show you an argument which includes the Fraser and Shane result by considering the following thing so what you learn from that first step that's a conformal map you also learn that you can rewrite the boundary equation now I write little u here in a very interesting way are people from harmonic maps know such a thing that you would maybe expect here a square but this is not true in the fractional case it's it's not a square it's a one and there's a deeper reason for that because if you test this equation against u then this term of course drops out and u times u scalar wise is one so you get this okay okay what is this geometrically the right hand side corresponds to the length of the curve and this if you use the harmonic extension the lift corresponds to the area you span okay and you will see that it will actually satisfy an isoparametric inequality set set rate an isoparametric it has satisfied of course the isoparametric inequality otherwise would be bad set rates and from that you could also conclude that you have to have plain disks but you don't know yet what the value of the left hand side is but this is just a side remark so the next step is and this is then the little innovation maybe is to consider another hopf differential differential well what is that take now this okay fine you can now I leave the details because this is maybe too much on the blackboard write down what that is again it's a holomorphic function you consider you consider study what's on the boundary you want to make may maybe run a trick like that but of course you have to work a little bit harder but using this formulation of the Euler Lagrange equation you also see that this is a holomorphic function which is identically zero but that's too much a minimal surface which of course satisfies the first thing there are lots of them however it also has to satisfy this okay and then you can use what's called the Weierstrass inner power representation of a minimal surface just want to say these things here so that you can deduce always one gets things right it's half so what's that so it means that you can this is classical thing parameterized the Wieltinger derivative of this u in terms of a holomorphic function f and the meromorphic function which satisfies some compatibility condition but if you plug this into this equation you work a little bit you will see ultimately that u the image of u can only lie in a fixed plane in R3 okay so so what you get from that finally I leave the details of course that the image lies in the plane in R3 so it's a flat really flat minimal surface so it's a disc so you will ultimately see that this is a disc and of course the boundary of the disc is a circle on the sphere now you can undo this transformation of course and go back to this case where it's the unboosted case okay and then you're almost done because now you can invoke some complex analysis because in the unboosted case say you transform it like this that it's in the xy plane then you will see with this analysis that f and g have to be say the real and the imaginary part of a complex function f i g on the upper half complex plane with complex values and because u lies on the unit sphere the modulus squared of f is one on the boundary so that's this very special holomorphic function on the upper half plane such that the modulus is one identically on the boundary and there's a classification for that in terms of plasco products and because also f is in h dot one it's not hard it's a sobolev space here you can conclude that this is actually a finite plasco product which i wrote down so they only have finitely many factors otherwise you would have infinite energy which is a bit formal okay finite plasco product hence you get the complete classification i wrote to you and i finally should say that the last idea of using these plasco products for this kind of problem is also something that already mironescu and pisante used for something which is related where you look for what's called maybe a half harmonic map from s1 into s1 okay i should pay also credit to this worker somewhat okay so what i want to say is now okay if you work out these real imagining parts of these finite plasco products it's just kind of rational functions you have so it's very explicit you can now study stability or instability you have very explicit formulas for your solitary waves okay and this is also the point i like to end because it's freaking warm here so thanks oh no this is what i this is a byproduct of the proof so if you have a stationary solution my terminology v is equal to zero and it has should have finite energy other okay modular rotations on the sphere of course but they all have to but they are all okay yeah yeah yes this is a yeah yeah that's a byproduct actually so and and i should maybe say about a i thought you're looking for things of a special form right that's why i'm asking the question with p on that i told you last component yeah but i cannot okay what i show finally is that i'm allowed to do this because they're always equatorial discs so i can always rotate it that the last is identically zero okay and i should say a is is a funny proof uh you test the equation against the hillbott transform of you and do some magic uh and the in the l2 critical half wave case we cannot actually rule out solitary waves which have