Nonlinear dispersion equations: smooth deformations, compactons, and extensions to higher orders

TL;DR

This paper investigates third-order nonlinear dispersion equations, demonstrating their ability to admit shock and rarefaction waves through smooth deformations, and establishes the existence of compacton solutions, extending the analysis to higher-order equations.

Contribution

It introduces a smooth deformation approach to distinguish wave types and proves compacton solutions are delta-entropy and G-admissible, extending the theory to higher-order NDEs.

Findings

Shock and rarefaction waves are admitted as weak solutions.

Compacton solutions are delta-entropy and G-admissible.

Extensions to higher-order NDEs are successfully performed.

Abstract

Third-order nonlinear dispersion equations (NDEs) are shown to admit both shock and rarefaction waves (as weak solutions), which are distinguished by a smooth deformation approach. Compacton-type travelling wave solutions are proved to be both delta-entropy and G-admissible (in the sense of I.M. Gel'fand, 1963). Extensions to some higher-order NDEs are performed.