Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose--Einstein condensates. As in the classical theory of hyperbolic equations where a nonconvex flux leads to nonclassical solution structures, a nonconvex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth order Korteweg--de Vries (KdV) equation, also known as the Kawahara equation, a classical model for shallow water waves, is shown to be a universal model of Eulerian hydrodynamics with higher order dispersive effects. Utilizing asymptotic methods and numerical computations, this work classifies the long-time behavior of solutions for step-like initial data. For convex dispersion, the result is a dispersive shock wave (DSW), qualitatively and quantitatively bearing close resemblance to the KdV DSW. For nonconvex dispersion, three distinct dynamic regimes are observed. For small amplitude jumps, a perturbed KdV DSW with ...
We consider the step Riemann problem for the system of equations describing the propagation of a coherent light beam in nematic liquid crystals, which is a general system describing nonlinear wave propagation in a number of different physical applications. While the equation governing the light beam is of defocusing nonlinear Schrodinger (NLS) equation type, the dispersive shock wave (DSW) generated from this initial condition has major differences from the standard DSW solution of the defocusing NLS equation. In particular, it is found that the DSW has positive polarity and generates resonant radiation which propagates ahead of it. Remarkably, the velocity of the lead soliton of the DSW is determined by the classical shock velocity. The solution for the radiative wavetrain is obtained using the Wentzel–Kramers–Brillouin approximation. It is shown that for sufficiently small initial jumps the nematic DSW is asymptotically governed by a Korteweg–de Vries equation with the fifth-order dispersion, which explicitly shows the resonance generating the radiation ahead of the DSW. The constructed asymptotic theory is shown to be in good agreement with the results of direct numerical simulations.  We show that pulses undergoing wave breaking in nonlinear weakly dispersive fibers radiate, owing to phase-matching (assisted by higher-order dispersion) of linear dispersive waves with the shock-wave front. Our theoretical results perfectly explain the radiation observed recently from pulses propagating in the normal dispersion (i.e., nonsolitonic) regime. Fundamental physical phenomena such as laser-induced ionization, driven quantum tunneling, Faraday waves, Bogoliubov quasiparticle excitations, and the control of new states of matter rely on time-periodic driving of the system. A remarkable property of such driving is that it can induce the localized (bound) states to resonantly couple to the continuum. Therefore experiments that allow for enlightening and controlling the mechanisms underlying such coupling are of paramount importance. We implement such an experiment in a special optical fiber characterized by a dispersion oscillating along the propagation coordinate, which mimics “time”. The quasi-momentum associated with such periodic perturbation is responsible for the efficient coupling of energy from the localized wave-packets (solitons in anomalous dispersion and shock fronts in normal dispersion) sustained by the fiber nonlinearity, into free-running linear dispersive waves (continuum) at multiple resonant frequencies. Remarkably, the observed resonances can be explained by means of a unified approach, regardless of the fact that the localized state is a soliton-like pulse or a shock front. We show that passive Kerr resonators pumped close to zero dispersion wavelengths on the normal dispersion side can develop the resonant generation of linear waves driven by cavity (mixed dispersive–dissipative) shock waves. The resonance mechanism can be successfully described in the framework of the generalized Lugiato–Lefever equation with higher-order dispersive terms. Substantial differences with radiation from cavity solitons and purely dispersive shock waves dispersion are highlighted. The viscously dominated, low Reynolds' number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless dispersive behavior when viewed appropriately. In these systems, nonlinear self-steepening competes with wave dispersion, giving rise to dispersive shock waves (DSWs). Example systems considered here include magma migration through the mantle as well as the buoyant ascent of a low density fluid through a viscously deformable conduit. These flows are modeled by a third-order, degenerate, dispersive, nonlinear wave equation for the porosity (magma volume fraction) or cross-sectional area, respectively. Whitham averaging theory for step initial conditions is used to compute analytical, closed form predictions for the DSW speeds and the leading edge amplitude in terms of the constitutive parameters and initial jump height. Novel physical behaviors are identified including backflow and DSW implosion for initial jumps sufficient to cause gradient catastrophe in the Whitham modulation equations. Theoretical predictions are shown to be in excellent agreement with long-time numerical simulations for the case of small to moderate amplitude DSWs. Verifiable criteria identifying the breakdown of this modulation theory in the large jump regime, applicable to a wide class of DSW problems, are presented. There is growing physical and mathematical interest in the hydrodynamics of dissipationless dispersive media. Since G.B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrodinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs. We show that dispersive shock waves resulting from the nonlinearity overbalancing a weak leading-order dispersion can emit resonant radiation owing to higher-order dispersive contributions. We analyze such phenomenon for the defocusing nonlinear Schroedinger equation, giving criteria for calculating the radiated frequency based on the estimate of the shock velocity, revealing also a diversity of possible scenarios depending on the order and magnitude of the dispersive corrections.