speed more than one we don't know how to do this in this model it's a magic thing to really get the sharp limit for the do you have a question take me first to understand okay now we have done the job okay next oh okay i go home the next question that ability and just existence of the flow by the way yes i have not talked about existence of the flow yes of course yeah that's a lot of things to do yes stability i would expect the forward why not in your project when you're just i would expect that but also because formally you see if you think of the schrodinger map blow up you have an unstable one because you have a slow decay of a resonance i mean if a slow decay in the linearization a zero mode which is very slow decay and i think here it does not happen that you have i don't i don't think so but i would not bet much money i have to look into after the euro look into the things more closely yeah maybe yes maybe next week i have time more questions please so what is your momentum functional ah yeah that is a good question it's a bit like this i draw a picture so the momentum of a configuration u a closed curve is the solid angle that this curve so to say in the unit sphere gives you but this is of course not well defined up to four pies because you have ambiguities so um so in the physical literature um there's a kind of a dispute about what is whether there is a true momentum or not for such a model but you have a you have a conserved quantity modulo a value of four pie you could consider an exponential of that but it's not it's a it's a it's a it's a funny functional the momentum is is really not that straightforward say for these models of course also you might wonder whether the traveling solitary waves you get are also existing in the discrete models and this is always a delicate question for nls there are might some traveling waves in the discrete lattice model or not and so on and here it's also not so clear what survives or what gets created by this continuum limit but sorry i was going off the tension there's there's so much structure it's actually very beautiful there's a lot of structure and it does come from not just one iterable system but a class yes so what would you speculate i would speculate it's also completely integral but i don't i mean i try to to cook up lex pairs but it's not my prime education so uh take some time maybe but i don't know maybe there's maybe i'm wrong maybe you see really uh serious blow up yeah i don't yeah yeah that maybe in some weird way you see like for the chigo equation you might have infinitely many conservation laws in evolution up to a certain kind of regularity in in higher regularity you might as have at least grow up for infant i mean some small kind of infinite tight blow up or something like that i don't know but conservation laws need to be need to be coercive so for example bluzonesque is nicely integral that solutions just don't exist there's no incompatibility between yeah okay you will teach me more than i know it's good but yeah yeah it depends on how you define this i don't know we should waste more
 Thank you so much and thanks to the organizers for giving me the opportunity to give a talk.  And actually my original plan was as said in the title to talk about critical half-wave  problems which would mean two the L2 critical problem that Patrick Girard was talking about on Monday  and another new problem which I will now try to focus on because Patrick did such a great job,  it's hard to really add something to this. So let me explain what I want to discuss is what I  phrase is a half-wave maps problem. Okay so what's that? So it's an equation that is extremely  easy to write down. So let's see and I will say a few words in a moment. So the wedge means the  cross product in R3 and U is our unknown function depending on time on an interval say from 0 to  capital T and say it's defined on Rd with values in S2 which we always of course think being embedded  in R3. Yeah so this is a quite I mean simple minded at first glance generalization of the Schrodinger  maps problem. So you just take the say Landau-Lichtschitz equation. This would be the Laplacian or it  would be minus the Laplacian but minus signs can be easily so to say absorbed by redefining  direction of time. This is in Hamiltonian equation by the way so I will show to you soon.  So I might get some various sign mistakes in my formulas but they don't matter so that's  only where it matters I think I have them right. So let's see. Okay so this is the half-wave maps  equation and so there are two reasons why I recently started to work on that. First of all  I've seen this equation before but so only recently I realized that this has a really  somewhat very nice physical interpretation that's the one bonus thing about it which I tried to make  somewhat clear to you here and the second thing is which I was always also afraid about is that you  cannot do much with that that is rather quite unexplicit what you can do compared to what like  Schrodinger maps or wave maps equation where you have a nice explicit structure say of  a covariant or wave maps and so on it studies stability and problems like that but it turns  out to be wrong this statement that I will show to you that you can do a lot of it explicitly  because I will as my main result today discuss that in