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Bobkov, Vladimir; Drábek, Pavel; Ilyasov, Yavdat

On full Zakharov equation and its approximations. (English) Zbl 1453.35004

Physica D 401, Article ID 132168, 8 p. (2020).
Summary: 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.

MSC:

35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35P05 General topics in linear spectral theory for PDEs

Keywords:

Zakharov equations; Langmuir collapse; ground state; variational methods
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References:

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