Transonic limit of traveling waves of the Euler-Korteweg system

TL;DR

This paper proves that two-dimensional traveling waves of the Euler-Korteweg system converge to Kadomtsev-Petviashvili ground states, and one-dimensional solitons converge to Korteweg-de Vries solitons in the transonic limit.

Contribution

It establishes the transonic limit convergence of Euler-Korteweg traveling waves to integrable system solitons, linking fluid dynamics and soliton theory.

Findings

Convergence of 2D Euler-Korteweg waves to KP ground states

Convergence of 1D Euler-Korteweg solitons to KdV solitons

Results hold in the transonic limit with appropriate rescaling

Abstract

We prove the convergence in the transonic limit of two-dimensional traveling waves of the E-K system, up to rescaling, toward a ground state of the Kadomtsev-Petviashvili Equation. Similarly, in dimension one we prove the convergence in the transonic limit of solitons toward the soliton of the Korteweg de Vries equation.