the one dimensional case which is the  energy critical case I will show to you how to completely classify all traveling solitary  wave solutions in an explicit form in terms of so-called Blaschke products. Okay so this is  something I recently found and something that amazed me and gives me hope to continue this study  of this energy critical problem. Okay so as I said this is an Hamiltonian equation so  there's an energy functional attached to it  which of course looks like this  so I guess what this is going to be so maybe there's a half here if you like that  and for later purposes I will just do this here for record that this is of course  as expressible as a double integral which will as you see make a connection to a very  interesting physical application of this model and the Poisson bracket attached to that so  yet you can see that you can write this equation of as a Hamiltonian equation of motion is the one  that is given by functions with values in s2 say the components at x and y satisfy  a Poisson bracket structure which is given by this expression so this is the  antisymmetric Levy-Civita symbol and the delta function so if you if you use this structure  at least formally you see that the half-wave maps equation is formally equivalent to this  evolution equation given by the Poisson bracket given with h okay so in particular h is at least  a formally conserved quantity so let me now start and discuss the motivation behind this  equation so first of all it is clear how that you get to each other equation? oh yeah if you are  clear I mean you have to plug in what this Poisson structure is and you work out what this is by  plugging this in and then you finally see that this is the equation that you get I mean if you take  for instance here the Dirichlet integral you get the Landau-Liffchitz equation with that Poisson  structure the Poisson structure is some universal thing it's just the energy function of it  determines whether you have the half-wave maps equation as I call it or the Landau-Liffchitz  equation. What is the Landau-Liffchitz equation? The Landau-Liffchitz is the Schrödinger maps equation.  I don't have to write it down yeah Schrödinger map or Landau-Liffchitz  coming from physics like the continuous version of the Heisenberg equation for a  ferromagnetic system okay yeah so d equal to 1 by the way would correspond to the energy  critical case everything above d equal to 2 and higher is of course an energy sub-critical a  supercritical sorry by the way and the energy critical case it's also physically most relevant  as I will explain in a second and there's also another thing attached to it it has a  conformal invariance property  okay so this is the case I will focus on okay so the motivation  so one motivation  comes from differential geometry I mean recently there was some  some growing interest in what is called a half harmonic map  with target s2 okay so that would correspond of course obviously to static solutions to my flow  equation huh so in particular these static solutions will play also an important role for this  time dependent equation of course you might also think about something like a half harmonic  map heat flow which would be a parabolic analog to this equation which I don't discuss here so  this is something that was recently introduced Riviera and Daliou  so in particular the regularity theory for these equations like say if you're in the critical  dimension that an h1 half map is actually a smooth map like an analog of Elan's regularity theory  for harmonic maps in two dimensions yeah but it's a different thing because Riviera used some kind  of commutator type arguments to get this regularity result and also from a more geometric point of view  in a very recent work by Fraser and Shane it is seen that this equation corresponds to a free boundary  minimal disk problem I will come to this later that is to say it is also naturally connected  that you look for something which is a minimal surface say inside the ball the unit ball with  a certain condition how the boundary of this minimal surface hits the boundary of the ball  which is s2 and in a in this setting it has to hit perpendicular and you can actually see that this  problem is equivalent to this to study the free boundary minimal just yes this is a boundary  equation I will come to this point where you see that it's a completely natural connection to the  theory of minimal surfaces okay I come to this and B this is the physical motivation behind that  of course I will later look for traveling solitary waves which will be generalizing these  half harmonic maps in the in the sense that there's another non-zero right hand side okay  and I will classify also all these solutions and B is physics and this is a typical thing you have  to know the right names to look for because physicists like these names you see weak ordering  Gibbs state if you don't know what who weak was who Gibbs was it's hard to find out in the literature  what's this about once you have the right code words it works so the first two names that  somewhat play a role in these business are Haldane and Chastry two physicists who wrote down a  Hamiltonian for a spin chain and I will be a little bit formal here of course which is some  object like that okay these are spin operators I won't go into any details and there