Jaime Angulo Pava

Nonlinear dispersive equations : existence and stability of solitary and periodic travelling wave solutions

This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied include Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.

Positivity Properties of the Fourier Transform and the Stability of Periodic Travelling-Wave Solutions

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg–de Vries-type

(Non)linear instability of periodic traveling waves: Klein-Gordon and KdV type equations

u_t+u^pu_x-Mu_x=0

Stability of periodic optical solitons for a nonlinear Schrödinger system

, with M being a general pseudodifferential operator and where

Stability of standing waves for NLS-log equation with \(\varvec{\delta }\)-interaction

p\geq1

On the orbital instability of excited states for the NLS equation with the δ -interaction on a star graph

is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin–Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg–de Vries and modified Korteweg–de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

Stability properties of periodic traveling waves for the intermediate long wave equation

Abstract. We prove the existence and nonlinear instability of periodic traveling wave solutions for the critical one-dimensional Klein–Gordon equation. We also establish a linear instability criterium for a KdV type system. An application of this approach is made to obtain the linear/nonlinear instability of vector cnoidal wave profiles. Finally, via a theoretical and numerical approach we show the linear stability or instability of periodic positive and sign changing waves, respectively, for the critical Korteweg–de Vries equation.

STABILITY OF CNOIDAL WAVES

This paper is concerned with the existence and nonlinear stability of periodic travelling-wave solutions for a nonlinear Schrodinger-type system arising in nonlinear optics. We show the existence of smooth curves of periodic solutions depending on the dnoidal-type functions. We prove stability results by perturbations having the same minimal wavelength, and instability behaviour by perturbations of two or more times the minimal period. We also establish global well posedness for our system by using Bourgains approach.

Stability and instability of periodic travelling wave solutions for the critical Korteweg–de Vries and nonlinear Schrödinger equations

We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrödinger equation with