Cauchy problem for higher-order Schrödinger equations in anisotropic Sobolev space

Abstract

We study the solvability in anisotropic Sobolev spaces for the Cauchy problem of the nonisotropically perturbed sixth-order nonlinear Schrödinger equation: , where a, b are real constants, 1 ≤ d < n and x = (x ₁, x ₂, …, x _n ) ∈ R ⁿ , t ∈ R and f(u) is a given complex function. By using Kato's method, we establish local well-posedness for the equation. We also obtain its global solvability by the anisotropic Sobolev embedding relations and the energy method.

Keywords:

• non-isotropic Schrödinger equation
• Cauchy problem
• local well-posedness
• global solvability
• anisotropic Sobolev embedding

AMS Subject Classification:

• 35Q55

Acknowledgements

C. Guo of this work is supported by the China National Science Foundation (No. 10726041, No. 10771128 and No. 10911140114) and by the Youth Science and Technology Research Foundation of Shanxi (No. 2008021001-2). X. Wei is supported by the China National Natural Science Foundation (No. 10726017) and the PhD Foundation of Guangdong University of Technology (No. 063043). We would like to thank the referees for valuable corrections and suggestions.