Nonlinear nonlocal Schrödinger type equations on a segment

Abstract

We study the global existence and large time asymptotic behavior of solutions to the initial-boundary value problem for the nonlinear nonlocal Schrödinger equation on a segment $\left(0,a\right)$

(0.1) $$\{\begin{array}{c}{u}_t+i|u{|}^{2}u+\mathbb{K}u=0,t>0,x\in (0,a)\\ u(x,0)=u_0(x),x\in (0,a),\end{array}$$

where the pseudodifferential operator $\mathbb{K}$ has the dissipation propery and the symbol of order $\alpha \in (0,1)$. We prove that if the initial data $u_0\in {\mathbf{L}}^{\infty }$ are small, then there exists a unique solution $u\in \mathbf{C}([0,\infty );{\mathbf{L}}^{\infty })$ of the initial-boundary value problem (0.1) Moreover there exists a function $A\in {\mathbf{L}}^{\infty }$ such that the solution has the following large time asymptotics

$$u(x,t)=A(x)t^{-\frac{1}{\alpha }}\text{Λ}\left(\frac{x}{t^{\frac{1}{\alpha }}}\right)+O\left(t^{-\frac{1+\delta }{\alpha }}\right),$$

where $\text{Λ}(x)=\frac{1}{2\text{πi}}{\displaystyle {\int }_{-i\infty }^{i\infty }{e}^{-z^{\alpha }+z\text{x}}dz}$.