论文标题
具有逐步振荡背景的聚焦NLS方程：长期渐近学的场景
The focusing NLS equation with step-like oscillating background: scenarios of long-time asymptotics
论文作者
论文摘要
我们考虑了焦点非线性schrödinger方程的库奇问题，初始数据接近两个不同的平面波$ a_j \ mathrm {e}^{\ mathrm {\ mathrm {i} ϕ_j} \ mathrm {e} $ x \ to \ pm \ infty $。使用Riemann-Hilbert技术和Deift-Zhou最陡峭的下降论证，我们研究了溶液的长期渐近学。我们发现每种情况$ b_1 <b_2 $，$ b_1> b_2 $，$ b_1 = b_1 = b_2 $都应进行单独的分析。我们主要关注第一种情况，即所谓的冲击案例，我们表明存在广泛的渐近场景。我们还提出了一种严格确定某些高级渐近部门存在的方法。
We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching two different plane waves $A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. Using Riemann-Hilbert techniques and Deift-Zhou steepest descent arguments, we study the long-time asymptotics of the solution. We detect that each of the cases $B_1<B_2$, $B_1>B_2$, and $B_1=B_2$ deserves a separate analysis. Focusing mainly on the first case, the so-called shock case, we show that there is a wide range of possible asymptotic scenarios. We also propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors.
