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Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation. (English) Zbl 0764.35096

Summary: We give the approximate symmetries for the multidimensional Landau- Ginzburg equation \(\sum_{i=1}^ 3 \partial^ 2 u/\partial x_ i^ 2+\partial u/\partial x_ 4=a_ 1+a_ 2u+\varepsilon u^ n\) where \(n\in{\mathcal R}\) and \(0<\varepsilon\ll 1\). We also construct approximate solutions for this nonlinear equation using the approximate symmetries.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
58J70 Invariance and symmetry properties for PDEs on manifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs

Keywords:

Lie symmetries
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