论文标题
具有逐步振荡背景的聚焦NLS方程：过渡区的渐近学
The focusing NLS equation with step-like oscillating background: asymptotics in a transition zone
论文作者
论文摘要
在最近的一篇论文中，我们介绍了长期渐近学的场景，以解决焦点非线性schrödinger方程的解决方案，该方程的初始数据接近两种不同的平面波$ a_j \ m a_j \ mathrm {e}^{\ mathrm {i} ϕ_j} ϕ在负和无穷大。 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 $3$.在最近的另一篇论文中，在此类部门进行了长期的渐近分析。本文涉及两个属$ 3 $部门之间的过渡区中的渐近分析$ξ_1<ξ<ξ_0$和$ξ_0<ξ_2<ξ_2$。主术语用附加在Riemann Surface $ \ tilde {m} $ $ 1 $的椭圆形功能方面表示。派生中的一个核心步骤是在两个合并分支点附近的局部参数构建。我们通过解决与PainlevéIV方程相关的Riemann-Hilbert问题相似的模型问题来构建此参数。
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.