is a so to  say interaction among spins this is on a discrete lattice say in one dimension so you have fixed  lattice sides and xk xk plus one and fourth are the lattice sides and to each lattice sides you  attach a quantum spin I don't want it to get go into details what this is about and they interact  with certain what's called long-range potentials  that might be a sine squared of the difference of the lattice sides or there's also the so to say  algebraic model where this is a square of this and so on there's a whole zoo of this and the  point about this these are exactly solvable models in the sense you can calculate the spectrum and so  forth and you can also find lax pairs and so on these are long-range quantum chains which have a  very nice particular structure from the physical point of view so it's hard to draw what these  quantum operators do actually but I installed Planck's constant here just to indicate that in  a certain limiting regime which you can think of the semi-classical limit or also a large spin limit  you are at least formally led to what is called a system of classical spins so then this Hamiltonian  becomes something which you could phrase as a calogero mosa type Hamiltonian with spin which  would look like this that formally at least the spin operators get replaced by something of this form  where the sk's are now unit vectors in three space okay so in the classical regime the spins are  given by an arrow which lies on the unit's phi in three-dimensional space that's a at least for a  physicist a typical procedure to go from a quantum spin system to a classical spin system  so if you just had just nearest neighbor interactions so this was in this function but  just only nearest neighbors couple you get the Heisenberg chain and the continuum limit of the  Heisenberg chain in the sense that this parameter a shrinks to zero would be the classical Landau-Lischitz  Schrodinger maps equation however here you can of course somewhat anticipate that in the next  step of taking this limit you get that this sum leads to a double integral of that form  that you have a continuous spin variable which I call now s  which is of this form and we are here on one space dimension making this rigorous makes it of  course one step is to study actually that the dynamics converge in a certain sense and of course  in higher dimensions might be even more difficult in the critical case I think it's still doable but  it's it's it's not so simple and on account of the fact that this is of unit lengths you can of  course rewrite it as this which is the Hamiltonian I wrote above in one space dimension okay again this  Kaluger-Romosa type Hamiltonian also shares the magic property that you have lax pairs you have  a very explicit structure here and you might wonder if this survives actually for this limiting PDE  there's a strong indication for that because I will show to you an explicit classification of  all solitary waves it might be an hint that this is a completely integrable system which is yet  an energy critical so it shows a criticality which in principle can also say that you have  a finite time blow up and so on so let's see so now I come to the more  yes exactly in dimension two and larger it's super critical and in one dimension it's energy  critical and in principle you can think like for energy critical Schrodinger maps you have also  blob solutions that in this case you might have because of an integrable structure too many  conservation laws which forbid such a thing but it could also be like for the cubic chigo equation  that they only live up to certain level of regularity and you still see some kind of  solo left type growth phenomena which are a little bit counter to it the first if you think of a  completely into if I think of a completely integrable system so what I'm here interested in is to  consider the traveling solitary waves for this model and this is interesting in one respect  first that for the Landau-Liffchitz equation such a thing does not exist and you mentioned  they are only static finite energy solutions you don't cannot create a traveling solitary waves  for the Landau-Liffchitz equation you might think you can apply a Galilean boost but this is not  working but of course for the wave maps equation you can do a Lorentz boost to get a traveling  wave map if you wish now but here this is not a Lorentz invariant system so we cannot apply  something but you will see that we get a rather explicit way of constructing these things and  I'll show to you that this is the only way to get them okay so you make an ansatz of course that you  say I have a velocity given a real number v and you say I look for solutions of the form utx is  equal to say some profile uv of x minus vt and this is simple to see that this now satisfies  an equation of this form okay there's a minus here but it's inessential okay  and I have to be a little bit more precise what I mean here I mean a solution which is an h dot  one-half going from the real line to the unit sphere okay and there's a certain  boundary condition at infinity I fixed say this south pole if you wish maybe huh okay this is  always understood implicitly one thing which I will completely skip is that you might ask for a  regularity theory for this equation of course we want to ultimately that any solution is