标题： 聚焦非线性薛定谔方程半经典极限解在第一个破裂曲线处的普遍性 显示英文标题

标题： Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve

摘要： 我们考虑一维聚焦非线性薛定谔方程（NLS）在无色散极限下的半经典情况，其中势能是衰减的。 如果势能是一个快速振荡的波（周期与半经典参数epsilon的量级相当），其振幅和相位被调制，那么时空平面会被划分为具有定性不同行为的区域，它们之间的边界通常由一系列分段光滑的弧线（断裂曲线）组成。 在第一个区域中，势能的演化由调制方程（Whitham方程）控制，但对于每个空间变量x的值，存在一个转变时刻（断裂时刻），此时解表现出快速的准周期行为，即振幅在epsilon量级的尺度上也变得快速振荡。 这种转变的最初始点被称为梯度灾难点。 我们研究了在梯度灾难之后任何时间点，这两个区域之间界面左右边缘的详细渐近行为。 主要发现是，即使当epsilon趋近于零时，振幅中的第一个振荡仍具有非零的渐近大小，并且显示出两个独立的自然尺度；在(x,t)平面上断裂曲线的平行方向上为epsilon量级，在垂直方向上为epsilon ln(epsilon)量级。 这项研究基于逆散射方法和非线性最陡下降方法。 显示英文摘要

摘要： We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the semiclassical parameter epsilon) with modulated amplitude and phase, the space-time plane subdivides into regions of qualitatively different behavior, with the boundary between them consisting typically of collection of piecewise smooth arcs (breaking curve(s)). In the first region the evolution of the potential is ruled by modulation equations (Whitham equations), but for every value of the space variable x there is a moment of transition (breaking), where the solution develops fast, quasi-periodic behavior, i.e., the amplitude becomes also fastly oscillating at scales of order epsilon. The very first point of such transition is called the point of gradient catastrophe. We study the detailed asymptotic behavior of the left and right edges of the interface between these two regions at any time after the gradient catastrophe. The main finding is that the first oscillations in the amplitude are of nonzero asymptotic size even as epsilon tends to zero, and they display two separate natural scales; of order epsilon in the parallel direction to the breaking curve in the (x,t)-plane, and of order epsilon ln(epsilon) in a transversal direction. The study is based upon the inverse-scattering method and the nonlinear steepest descent method.