Formation of compression waves with multiscale asymptotics in the Burgers and KdV models

TL;DR

This paper develops multiscale asymptotic methods to analyze the formation of shock and rarefaction waves in Burgers and KdV equations, providing insights into wave dynamics with small dissipation and large gradients.

Contribution

It introduces new multiscale asymptotic techniques for constructing solutions related to shock and rarefaction waves in nonlinear PDEs.

Findings

Constructed asymptotics for shock wave formation.

Applied methods to Burgers and KdV models.

Results enhance understanding of wave evolution in nonlinear systems.

Abstract

The Cauchy problem for the Burgers equation with a small dissipation and an initial weak discontinuity and the Cauchy problem with a large initial gradient for a quasilinear parabolic equation and for the Korteweg-de Vries (KdV) equation are considered. Multiscale asymptotics of solutions corresponding to shock waves are constructed. Some results can also be applied to rarefaction waves.