论文标题
具有逐步振荡背景的聚焦NLS方程：3属群
The focusing NLS equation with step-like oscillating background: the genus 3 sector
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，其初始数据接近不同的平面波$ a_j \ mathrm {e}^{\ mathrm {\ mathrm {i} ϕ_j} \ mathrm {e}目标是根据$ξ= x/t $的值确定解决方案的长期渐近学。在[7]中分析了总体情况，我们开发了Riemann-Hilbert方法并检测到渐近分析的不同方案，具体取决于参数之间的关系$ a_1 $，$ a_2 $，$ b_1 $和$ b_2 $。特别是，在冲击情况下，$ b_1 <b_2 $，某些场景包括$ 3 $部门的属，即$ξ$的值范围，其中渐近函数的主要术语是根据三分之三的Riemann Surface $ M（ξ）$的高elliriptic函数给出的。本文致力于此类部门的完整渐近分析。
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.
