Normal form and solitons

TL;DR

This paper reviews the normal form theory for weakly dispersive nonlinear wave equations, focusing on soliton interactions, with applications to various physical models and comparisons with numerical simulations.

Contribution

It extends normal form theory to infinite-dimensional wave equations and analyzes solitary wave interactions with explicit examples and numerical validation.

Findings

Normal form theory effectively describes weakly dispersive wave phenomena.

Explicit analysis of solitary wave interactions in several models.

Comparison shows good agreement between theory and simulations.

Abstract

We present a review of the normal form theory for weakly dispersive nonlinear wave equations where the leading order phenomena can be described by the KdV equation. This is an infinite dimensional extension of the well-known Poincar\'e-Dulac normal form theory for ordinary differential equations. We also provide a detailed analysis of the interaction problem of solitary wavesas an important application of the normal form theory. Several explicit examples are discussed based on the normal form theory, and the results are compared with their numerical simulations. Those examples include the ion acoustic wave equation, the Boussinesq equation as a model of the shallow water waves, the regularized long wave equation and the Hirota bilinear equation having a 7th order linear dispersion.