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

Hayashi, Nakao; Naumkin, Pavel; Yamazaki, Yasuko

doi:10.1090/S0002-9939-01-06111-1

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations
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by Nakao Hayashi, Pavel I. Naumkin and Yasuko Yamazaki PDF

Proc. Amer. Math. Soc. 130 (2002), 779-789 Request permission

Abstract:

We consider the derivative nonlinear Schrödinger equations \begin{equation*} \left \{ \begin {split} iu_{t}+\tfrac {1}{2}&u_{xx}=a(t)F(u,u_{x}),\quad (t,x)\in {\mathbf {R}}^{2} , &u(0,x)=\epsilon u_{0}(x),\quad x\in {\mathbf {R}}, \end{split} \right . \end{equation*} where the coefficient $a\left ( t\right )$ satisfies the time growth condition \[ \left | a\left ( t\right ) \right | \leq C\left ( 1+\left | t\right | \right ) ^{1-\delta },\qquad 0<\delta <1,\] $\epsilon$ is a sufficiently small constant and the nonlinear interaction term $F$ consists of cubic nonlinearities of derivative type \begin{align*} F(u,u_{x})\ \ =\ \ &\lambda _{1}\left | u\right | ^{2}u+i\lambda _{2}\left | u\right | ^{2}u_{x}+i\lambda _{3}u^{2}\bar {u}_{x} &+\lambda _{4}\left | u_{x}\right | ^{2}u+\lambda _{5}\bar {u} u_{x}^{2}+i\lambda _{6}\left | u_{x}\right | ^{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 $b(x)\neq 0.$

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