Integrability of a Generalized Short Pulse Equation Revisited

In this article we present a brief overview of the nature of localized solitary wave structures/solutions underlying integrable nonlinear dispersive wave equations with specific reference to shallow water wave propagation and explore their possible connections to tsunami waves. In particular, we will discuss the derivation of Korteweg-de Vries family of soliton equations in unidirectional wave propagation in shallow waters and their integrability properties and the nature of soliton collisions.

Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operator

The bifurcation theory of dynamical systems is applied to an integrable nonlinear wave equation. As a result, it is pointed out that the solitary waves of this equation evolve from bell-shaped solitary waves to W/M-shaped solitary waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitary wave solutions are obtained.

Bifurcation studies on travelling wave solutions for an integrable nonlinear wave equation

Dynamical system theory is applied to the integrable nonlinear wave equation ut±(u3–u2)x+(u3)xxx=0. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation correspond to the case of wave speed c=0. In the case of c≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.

Nonlinear Fourier Methods for Ocean Waves

The Numerical Solution of Integral Equations on the Half-Line

Soliton physics and the periodic inverse scattering transform

In this review paper we discuss the range of validity of nonlinear dispersiveintegrable equations for the modelling of the propagation of tsunami waves. For the 2004tsunami the available measurements and the geophyiscal scales involvedrule out a connection between integrable nonlinear wave equations and tsunami dynamics.

This article is a summary of our numerical and theoretical studies (which were done in various collaborations with Alan Bishop, Nick Ercolani, Greg Forest, and Steve Wiggins) of near integrable nonlinear wave equations under periodic boundary conditions. Two examples, a damped driven sine-Gordon equation and a perturbed nonlinear Schrodinger equation, are discussed in detail. The article begins with a thorough description of numerical experiments on the two systems in a parameter regime for which the response is spatially coherent, yet temporally chaotic. In addition to the description of this qualitative behavior in the pde’s, numerical and statistical issues are emphasized. Next, the spectral transform for the integrable nonlinear Schrodinger equation is developed in sufficient detail for use in both theoretical and numerical analysis of the perturbed system. This integrable theory includes the introduction of a Morse function which unveils a hyperbolic or saddle structure in the constants of the motion, the association of this saddle structure with complex double periodic eigenvalues for the spectral transform, and the use of Backlund transformations to produce from these complex double points analytical representations of homoclinic orbits and whiskered tori. Next, the spectral transform is used as a numerical diagnostic to monitor the chaotic attractors in the perturbed system. Finally, a Melnikov analysis of a perturbed model system is described. This geometric perturbation theory is based upon the analytical representations of whiskered tori in the nearby integrable system. Open problems are discussed throughout the text and summarized in the conclusion.

We have obtained a class of solitary wave solutions to novel exactly integrable nonlinear wave equations. Conservation laws can be identified and velocities of propagation predicted. We propose to test our predictions in the optical domain with two-color experiments.

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schrödinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.

We show a direct connection between a cellular automaton and integrable nonlinear wave equations. We also present the $N$-soliton formula for the cellular automaton. Finally, we propose a general method for constructing such integrable cellular automata and their $N$-soliton solutions.

New exact solutions for a generalization of the Korteweg–de Vries equation (KdV6)