Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities

Nakao Hayashi; Pavel I. Naumkin

doi:10.3934/ipi.2007.1.391

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2007, Volume 1, Issue 2: 391-398. Doi: 10.3934/ipi.2007.1.391

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Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities

Nakao Hayashi^1, and
Pavel I. Naumkin^2,

1.
Department of Mathematics, Graduate School of Science, Osaka University, Osaka Toyonaka 560-0043
2.
Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán

Received: October 31, 2006

Published: April 2007

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Abstract

We construct the modified wave operator for the nonlinear Schrödinger type equations

$u_{t}-\frac{i}{\beta }\| partial _{x} |^{\beta }u=i\lambda \ |u| ^{\rho -1}u,$

for $\( t,x ) \in \mathbf{R}\times \mathbf{R.}$ We find the solutions in the neighborhood of suitable approximate solutions provided that $\beta \geq 2$, $\Im \lambda >0$ and $\rho <3$ is sufficiently close to $3.$ Also we prove the time decay estimate of solutions

$\ ||u ( t )| |\ _{\mathbf{L}^{2}}\leq Ct^{\frac{1}{2} -\frac{1}{\rho -1}}.$

When we prove the existence of a modified scattering operator, then a natural inverse problem arises to reconstruct the parameters $\beta ,\lambda $ and $\rho $ from the modified scattering operator.

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Mathematics Subject Classification: 65N21, 33F05.

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Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities

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