On full Zakharov equation and its approximations
Abstract
We study the solvability of the Zakharov equation Δ^{2} u +(κ ω^{2}) Δu  κdiv(^{e   ∇ u2} ∇ u) = 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.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 January 2020
 DOI:
 10.1016/j.physd.2019.132168
 arXiv:
 arXiv:1801.00803
 Bibcode:
 2020PhyD..40132168B
 Keywords:

 Zakharov equations;
 Langmuir collapse;
 Ground state;
 Variational methods;
 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 17 pages