Shock waves and compactons for fifth-order nonlinear dispersion equaitons

TL;DR

This paper investigates fifth-order nonlinear dispersion equations, demonstrating shock wave formation, rarefaction waves, and nonuniqueness issues post-blow-up, with a focus on entropy shocks and smooth deformations.

Contribution

It introduces a framework to distinguish entropy shocks from rarefaction waves and analyzes the nonuniqueness arising after gradient catastrophe in fifth-order nonlinear dispersion equations.

Findings

Shock waves and rarefaction waves can form in fifth-order nonlinear dispersion equations.

Single point gradient catastrophe leads to nonuniqueness after blow-up.

Smooth deformations help differentiate between entropy shocks and rarefaction waves.

Abstract

Fifth-order 1D nonlinear dispersion equations are shown to admit blow-up formation of shock waves as well as rarefaction waves. The concepts of smooth deformations are applied to distinguish "entropy" shocks from smooth rarefaction waves. Single point gradient catastrophe is shown to lead to nonuniqueness after blow-up.