Type: Article

Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrodinger equation in the critical case

Hayashi, Nakao; Naumkin, Pavel I.;

Abstract:

We consider the Cauchy problem for the nonlinear fourth-order nonlinear Schrodinger equation {i partial derivative(t)u + 1/4 partial derivative(4)(x)u = i lambda partial derivative(x)(vertical bar u vertical bar(3)u), t > 0, x is an element of R, u(0, x) = u(0) (x), x is an element of R with critical nonlinearity, where lambda is an element of R. We assume that the initial data are such that u(0) is an element of H-1,H-1, with sufficiently small norm vertical bar u(0)vertical bar(H1,1). We prove that there exists a unique global solution e(-it/4 partial derivative x4u) is an element of C([0, infinity); H-1,H-1) of the Cauchy problem for the nonlinear fourth-order nonlinear Schrodinger equation such that vertical bar u(t)vertical bar(L infinity) <= C(1 + t)(-1/4). Moreover we show that if the total mass 1/root 2 pi integral(R)u(0)(x) dx not equal 0, then the large time asymptotics is determined by the self-similar solution