阶跃振荡背景下的聚焦NLS方程：亏格3扇区
The focusing NLS equation with step-like oscillating background: the genus 3 sector
摘要：
本文考虑聚焦非线性薛定谔方程的Cauchy问题，其初值逼近不同的平面波$A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$，$j=1,2$为$x\to\pm\infty$ .目标是根据$ξ=x/t$的值确定解的长时间渐近性。一般情况在[7]中进行了分析，其中我们开发了Riemann-Hilbert方法，并根据参数$A_1$、$A_2$、$B_1$和$B_2$之间的关系检测了渐近分析的不同场景。具体地，在电击情况下$B_1<B_2$，一些场景包括属$3$扇区，即，$ξ$的值的范围，其中渐近性的首项以附接到亏格为三的黎曼曲面$M(ξ)$的超椭圆函数的形式给出。本文致力于在这样一个部门的完整的渐近分析。
英文摘要：
We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves $A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. The goal is to determine the long-time asymptotics of the solution, according to the value of $ξ=x/t$. The general situation is analyzed in [7] where we develop the Riemann-Hilbert approach and detect different scenarios of asymptotic analysis, depending on the relationships between the parameters $A_1$, $A_2$, $B_1$, and $B_2$. In particular, in the shock case $B_1<B_2$, some scenarios include genus $3$ sectors, i.e., ranges of values of $ξ$ where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface $M(ξ)$ of genus three. The present paper is devoted to the complete asymptotic analysis in such a sector.
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