Long-time asymptotics for Toda shock waves in the modulation region

How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave

We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this solution. We also study the effect of possible resonances at the endpoints of the gap on this phase. This paper is a continuation of research started in [arXiv:2001.05184].

Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra

We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regi…

Toda晶格长波极限的数值研究

Numerical study of the long wavelength limit of the Toda lattice

我们提出了第一个详细的数值研究户田方程的$2+1$尺寸在长波长的限制，双曲和椭圆的情况下。我们首先研究Toda方程的形式无色散极限，并解决由此产生的系统的梯度突变点的初值问题。结果表明，双曲型方程解的破裂类似于Hopf方程中激波的形成，是一个$1+1$维奇异性.在椭圆情形下，发现对于$1+1$维的聚焦非线性薛定谔方程的半经典系统，分裂是由尖点给出的。完整的Toda系统，然后研究有限的小值的色散参数$ε$在附近的无色散Toda方程的冲击。我们确定了小$ε$的Toda解和无色散Toda系统的奇异解之间的差异在$ε$中的标度。在双曲线的情况下，相同的缩放比例$ε^{2/7}$被发现在小色散极限的Korteweg-de Vries和散焦非线性薛定谔方程。在椭圆情形下，我们得到了与聚焦非线性薛定谔方程的半经典极限相同的与$ε^{2/5}$成比例的标度。我们还研究了在双曲情形下，比破裂时间大得多的频散激波的形成。在椭圆情况下，观察到的是$L_{\infty}$爆破，而不是大于破裂时间的有限倍的色散激波。爆破时间的$ε$依赖性被确定。

阶跃初始数据下Toda晶格的波动现象

Wave phenomena of the Toda lattice with steplike initial data

我们给出了一个调查的长时间渐近的Toda格与阶跃常数的初始数据使用非线性最速下降分析及其扩展的基础上适当选择的$g$ -功能。给出了Toda激波和稀疏问题（包括背景谱重叠的情况）渐近解首项的解析公式，并通过数值模拟加以补充。我们提供了一个明确的公式调制的解决方案的阿贝尔积分的基础超椭圆黎曼曲面。

Trigonometric shock waves for the Kaup-Boussinesq system

We consider the modulationally stable version of the Kaup-Boussinesq system which models propagation of nonlinear waves in various physical systems. It is shown that the Whitham modulation equations for this model have a new type of solutions which describe trigonometric shock waves. In the Gurevich-Pitaevskii problem of evolution of an initial discontinuity, these solutions correspond to a non-zero wave excitation on one of the sides of the discontinuity. Our analytical results are confirmed by numerical calculations.

具有盒型初始数据的 KdV 方程中孤子-平均流的调制理论

Modulation theory of soliton-mean flow in KdV equation with box type initial data

针对盒型初值的 KdV 方程，通过退化双亏格 Whitham 调制系统推导孤子调制系统，理论预测并数值验证了孤子与色散平均流相互作用时的轨迹、振幅及相移变化。