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Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

Authors:	Nakao Hayashi Pavel I Naumkin Yasuko Yamazaki

Institution:	Department of Applied Mathematics, Science University of Tokyo, Tokyo 162-8601, Japan ; Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacán, México ; Department of Applied Mathematics, Science University of Tokyo, Tokyo 162-8601, Japan

Abstract:	We consider the derivative nonlinear Schrödinger equations $\begin{displaymath}\left\{ \begin{split} iu_{t}+\tfrac{1}{2}&u_{xx}=a(t)F(u,u_{... ...epsilon u_{0}(x),\quad x\in {\mathbf{R}}, \end{split} \right. \end{displaymath}$ where the coefficient $a\left( t\right)$ satisfies the time growth condition $\begin{displaymath}\left\vert a\left( t\right) \right\vert \leq C\left( 1+\left\vert t\right\vert \right) ^{1-\delta },\qquad 0<\delta <1,\end{displaymath}$ $\epsilon$ is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type $\begin{align}F(u,u_{x}) = &\lambda _{1}\left\vert u\right\vert ^{2}u+i\lambda _... ...ar{u} u_{x}^{2}+i\lambda _{6}\left\vert u_{x}\right\vert ^{2}u_{x}, \end{align}$ where $\lambda _{1},\lambda _{6}\in \mathbf{R},$ $\lambda _{2},\lambda _{3},\lambda _{4},\lambda _{5}\in \mathbf{C},$ $\lambda _{2}-\lambda _{3}\in \mathbf{R,}$ and $\lambda _{4}-\lambda _{5}\in \mathbf{R}$ . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate $\Vert u(t)\Vert _{\mathbf{L}^{p}}\leq C\epsilon t^{ \frac{1}{p}-\frac{1}{2}}$ , for all $t\geq 1$ , where $2\leq p\leq \infty$ . Furthermore we show that for $\frac{1}{2}<\delta <1$ there exist the usual scattering states, when $b(x)=\lambda _{1}-\left( \lambda _{2}-\lambda _{3}\right) x+\left( \lambda _{4}-\lambda _{5}\right) x^{2}-\lambda _{6}x^{3}=0,$ and the modified scattering states, when $b(x)\neq 0.$

Keywords:	Subcritical nonlinear Schr\"{o}dinger equations large time asymptotics scattering problem

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