Asymptotics of solutions to the fourth order Schrodinger type equation with a dissipative nonlinearity
Asymptotics of solutions to the fourth order Schrodinger type equation with a dissipative nonlinearity
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DOI：
10.1215/kjm/1250281786
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发表时间：
2006
影响因子：
--
通讯作者：
J. Segata;A. Shimomura
中科院分区：
文献类型：
--
作者：
J. Segata;A. Shimomura
本文研究了具有立方耗散非线性项λ的一维四阶非线性Schrodinger型方程解的时间渐近性态|u| 2 u，w，其中λ是满足Im λ< 0的复常数。这种非线性是一种长程相互作用。对于给定的最终状态，在没有大小限制的情况下，求解了该方程在无穷初值的局部Cauchy问题（终值问题），这意味着当t → +∞时，在适当的函数空间中，该方程逼近某种修正的自由动力学时存在唯一解.我们修正的自由动力学随着t →∞像（tlog t）−1/2那样衰减。
In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrodinger type equation with a cubic dissipative nonlinearity λ|u| 2 u ,w hereλ is a complex constant satisfying Im λ< 0, is studied. This nonlinearity is a long-range inter- action. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suit- able function space. Our modified free dynamics decays like (tlog t) −1/2 as t →∞ .