On two systems of Burgers type arising in nonlinear wave interaction

TL;DR

This paper investigates the well-posedness and singularity formation in two Burgers-type systems modeling nonlinear wave interactions, including a bore-KdV interaction and weak wave interactions, with results on existence, uniqueness, and finite-time blow-up.

Contribution

It establishes local well-posedness for one system and demonstrates finite-time singularity for the other, advancing understanding of wave interaction models.

Findings

Proved local existence and uniqueness in Sobolev and Wiener spaces for the first system.

Established finite-time singularity formation in the second system.

Provided elementary proof techniques for wave interaction phenomena.

Abstract

In this note we study the well-posedness of two systems of Burgers type arising in nonlinear wave interactions. The first model describes the interaction of a Burger's bore with the classical Korteweg-de Vries equation while the second exemplify the interaction of weak sound waves and entropy waves with small amplitudes. For the former, we show the local existence and uniqueness of solutions in Sobolev spaces and Wiener-type spaces. For the latter, we provide an elementary proof of finite time singularity.