Nonexistence of multi-dimensional solitary waves for the Euler-Poisson system

TL;DR

This paper proves that multi-dimensional Euler-Poisson systems do not admit localized solitary wave solutions, contrasting with the one-dimensional case where such waves exist, using Pohozaev identities and energy integrals.

Contribution

It establishes the nonexistence of multi-dimensional solitary waves for the Euler-Poisson system, extending results to two-species models and general pressure laws.

Findings

No nontrivial irrotational solitary waves in 2D and 3D Euler-Poisson systems.

Pohozaev identities are key to proving nonexistence.

Results extend to two-species ion-electron models.

Abstract

We study the nonexistence of multi-dimensional solitary waves for the Euler-Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler-Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves for any traveling velocity and for general pressure laws. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler-Poisson system for ions and electrons.