论文标题
长期渐近楔形物的弯曲楔块，用于可集成的非局部非线性schrödinger方程
Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
论文作者
论文摘要
我们考虑可集成的非本地非线性schrödinger（nnls）方程的库奇问题$类似阶梯的边界值：$ q（x，t）\ to 0 $ as $ x \ to- \ to- \ to-\ infty $和$ q（x，t）\ to a $ as a as $ x \ to $ x \ to \ infty $ for All $ t \ geq0 $，其中$ a> a> 0 $是常数。在\ cite {rs2}中显示了解决方案$ q（x，t）$的长期渐近学$ q（x，t）$。在本文中，我们将渐近学扩展到一个与任何非零常数$ c $的射线更接近射线$ x = 0 $的区域。我们在$ x，t $平面中指定了一个单参数楔形，具有弯曲的边界，其特征在于$ q（x，t）$的质量不同的渐近行为，并给出了每个楔形的主要渐近术语。特别是，对于$ x <0 $的楔子，我们表明该解决方案衰减为$ t^{p} \ sqrt {\ ln t} $，带有$ p <0 $，具体取决于楔子。对于$ x> 0 $的楔子，我们表明渐近学具有振荡性质，相位函数针对每个楔形物具有特定的相位功能，并且取决于缓慢的可变参数参数。这项工作中使用的主要工具是对非线性陡峭的体面方法的适应，当相关的riemann-hilbert问题的跳跃中，相位函数的固定相位点与相应光谱函数是单数的点合并时。
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \, x\in\mathbb{R},\,t>0,$ with a step-like boundary values: $q(x,t)\to 0$ as $x\to-\infty$ and $q(x,t)\to A$ as $x\to\infty$ for all $t\geq0$, where $A>0$ is a constant. The long-time asymptotics of the solution $q(x,t)$ of this problem along the rays $x/t=C\ne 0$ is presented in \cite{RS2}. In the present paper, we extend the asymptotics into a region that is asymptotically closer to the ray $x=0$ than these rays with any nonzero constant $C$. We specify a one-parameter family of wedges in the $x,t$-plane, with curved boundaries, characterized by qualitatively different asymptotic behavior of $q(x,t)$, and present the main asymptotic terms for each wedge. Particularly, for wedges with $x<0$, we show that the solution decays as $t^{p}\sqrt{\ln t}$ with $p<0$ depending on the wedge. For wedges with $x>0$, we show that the asymptotics has an oscillating nature, with the phase functions specific for each wedge and depending on a slow variable parametrizing the wedges. The main tool used in this work is an adaptation of the nonlinear steepest decent method to the case when the stationary phase point of the phase function in the jump of the associated Riemann-Hilbert problem merges with a point which is singular for the corresponding spectral functions.
