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Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities

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  • 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.

    Mathematics Subject Classification: 65N21, 33F05.


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