Solitary waves with intensity-dependent dispersion: variational characterization

TL;DR

This paper characterizes a family of singular solitary waves with intensity-dependent dispersion, demonstrating their variational derivation, stability, and numerical confirmation, highlighting their unique cusped and bell-shaped structures.

Contribution

It introduces a variational framework for singular solitary waves with intensity-dependent dispersion and proves their stability under certain perturbations.

Findings

Existence of a continuous family of singular solitary waves.

Variational characterization via mass minimization at fixed energy and length.

Numerical simulations confirm stability of the waves.

Abstract

A continuous family of singular solitary waves exists in a prototypical system with intensity-dependent dispersion. The family has a cusped soliton as the limiting lowest energy state and is formed by the solitary waves with bell-shaped heads of different lengths. We show that this family can be obtained variationally by minimization of mass at fixed energy and fixed length of the bell-shaped head. We develop a weak formulation for the singular solitary waves and prove that they are stable under perturbations which do not change the length of the bell-shaped head. Numerical simulations confirm the stability of the singular solitary waves.