Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

Abstract

The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude-frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, that is, this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge. K E Y W O R D S asymptotic analysis, nonlinear waves, partial differential equationsThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Thermodynamic limit and dispersive regularization in matrix models

“…for suitable constants C k , the solution of which, for the assigned initial condition, is implicitly given by the polynomial equation (12). The effect of the corrections in the parameter will be discussed in the following section.…”

“…In particular, local minima and maxima depend on the signature of the discriminant ∆(x, T 2 , T 4 , T 6 ) of the cubic equation (12). If ∆ > 0 the free energy has two local minima and one local maximum, if ∆ < 0 the free energy presents one local minimum only.…”

“…1 we plot the set ∆ = 0 in the x-T 6 plane for a given choice of T 2 and T 4 (notice that in order to ensure convergence of the integral (1), we have T 6 < 0). The convex sector corresponds to the region where the equation of state (12) admits three real solutions that correspond to the stationary points of the free energy density. Fig.…”

“…Along with solitons, another important class of solutions of the KdV equation is constituted by Dispersive Shock Waves (DSW) [7,8]. DSWs occur as a universal regularisation mechanism of singularities in dispersive hydrodynamics, and effectively explain the emergence of a variety of complex behaviours in hydrodynamic systems [7][8][9][10][11][12]. The remarkable mathematical structure of completely integrable systems, of which the KdV equation is possibly the most celebrated example, such as the existence of an infinite number of conservation laws, allows for an effective and detailed description of dispersive shock waves (see e.g.…”

Preface: Nonlinear waves in fluids in honor of Roger Grimshaw on the occasion of his 80th birthday

“…The traveling waves are shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. In, the authors develop a universal analytical description of dispersive shock waves generated in Riemann problems for a broad class of integrable and non‐integrable nonlinear dispersive equations. Several representative, physically relevant examples are considered to illustrate the efficiency of the developed general theory.…”