a smooth  solution because the arguments I will present you will in some way need this kind of property but I  will completely skip this and just mention that joint work with my former postdoc Armin Cicorah  you get this smooth okay naturally of course it's not obvious at all because the perturbation  so to say that you add is also first order operator and it's a massive so to say change of the equation  we know already smoothness for the v is equal to zero case okay but let's leave this issue aside for a moment  so what you can do first is of course you can try to cook up a solution okay how could you maybe  try to do that first there's one way of course you can think you attack this variationally this is  critical points of the at least formally of the energy and then you have to find another functional  which is a side condition maybe it's such that the left hand side comes from that side condition in  v is a Lagrange multiplier okay call this functional p of u v constant which in a sense would correspond  to a linear momentum but this is a tricky business in this setting I will not write down you can in  fact find a functional at least formally so that you get this the problem is as you will see is that  both functionals show a conformal invariance property and hence discussing minimizing say  sequences is not so simple it's a little bit like the plateau problem so it might be a little bit  painful to attack this problem variationally but in principle it should be doable okay one thing  so we won't follow this road second road is across an implicit function theorem at least for small v  you might be able to construct in the neighborhood of these whatever are the the half harmonic maps  at least slow moving traveling waves okay but we won't do that so there's a lucky punch here  which does the job okay at least you get solutions you make an ansatz  so you start with a half harmonic map  which corresponds to a static solution  and you can find such things like just living say in the equator plane  so the third component is zero of course then I have to make sure that this point also lies in  that plate but it's fine okay and now you make an ansatz that for uv and that's sort of a magic thing  here but you will see a geometric reason for that later  of course I made a much more general ansatz first because I was thought I was thinking about  kind of mimicking Lorentz boosts however alpha and beta are just constants and if and only if alpha  and that it's not true there's a there's a sign so if alpha is 1 minus v squared so this of course  assumes that v is less than unity which means the solitary waves cannot propagate faster than the  speed one in my units okay what let's assume that for a moment and b is equal to v you get a solution  that's quite fascinating that you get an explicit cheap trick to construct so to say boosted solutions  so it's it's even is there any variance in the equations that leads to this? I don't think so  I don't think so there's no Lorentz invariance there's no Galilean invariance and the funny thing  is also that this transform which comes from the static to a traveling one is just acting so to say  from the outside on the target I mean you don't have to do anything in the arguments of x  what if it's just one verb yeah yeah but still I mean you could I mean okay let me draw a picture  the picture is this don't you have to adjust the boundary conditions?  huh okay I changed the boundary conditions but I can rotate back so I can always that's everything  I say is modular rotation on the target that's true I violate the boundary condition but I could  rotate back and then I would get the same boundary condition but I'm gentle about that so so what I  did is that this half harmonic map and of course f and g are certain special functions but it's a  certain kind of parametrization of the great of a great circle gets in a sense boosted to  some other circle which lives here and in the extreme case when v say tends to 1 you just  on the get the constant which just lives on the tip of the sphere okay okay so the theorem  that I'm going to talk about now says this is the only way to get the solutions okay the boosted  solutions and gives an explicit characterization of all the possible functions f and g you can  have here so theorem  so let you let v be a real number and uv  so this equation makes perfect sense if it's an h dot  one-half function and distribution of sense solve that then you have two cases  if v is bigger and modulus than one or equal to one the only way is that you have a trivial  solution it's a constant so it's nothing else can happen and if v satisfies this  then what rotations of course are on the sphere on the target you have to have that uv of x  is of that form which I wrote down and now what is f and g  v sorry yes  where f  f  is the real part say of a function capital f and g is the imaginary part  it's the real part of a holomorphic function in fact on the boundary living on the upper  half plane and on the boundary you get the little f and g is the imaginary part of that function  and f  is of the form f of z is equal to what is called a finite Blush product  so there's a k running from one to say d and there's maybe a lambda k z minus ak there's a  shift of course that is maybe possible plus i and these lambda k's are real