Asymptotic behavior for large time of solutions to the nonlinear nonlocal Schrödinger equation on half-line

Abstract

We study the following initial-boundary value problem for the nonlinear nonlocal Schrödinger equation

with the compatibility condition ${\partial }_x^{j-1}\overline{u}(0)={\tilde{u}}_j(0),j=1,2\dots ,n$, where $n=\left[\frac{\alpha }{2}\right],\left[s\right]$ denotes the largest integer less than $s,b\ge 0$, the nonlinear term $\text{N(}u\text{)=}ia(t)|u{|}^{\rho }u,\rho >1$, the coefficient $a(t)\in {\text{C}}^1$ and $\mathbf{K}$ is the pseudodifferential operator on the half line ${\mathbf{R}}^{+}$ of order . We prove that if $x^{\delta }\overline{u}\in {\mathbf{L}}^1$, with and the norm $\Vert \overline{u}{\Vert }_{\mathbf{X}}$ of the initial data and the norms $\Vert {\tilde{u}}_j{\Vert }_{\mathbf{Y}}$, $j=1,\dots n$ of the boundary data are sufficiently small then there exists a unique solution $u\in \mathbf{C}\text{([0,}\infty \text{);}{\mathbf{L}}^2)\cap \mathbf{C}\text{(}{\mathbf{R}}^{+};{\mathbf{W}}_2^{[\alpha ]-1}\cap {\mathbf{C}}^{[\alpha ]-1})$ of the initial-value problem (NNS). Here $\mathbf{X}={\mathbf{W}}_{\infty }^{[\alpha ]}\cap {\mathbf{W}}_1^{[\alpha ]+1}$ and $\mathbf{Y}={\mathbf{W}}_{\infty }^1\cap {\mathbf{W}}_1^2$ and ${\mathbf{W}}_p^k$ is the Sobolev space with the norm $\Vert \varphi {\Vert }_{{\mathbf{W}}_p^k}=\Vert {(1-{\partial }_x^2)}^{k/2}\varphi (x){\Vert }_{{\mathbf{L}}^p}$. We also find the large time asymptotics of the solutions.