Compactons and their variational properties for degenerate KdV and NLS in dimension 1

TL;DR

This paper investigates degenerate KdV and NLS equations in one dimension, revealing the existence of compactly supported solutions due to degeneracy, and employs variational methods to analyze their properties.

Contribution

It introduces a variational framework for degenerate dispersive equations, demonstrating the existence of compactons unlike standard solitons.

Findings

Existence of compactly supported steady and traveling solutions.

Degeneracy enables solutions with finite support, contrasting with classical solitons.

Variational methods effectively analyze these solutions.

Abstract

We analyze the stationary and traveling wave solutions to a family of degenerate dispersive equations of KdV and NLS-type. In stark contrast to the standard soliton solutions for non-degenerate KdV and NLS equations, the degeneracy of the elliptic operators studied here allows for compactly supported steady or traveling states. As we work in $1$ dimension, ODE methods apply, however the models considered have formally conserved Hamiltonian, Mass and Momentum functionals, which allow for variational analysis as well.