numbers not zero  and the ak's are arbitrary points in r and d is a natural number of course  okay yes for v equal to zero you can characterize that but the point is and this is the  point that you can really reduce it to the case v is equal to zero so I will explain that the proof  is really an advancement of understanding the case when v is equal to zero okay  but so can this be understood directly on the stationary solution of the half harmonic map  or unique okay when I look at the stationary solution the half harmonic maps this is not  completely easy to see but doable okay with what's known in the literature in the recent  literature say the point is that you can so to say also undo the boost transformation I will explain  that okay yeah yeah it gives you in particular characterization of all half harmonic maps because  it's a case v is equal to zero okay and there's of course a saying there is no traveling wave  with speed larger than one okay this is also one point okay and the interesting fact is that the  energy of these guys is of this form there's a quantization of course involved like for  harmonic maps the d the index of the blusher product however by tuning the v you can make this  arbitrarily small so it's like the ill-to-critical half wave equation with patrick girard where you  have l2 criticality but you have solitary waves which have arbitrarily small l2 mass and now you have  solitary waves with arbitrary small energy energy is now the critical thing so it's completely  different say from the land of life shits where you have don't have that okay okay  okay all right so i have roughly 20 minutes what's the total  okay 17 it counts okay let me explain to you about the proof  okay  okay this is actually two statements a and b okay i will along say something about a  and but i will of course focus on the more interesting part b  so what's the initial situation  so think first that we are given a solution uv and think it's smooth okay to avoid any issues here  so that the image of course it raises us a closed curve on the unit sphere okay and  actually what we posteriori see that these are just great circles or in the case when v is equal  to zero or when v is not equal to zero it's it's a circle on the sphere okay but first we don't  know anything about this more than this so if you take so rewrite for simplicity little u  is equal to uv i want to skip that index little v because writing is a bit difficult okay first  thing of course that we do is also to see that you have a conformal invariance here is of course  you consider this as a two-dimensional problem in a sense because this arises as a boundary of  something yeah so you take the u to be the harmonic extension okay of little u that is  capital u is a map from r2 plus which you can also identify with the complex alpha half plane  into r3 it's a component wise harmonic function  yeah and the boundary condition star is a boundary condition then because as you know it's a classical  fact that operator just becomes the normal derivative with respect to this extension variable  minus the normal derivative so effectively you study the problem minus v of dx capital u  and there's a minus so it will change here but it's inessential of dy this is my new variable  that adds to my space like this on the boundary okay fine so far so good so by the maximum principle  actually there's a strong maximum principle that inside is strictly less than one well you see out  that you the image of capital u is something inside the sphere of course because of this  and the boundary is so to say given by this little u the boundary curve the profile of the  solitary wave okay so far so good now comes the first step let's say okay  capital u is a parametrization of a surface modulo there might be branching points and these things  it's maybe not embedded but we just say it's a minimal surface provides a minimal surface  that's the first step so how to see that the first idea is to use something which is called a  hopf differential so it's a function that you cook up  up depending on z say so z is just a notation of course for this  and dz is the wilting derivative that's that's a fairly standard step step here what i do but  i just want to explain it so i remember there should be maybe a fault to have everything nice  here so now you take the derivative of capital u with respect to the z so it's actually just this  you combine them so that's a c3 valued function it's a three vector with complex entries and you  take the scalar product not the Hermitian product so take this product this is a standard  thing in minimal surface theory so what you get is this  Jesus it's a worm here  nothing to do really with the questions it's the climate here okay okay so the the upshot is  that this is a because u is a harmonic function this is a holomorphic function okay so you either  really check that dz bar the anti- wilting derivative of this is zero hence it's a holomorphic  function it's easy to see because dz bar times dz is just times the constant the Laplace operator  okay so far so good and now comes the point that if you look at that function  and you consider the imaginary part of this hopf differential on the boundary  okay so then you use the boundary equation okay and now you have to explicitly work out what this  means actually it means