On full Zakharov equation and its approximations

TL;DR

This paper investigates the solvability of a nonlinear Zakharov equation related to plasma physics, analyzing existence, nonexistence, and multiplicity of solutions, and compares the original problem with its Taylor expansion approximations.

Contribution

It provides new insights into the existence and nonexistence of solutions for the Zakharov equation and its approximations, highlighting significant differences.

Findings

Existence and nonexistence results differ between original and approximate equations.

Conditions for ground state solutions are established.

Approximate models show substantially different solution behaviors.

Abstract

We study the solvability of the Zakharov equation $$\Delta^2 u + (\kappa-\omega^2)\Delta u - \kappa \,\text{div} \left(e^{-|\nabla u|^2} \nabla u\right) = 0$$ in a bounded domain under homogeneous Dirichlet or Navier boundary conditions. This problem is a consequence of the system of equations derived by Zakharov to model the Langmuir collapse in plasma physics. Assumptions for the existence and nonexistence of a ground state solution as well as the multiplicity of solutions are discussed. Moreover, we consider formal approximations of the Zakharov equation obtained by the Taylor expansion of the exponential term. We illustrate that the existence and nonexistence results are substantially different from the corresponding results for the original problem.