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Remarks on NLS with higher order anisotropic dispersion. (English) Zbl 1159.35424

Summary: We show that the introduction of third- and fourth-order anisotropic dispersion terms in the nonlinear Schrödinger equation enables to solve the global Cauchy problem for nonlinearities which are (super) critical in the standard case. Some blow-u results are proved in the supercritical case. Then, we prove the existence of solitary waves and discuss their regularity, their exponential decay rate at infinity, and their stability.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B35 Stability in context of PDEs
35A15 Variational methods applied to PDEs
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