Multi-solitons of one-dimensional Boussinesq equation

TL;DR

This paper proves the existence of multi-speed solitary wave solutions for the one-dimensional good Boussinesq equation with power nonlinearity, demonstrating their behavior at large times as pairs of scalar solitary waves traveling at different speeds.

Contribution

It introduces a novel method for constructing multi-speed solitary waves for the Boussinesq equation using backward-in-time approximations and energy estimates.

Findings

Existence of multi-speed solitary waves proven for all cases.

Solutions behave as pairs of solitary waves at large times.

Method applicable to both subcritical and supercritical cases.

Abstract

The existence of multi-speed solitary waves for the one-dimensional good Boussinesq equation with a power nonlinearity is proven. These solutions are shown to behave at large times as a pair of scalar solitary waves traveling at different speeds. Both subcritical and supercritical cases are treated. The proof is based on the construction of approximations of the multi-speed solitary waves by solving an equivalent system backward in time and using energy methods to obtain uniform estimates.