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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Nakao Hayashi, Pavel I. Naumkin and Yasuko Yamazaki
Journal: Proc. Amer. Math. Soc. 130 (2002), 779-789
MSC (2000): Primary 35Q55
Published electronically: August 29, 2001
MathSciNet review: 1866034
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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 $F$ 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 $b(x)\neq 0.$


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Additional Information

Nakao Hayashi
Affiliation: Department of Applied Mathematics, Science University of Tokyo, Tokyo 162-8601, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: nhayashi@rs.kagu.sut.ac.jp, nhayashi@math.wani.osaka-u.ac.jp

Pavel I. Naumkin
Affiliation: Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacán, México
Email: pavelni@zeus.ccu.umich.mx

Yasuko Yamazaki
Affiliation: Department of Applied Mathematics, Science University of Tokyo, Tokyo 162-8601, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Hokkaido University, Sapporo 060, Japan
Email: yamazaki@math.sci.hokudai.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06111-1
PII: S 0002-9939(01)06111-1
Keywords: Subcritical nonlinear Schr\"{o}dinger equations, large time asymptotics, scattering problem
Received by editor(s): May 22, 2000
Received by editor(s) in revised form: September 15, 2000
Published electronically: August 29, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society