具有阶跃振荡背景的聚焦NLS方程：过渡区的渐近性
The focusing NLS equation with step-like oscillating background: asymptotics in a transition zone
摘要：
在最近的一篇论文中，我们提出了聚焦非线性薛定谔方程的解的长时间渐近性的方案，其初始数据接近两个不同的平面波$A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$，$j=1,2$在负无穷大和正无穷大。在冲击情形$B_1<B_2$中，某些场景包括$3$亏格的扇区，即扇区$ξ_1<ξ<ξ_2$、$ξ:=\frac{x}{t}$，其中渐近的首项用附在$3$亏格的黎曼曲面$M(ξ)$上的超椭圆函数表示。在这样一个部门的长期渐近分析是在另一个最近的文件。本文讨论了两个亏格$3$扇区$ξ_1<ξ<ξ_0$和$ξ_0<ξ<ξ_2$之间过渡区的渐近分析.首项用附着在亏格为$1$的黎曼曲面$\tilde{M}$上的椭圆函数表示。在推导中的一个中心步骤是在两个合并分支点的邻域中构造局部参数。我们通过解决一个类似于与Painlevé IV方程相关的Riemann-Hilbert问题的模型问题来构造这个参数集。
英文摘要：
In a recent paper, we presented scenarios of long-time asymptotics for a solution of the focusing nonlinear Schrödinger equation whose initial data approach two different plane waves $A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ at minus and plus infinity. In the shock case $B_1<B_2$ some scenarios include sectors of genus $3$, that is sectors $ξ_1<ξ<ξ_2$, $ξ:=\frac{x}{t}$ where the leading term of the asymptotics is expressed in terms of hyperelliptic functions attached to a Riemann surface $M(ξ)$ of genus $3$. The long-time asymptotic analysis in such a sector is performed in another recent paper. The present paper deals with the asymptotic analysis in a transition zone between two genus $3$ sectors $ξ_1<ξ<ξ_0$ and $ξ_0<ξ<ξ_2$. The leading term is expressed in terms of elliptic functions attached to a Riemann surface $\tilde{M}$ of genus $1$. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points. We construct this parametrix by solving a model problem which is similar to the Riemann-Hilbert problem associated with the Painlevé IV equation.
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