论文标题
使用逐步的边界条件进行偏置非局部非线性schrödinger方程：转移的初始数据的长期行为
Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
论文作者
论文摘要
本文介绍了可通过非本地非线性schrödinger方程$的可集成散落的初始值问题的长期渐近分析。 iq_ {t}（x，t） +q_ {xx}（x，x，t）-2 q^{2}（x，x，x，t）\ bar {q}（ - x，x，t）= 0 $，带有逐步的初始数据：$ q（x，0）\ to $ q（x，0）\ to 0 $ x \ to $ x \ to-x \ to- \ fty $ x \ fty $和$ q infty和$ q（x（x）由于方程不是翻译不变的，因此该问题的解决方案对初始数据的变化敏感。我们考虑一个问题家族，参数为$ r> 0 $，其初始数据可以视为“移动步骤函数”的扰动。我们表明，在$（x，t）$平面的部门中，渐近学在质量上有所不同，其数量取决于$ a $ a $ a $ $ $：对于固定的$ a $，较大的$ r $，较大的扇区。此外，可以将扇区收集到2个替代组中：在第一组的扇区中，解决方案衰减至0，而在第二组的扇区中，解决方案接近常数（随$ x/t = const $而变化）。
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 $x/t=const$).