in terms of real parts that you have this minus maybe two i i'm not  100 sure whether there's a plus or minus but it doesn't matter here this okay so the imaginary part  on the real axis of that holomorphic function in the upper half plane is the scalar product of d  x u times scalar product y u but now look at the equation here this immediately tells you  that at least when v is not equal to zero this has to be perpendicular to that vector  and you see that the imaginary part of that holomorphic function on the real line is zero  so you can extend it to the whole complex plane plane the imaginary part by  odd reflection however the imaginary part of phi because on the boundary it's h1 half so this  is actually an h dot one function so this is this times this is an l1 function it's integrable  okay so then you can easily conclude that the imaginary part is identically zero okay so  because it's a holomorphic function the real part has to be a constant a real constant  however it's also an l1 so it's also identically zero  so this hopf differential is actually identically zero fine so that tells you this is always equal  to this and this is always zero so it's right angle so u capital u is a conformal map and this  is just saying it's an harmonic function with which is conformal hence it's a minimal surface  which you get now as a next step yes as a next step but there could be lots of minimal surfaces  inside the unit ball of course they are very funny ones but they have to meet the boundary  in a certain specific way they have to respect of course this boundary condition here yeah  and when v is equal to zero this is something which I first referred as a free minimal disc  which was studied by Fraser and Shane and they classified that this can only be a plain disc  and then you are in the situation that the boundary is a great circle and then you are in good shape  to get to get your classification theorem however they approve I could not see how to make it really  work in the case when v is not equal to zero because you have a more complicated boundary condition  okay but I show you an argument which includes the Fraser and Shane result by considering the  following thing so what you learn from that first step that's a conformal map you also learn that  you can rewrite the boundary equation now I write little u here in a very interesting way  are people from harmonic maps know such a thing that you would maybe expect here a square  but this is not true in the fractional case it's it's not a square it's a one and there's a deeper  reason for that because if you test this equation against u then this term of course drops out and  u times u scalar wise is one so you get this okay okay what is this geometrically the right  hand side corresponds to the length of the curve and this if you use the harmonic extension the  lift corresponds to the area you span okay and you will see that it will actually satisfy an  isoparametric inequality set set rate an isoparametric it has satisfied of course the isoparametric  inequality otherwise would be bad set rates and from that you could also conclude that you have to  have plain disks but you don't know yet what the value of the left hand side is but this is just  a side remark so the next step is and this is then the little innovation maybe is to consider  another hopf differential differential well what is that take now this okay fine you can now I  leave the details because this is maybe too much on the blackboard write down what that is again  it's a holomorphic function you consider you consider study what's on the boundary you want to  make may maybe run a trick like that but of course you have to work a little bit harder  but using this formulation of the Euler Lagrange equation you also see that this is a holomorphic  function which is identically zero but that's too much a minimal surface which of course satisfies  the first thing there are lots of them however it also has to satisfy this okay and then  you can use what's called the Weierstrass inner power representation of a minimal surface  just want to say these things here so that you can deduce always one gets things right  it's half so what's that so it means that you can this is classical thing parameterized the  Wieltinger derivative of this u in terms of a holomorphic function f and the meromorphic  function which satisfies some compatibility condition but if you plug this into this  equation you work a little bit you will see ultimately that u the image of u can only lie  in a fixed plane in R3  okay so so what you get from that  finally I leave the details of course  that the image lies in the plane in R3 so it's a flat really flat minimal surface  so it's a disc so you will ultimately see that this is a disc  and of course the boundary of the disc is a circle on the sphere now you can undo this  transformation of course and go back to this case where it's the unboosted case okay and then  you're almost done because now you can invoke some complex analysis because in the unboosted case  say you transform it like this that it's in the xy plane then you will see with this analysis that  f and g have to be say the real and the imaginary part of a complex function f  i g on the upper half complex plane with complex values and because u lies on the unit sphere  the modulus