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## Mini-Course——Asymptotic behavior of solutions to nonlinear Schr{o}dinger equations

2019-09-24 15:00-17:00 Room 1304, Sciences Building No. 1

Abstract: In a series of talks, we consider the one dimensional nonlinear Schr\"{o}dinger equations with a power nonlniearity of order $p$.

1. Nonlinear Schr\"{o}dinger equation with a super critical nonlinearity （I）
We consider asymptotic behavior of solutions to the linear problem and using results of the linear problem, we present a global existence of solutions to the nonlinear problem with a super critical
nonlinearity $p>3$.

2. Nonlinear Schr\"{o}dinger equation with a super critical nonlinearity （II）
Scattering problem is discussed. More precisely, we show  global in time of solutions to nonlinear problem with a super critical nonlinearity exist in the neighborhood of free solutions which implies sacttering operators are well defined in small ball of some Banach space with the center at the origin.

3.  Nonexistence of scattering states in the subcritical cases
We call the problem a critical case in the sense of scattering problem when $p=3$. Nonexistence of scattering states is proved when $1<p\leq 3,$ namely, it is impossible to find a solution in the neighborhood of the free solutions. This fact implies that we need some modification to the free solutions to get asymptotic behavior of solutions to nonlinear problem when $1<p\leq 3.$

4. Modified scattering operator in a critical case
We consider the problem when $p=3,$ and show existence of solutions to the nonlinear problem in the neighborhood of an approximate solution defined by a given final function. Moreover, we consider the Cauchy problem for the problem with a critical case $p=3$ and show asymptotic behavior of solutions. By these results, we find that the modified scattering operator is defined.