arXiv 2011.023002020-11-04math.APmath-phmath.MP

具有阶跃边界条件的非局部非线性薛定谔方程：位移初值的长时间行为

Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data

Yan Rybalko, Dmitry Shepelsky

DOI：: 10.15407/mag16.04.418

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摘要：

本文研究了可积散焦非局部非线性薛定谔方程$ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $的初值问题的长时间渐近分析，其中$q(x,0)\to 0$为$x\to -\infty$，$q(x,0)\to A$为$x\to +\infty$ .由于该方程不是平移不变的，因此该问题的解对初始数据的移位敏感。我们考虑一个家庭的问题，参数化的$R>0$，与初始数据，可以被看作是扰动的“移位阶跃函数“$q_{R,A}(x)$：$q_{R,A}(x)=0$为$x<R$和$q_{R,A}(x)=A$为$x>R$，其中$A>0$和$R>0$是任意常数。我们发现，渐近性质是不同的部门的$(x,t)$平面，其数量取决于$A$和$R$之间的关系：对于一个固定的$A$，更大的$R$，更大的部门数量。此外，扇区可以被收集到2个交替的组中：在第一组的扇区中，解衰减到0，而在第二组的扇区中，解接近常数（随着方向$x/t=const$而变化）。

英文摘要：

The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schrödinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a step-like initial data: $q(x,0)\to 0$ as $x\to -\infty$ and $q(x,0)\to A$ as $x\to +\infty$. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by $R>0$, with the initial data that can be viewed as perturbations of the "shifted step function" $q_{R,A}(x)$: $q_{R,A}(x)=0$ for $x<R$ and $q_{R,A}(x)=A$ for $x>R$, where $A>0$ and $R>0$ are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the $(x,t)$ plane, the number of which depends on the relationship between $A$ and $R$: for a fixed $A$, the bigger $R$, the larger number of sectors. Moreover, the sectors can be collected into 2 alternate groups: in the sectors of the first group, the solution decays to 0 while in the sectors of the second group, the solution approaches a constant (varying with the direction $x/t=const$).

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