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

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 in the $L^1$ space for any traveling velocity and for general pressure laws. Our results provide theoretical evidence for the stability of line solitary waves in multi-dimensional Euler–Poisson flows. 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.