Global Small Solutions of the Cauchy Problem for Nonisotropic Schrödinger Equations

Abstract

Abstract. In this paper we study the existence of global small solutions of theCauchy problem for the non-isotropically perturbed nonlinear Schr¨odinger equation:iu t + ∆u + |u| α u + aP di u x i x x x = 0, where a is real constant, 1 ≤ d < n is a inte-ger, α is a positive constant, and x = (x 1 ,x 2 ,··· ,x n ) ∈ R n . For some admissible α weshow the existence of global(almost global) solutions and we calculate the regularity ofthose solutions. 1. IntroductionThis paper is concerned with the Cauchy problem of the following fourth-ordernonlinear dispersive equation in R n ×R:(1)ˆiu t +∆u+|u| α u+aP di u x i i i i = 0, x ∈ R n , t ∈ R.u(x,0) = ϕ(x), x ∈ R n .where d is an integer, 1 ≤ d < n, a is a nonzero real constants, and α is a pos-itive constant. This equation is a modified version of the semi-discrete nonlinearSchrodinger equation (see [1]), or a non-isotropic higher-order perturbation of thesecond-order nonlinear Schrodinger equation:(2) iu t +∆u+|u| α u = 0, x ∈ R n , t ∈ R.Clearly, the following equations are special cases of (1)iu