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We consider the Schrodinger equation with nonlinear dissipation begin{equation*} i partial _t u +Delta u=lambda|u|^{alpha}u end{equation*} in ${mathbb R}^N $, $Ngeq1$, where $lambdain {mathbb C} $ with $Imlambda<0$. Assuming $frac {2} {N+2}<alpha<frac2N$, we give a precise description of the long-time behavior of the solutions (including decay rates in $L^2$ and $L^infty $, and asymptotic profile), for a class of arbitrarily large initial data.
We study the time-asymptotic behavior of solutions of the Schrodinger equation with nonlinear dissipation begin{equation*} partial _t u = i Delta u + lambda |u|^alpha u end{equation*} in ${mathbb R}^N $, $Ngeq1$, where $lambdain {mathbb C}$, $Re la
In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrodinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(mathbb{R})$, begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2sigma q^2(x,t)bar q(-x,t)=
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic regio
We prove sharp $L^infty$ decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schrodinger equation with small initial data chosen from a weighted Sobolev space. Specifically, we work with an averaged version of the d
Large deviation principle by the weak convergence approach is established for the stochastic nonlinear Schrodinger equation in one-dimension and as an application the exit problem is investigated.