Nonlinear surface waves on the plasma-vacuum interface

TL;DR

This paper investigates the weakly nonlinear propagation of surface waves at a plasma-vacuum interface, deriving an asymptotic nonlocal Hamiltonian equation and establishing local existence and blow-up criteria for solutions.

Contribution

It introduces a novel asymptotic nonlocal Hamiltonian evolution equation for surface waves on a plasma-vacuum interface and proves local existence and blow-up conditions.

Findings

Derived an asymptotic nonlocal Hamiltonian equation for surface wave amplitude.

Proved local-in-time existence of smooth solutions to the amplitude equation.

Established a blow-up criterion for solutions of the amplitude equation.

Abstract

In this paper we study the propagation of weakly nonlinear surface waves on a plasma-vacuum interface. In the plasma region we consider the equations of incompressible magnetohydrodynamics, while in vacuum the magnetic and electric fields are governed by the Maxwell equations. A surface wave propagate along the plasma-vacuum interface, when it is linearly weakly stable. Following the approach of Ali and Hunter, we measure the amplitude of the surface wave by the normalized displacement of the interface in a reference frame moving with the linearized phase velocity of the wave, and obtain that it satisfies an asymptotic nonlocal, Hamiltonian evolution equation. We show the local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables, and we derive a blow up criterion.