We consider two physically and mathematically distinct regularization mechanisms of scalar
hyperbolic conservation laws. When the flux is convex, the combination of diffusion and …

There is growing physical and mathematical interest in the hydrodynamics of
dissipationless/dispersive media. Since GB Whitham's seminal publication fifty years ago …

The inverse scattering transform for the focusing nonlinear Schrödinger equation with non-
zero boundary conditions at infinity is presented, including the determination of the …

We review the spectral theory of soliton gases in integrable dispersive hydrodynamic
systems. We first present a phenomenological approach based on the consideration of …

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-
wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of …

In this paper we consider soliton and breather gases for one dimensional integrable
focusing nonlinear Schrödinger equation (fNLS). We derive average densities and fluxes for …

The nonlinear Fourier transform, which is also known as the forward scattering transform,
decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common …

The semiclassical (zero‐dispersion) limit of solutions q=q(x,t,ϵ) to the one‐dimensional
focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a …

We calculate the leading-order term of the solution of the focusing nonlinear (cubic)
Schrödinger equation (NLS) in the semiclassical limit for a certain oneparameter family of …

We argue that the critical behavior near the point of “gradient catastrophe” of the solution to
the Cauchy problem for the focusing nonlinear Schrödinger equation iϵ\varPsi_t+ϵ^22 …