squared of f is one on the boundary so that's this very special holomorphic function  on the upper half plane such that the modulus is one identically on the boundary and there's a  classification for that in terms of plasco products and because also f is in h dot one it's not hard  it's a sobolev space here you can conclude that this is actually a finite plasco product  which i wrote down so they only have finitely many factors otherwise you would have infinite energy  which is a bit formal okay finite plasco product hence you get the complete classification i wrote  to you and i finally should say that the last idea of using these plasco products for this kind of  problem is also something that already mironescu and pisante used for something which is related  where you look for what's called maybe a half harmonic map from s1 into s1 okay i should pay  also credit to this worker somewhat okay so what i want to say is now okay if you work out these  real imagining parts of these finite plasco products it's just kind of rational functions  you have so it's very explicit you can now study stability or instability you have very explicit  formulas for your solitary waves okay and this is also the point i like to end because it's  freaking warm here so thanks  oh no this is what i this is a byproduct of the proof so if you have a stationary solution  my terminology v is equal to zero and it has should have finite energy other okay modular  rotations on the sphere of course but they all have to but they are all okay yeah yeah yes  this is a yeah yeah that's a byproduct actually so and and i should maybe say about a i thought  you're looking for things of a special form right that's why i'm asking the question with p on that  i told you last component yeah but i cannot okay what i show finally is that i'm allowed to do this  because they're always equatorial discs so i can always rotate it that the last is identically zero  okay and i should say a is is a funny proof uh you test the equation against the hillbott transform  of you and do some magic uh and the in the l2 critical half wave case we cannot actually rule out  solitary waves which have speed more than one we don't know how to do this in this model it's a  magic thing to really get the sharp limit for the do you have a question  take me first to understand okay now we have done the job okay next oh okay i go home the next  question that ability and just existence of the flow by the way yes i have not talked about  existence of the flow yes of course yeah that's a lot of things to do yes  stability i would expect the forward why not in your project when you're just i would expect that  but also because formally you see if you think of the schrodinger map blow up you have an unstable one  because you have a slow decay of a resonance i mean if a slow decay in the linearization a zero  mode which is very slow decay and i think here it does not happen that you have i don't i don't  think so but i would not bet much money i have to look into after the euro  look into the things more closely yeah maybe yes maybe next week i have time  more questions please so what is your momentum functional ah yeah that is a good question  it's a bit like this  i draw a picture so the momentum of a configuration u a closed curve is the solid angle  that this curve so to say in the unit sphere gives you but this is of course not well defined up to  four pies because you have ambiguities so um so in the physical literature um there's a kind of a  dispute about what is whether there is a true momentum or not for such a model but you have a  you have a conserved quantity modulo a value of four pie you could consider an exponential of that  but it's not it's a it's a it's a it's a funny functional the momentum is is really not that  straightforward say for these models of course also you might wonder whether the traveling  solitary waves you get are also existing in the discrete models and this is always a delicate  question for nls there are might some traveling waves in the discrete lattice model or not and so  on and here it's also not so clear what survives or what gets created by this continuum limit  but sorry i was going off the tension there's there's so much structure it's actually very  beautiful there's a lot of structure and it does come from not just one iterable system but a class  yes so what would you speculate i would speculate it's also completely integral but i don't  i mean i try to to cook up lex pairs but it's not my prime education so uh take some time maybe  but i don't know maybe there's maybe i'm wrong maybe you see really uh serious blow up yeah i don't  yeah yeah that maybe in some weird way you see like for the chigo equation you might have infinitely  many conservation laws in evolution up to a certain kind of regularity in in higher  regularity you might as have at least grow up for infant i mean some small kind of infinite  tight blow up or something like that i don't know but conservation laws need to be need to be coercive  so for example bluzonesque is nicely integral that solutions just don't exist  there's no incompatibility between yeah okay you will teach me more than i know it's good  but yeah yeah it depends on how you define this i don't know we should waste more
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