Asymptotic behaviour for Schrödinger equations with a quadratic nonlinearity in one-space dimension

Hayashi, Nakao ; Naumkin, Pavel I.

Electronic Journal of Differential Equations, Tome 2001 (2001).

Résumé

Summary: We consider the Cauchy problem for the Schrödinger equation with a quadratic nonlinearity in one space dimension $$ iu_{t}+\frac{1}{2}u_{xx}=t^{-\alpha}| u_x| ^2,\quad u(0,x) = u_0(x), $$ where $\alpha \in (0,1)$. From the heuristic point of view, solutions to this problem should have a quasilinear character when $\alpha \in (1/2,1)$. We show in this paper that the solutions do not have a quasilinear character for all $\alpha \in (0,1)$. due to the special structure of the nonlinear term. We also prove that for $\alpha \in [1/2,1)$ if the initial data $u_0\in H^{3,0}\cap H^{2,2}$ are small, then the solution has a slow time decay such as $t^{-\alpha /2}$. For $\alpha \in (0,1/2)$, if we assume that the initial data $u_0$ are analytic and small, then the same time decay occurs.

Détail

Classification : 35Q55, 74G10, 74G25
Keywords: Schrödinger equation, large time behaviour, quadratic nonlinearity