Renormalized asymptotic solutions of the Burgers equation and the Korteweg-de Vries equation

TL;DR

This paper develops uniform renormalized asymptotic solutions for the Burgers and Korteweg-de Vries equations, addressing large initial gradients and weak discontinuities to improve understanding of their long-term behavior.

Contribution

It introduces a novel method for constructing uniform asymptotic solutions for these equations under challenging initial conditions.

Findings

Successful construction of asymptotic solutions for large initial gradients

Effective handling of perturbed weak discontinuities

Enhanced understanding of solution behavior in complex initial scenarios

Abstract

The Cauchy problem for the Burgers equation and the Korteweg-de Vries equation is considered. Uniform renormalized asymptotic solutions are constructed in cases of a large initial gradient and a perturbed initial weak discontinuity.