Traveling Waves for Conservation Laws with Cubic Nonlinearity and BBM Type Dispersion

TL;DR

This paper investigates traveling wave solutions in scalar conservation laws with cubic nonlinearity and BBM-type dispersion, providing explicit calculations and applications to the Riemann problem and elasticity models.

Contribution

It explicitly computes traveling waves for cubic flux with BBM dispersion, extending the analysis to non-convex fluxes and elasticity models, and explores their role in solving the Riemann problem.

Findings

Explicit traveling wave solutions for cubic flux with BBM dispersion.

Traveling waves can be used to select physically relevant solutions in non-convex conservation laws.

Extension of results to the p-system of elasticity with cubic stress-strain law.

Abstract

Scalar conservation laws with non-convex fluxes have shock wave solutions that violate the Lax entropy condition. In this paper, such solutions are selected by showing that some of them have corresponding traveling waves for the equation supplemented with dissipative and dispersive higher-order terms. For a cubic flux, traveling waves can be calculated explicitly for linear dissipative and dispersive terms. Information about their existence can be used to solve the Riemann problem, in which we find solutions for some data that are different from the classical Lax-Oleinik construction. We consider dispersive terms of a BBM type and show that the calculation of traveling waves is somewhat more intricate than for a KdV-type dispersion. The explicit calculation is based upon the calculation of parabolic invariant manifolds for the associated ODE describing traveling waves. The results extend to the p-system of one-dimensional elasticity with a cubic stress-strain law.