The generation, propagation, and extinction of multiphases in the KdV zero‐dispersion limit

Abstract

We study the multiphases in the KdV zero-dispersion limit. These phases are governed by the Whitham equations, which are 2g + 1 quasi-linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space-time, and how it collapses to a single phase in a finite time.The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler-Poisson-Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler-PoissonDarboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the EulerPoisson-Darboux solution.

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

“…It is known [35,30] that there exists a time T > t c such that for t c < t < T , the leading edge x − (t) is determined uniquely by the system of equations 13) with u(t) > v(t) and with…”

Dispersive shock wave interactions and asymptotics

“…The evolution of multiphase DSWs to a single-phase DSW was investigated in the two-phase case by Grava and Tian [13] using Whitham theory in the zero-dispersion limit (ε → 0) for finite time and by Ablowitz et al [14] using numerical and asymptotic methods in the fixed-dispersion, long-time limit. Both zero-dispersion and long-time are important, but different, limits.…”

“…This is similar to interacting viscous shock waves (VSW), where only the single, largest possible VSW remains after a long time. Grava and Tian [13] and Ablowitz et al [14] suggested this merging of multiphase to single-phase by their two-phase to onephase results -they used Whitham theory, which applies to slowly varying periodic wavetrains. We anticipate that the IST and matched-asymptotic procedure presented here will be applied to other important, nonlinear integrable systems for general, step-like data, such as the modified Korteweg-de Vries (mKdV) and the nonlinear Schrödinger (NLS) equations.…”

Self-similar solutions of the non-strictly hyperbolic Whithamequations for the KdV hierarchy

“…We now turn to case (2). By Lemma 3.5, the last two equations of (3.1) determine u 2 and u 3 as functions of the self-similarity variable x/t in the interval α ≤ x/t ≤ β.…”

“…The solution of equations (2.17-2.19) and that of (2.22) and (2.23) can be solved explicitly [2]. In particular, the solution of (2.22) and (2.23) is [10] …”