论文标题
具有非对称边界条件的焦点非本地非线性schrödinger方程：大型行为
Focusing nonlocal nonlinear Schrödinger equation with asymmetric boundary conditions: large-time behavior
论文作者
论文摘要
我们考虑可聚焦的非本地非线性非线性schrödinger方程\ [\ mathrm {i} q_ {t}（x，x，x，t）+q_ {xx}（x，x，x，t）+2q^{2}（x，x，x，x，x，x，x，x，t）\ bar {q}（q}（q}（x，x，x，x，x，x，x，x，pmermmite councy councy councy councy councy councy cornedage） a \ mathrm {e}^{ - 2 \ mathrm {i} a^2t} $ as $ x \ to \ pm \ pm \ infty $，其中$ a> 0 $是任意常数。这项工作的目的是研究该方程式的初始值问题解决方案的渐近学，为$ t \ to+\ infty $。对于一类初始值，我们表明在$（x，t）$平面中存在三个定性不同的渐近区域。也就是说，在某些区域中，参数被调制（取决于比率$ x/t $）和中心区域，其中参数未经调制。这种渐进的图片让人联想到散落的经典非线性schrödinger方程，但存在一些重要差异。特别是，在所有三个区域中，解决方案的绝对值都取决于初始数据的细节。
We consider the focusing integrable nonlocal nonlinear Schrödinger equation \[\mathrm{i}q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar{q}(-x,t)=0\] with asymmetric nonzero boundary conditions: $q(x,t)\to\pm A\mathrm{e}^{-2\mathrm{i}A^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. The goal of this work is to study the asymptotics of the solution of the initial value problem for this equation as $t\to+\infty$. For a class of initial values we show that there exist three qualitatively different asymptotic zones in the $(x,t)$ plane. Namely, there are regions where the parameters are modulated (being dependent on the ratio $x/t$) and a central region, where the parameters are unmodulated. This asymptotic picture is reminiscent of that for the defocusing classical nonlinear Schrödinger equation, but with some important differences. In particular, the absolute value of the solution in all three regions depends on details of the initial data.
