Kyrylo Andreiev, Iryna Egorova, Till Luc Lange, Gerald Teschl
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.
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We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice with steplike initial data corresponding to a rarefaction wave.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data.
We present a method to compute dispersive shock wave solutions of the Korteweg-de Vries equation that emerge from initial data with step-like boundary conditions at infinity. We derive two different Riemann-Hilbert problems associated with the inverse scattering transform for the classical Schrödinger operator with possibly discontinuous, step-like potentials and develop relevant theory to ensure unique solvability of these problems. We then numerically implement the Deift-Zhou method of nonlinear steepest descent to compute the solution of the Cauchy problem for small times and in two asympto…
Soliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamics
本文研究 KdV 方程中孤子与稀疏波及色散冲击波的相互作用，综合微扰理论、Whitham 调制和逆散射变换解释透射或陷俘机制。
Interactions of dispersive shock (DSWs) and rarefaction waves (RWs) associated with the Korteweg-de Vries equation are shown to exhibit multiphase dynamics and isolated solitons. There are six canonical cases: one is the interaction of two DSWs which exhibit a transient two-phase solution, but evolve to a single phase DSW for large time; two tend to a DSW with either a small amplitude wave train or a finite number of solitons, which can be determined analytically; two tend to a RW with either a small wave train or a finite number of solitons; finally, one tends to a pure RW.
Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation
本文证明KdV方程稠密孤子气体在凝聚极限下的动力学方程约化为$N$相KdV-Whitham调制方程，并构造Riemann问题显式解及数值揭示稀释凝聚体的可积湍流行为。
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg--de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the $x/t$ plane splits into $g+1$ soliton regions which are interlaced by $g+1$ oscillatory regions, where $g+1$ is the number of spectral gaps. In the soliton regions the solution is asymptotically given by a number of solitons travelling on top of finite-gap solutions which are in the same isospectral class as the backg…
We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of shock waves for the Korteweg-de Vries (KdV) equation in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as the region where they are valid.
A modified Korteweg-de Vries equation soliton gas under the nonzero background
本文研究非零背景下聚焦mKdV方程的孤子气体，通过建立Riemann-Hilbert问题并利用Deift-Zhou非线性最速下降法，分析了其在不同空间方向及大时间尺度下划分为三个区域的渐近行为。
The defocusing nonlinear Schrödinger (NLS) equation is studied for a family of step-like initial data with piecewise constant amplitude and phase velocity with a single jump discontinuity at the origin. Riemann-Hilbert and steepest descent techniques are used to study the long time/zero-dispersion limit of the solution to NLS associated to this family of initial data. We show that the initial discontinuity is regularized in the long time/zero-dispersion limit by the emergence of five distinct regions in the $(x, t)$ half-plane. These are left, right, and central plane waves separated by a rare…
Numerical inverse scattering transform for the coupled modified Korteweg-de Vries equation
本文提出基于关联Riemann-Hilbert问题的耦合修正KdV方程数值反散射变换框架，通过Chebyshev摄动法求解散射数据，并利用Deift-Zhou方法变形振荡问题，实现无需时间步进的直接解算，展示了其在长时间模拟中的有效性。
In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation (nonlocal mKdV) \[q_t(x,t)+q_{xxx}(x,t)-6q(x,t)q(-x,-t)q_x(x,t)=0,\] which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal mKdV equation. In contrast with the classical mKdV equation, we find some new and di…
In this paper, we consider Cauchy problem for the modified Korteweg-de Vries hierarchy on the real line with decaying initial data. Using the Riemann--Hilbert formulation and nonlinear steepest descent method, we derive a uniform asymptotic expansion to all orders in powers of $t^{-1/(2n+1)}$ with smooth coefficients of the variable $(-1)^{n+1}x((2n+1) t)^{-1/(2n+1)}$ in the self-similarity region for the solution of $n$-th member of the hierarchy. It turns out that the leading asymptotics is described by a family of special solutions of the Painlevé II hierarchy, which generalize the classica…
We present a new generalization of the steepest descent method introduced by Deift and Zhou for matrix Riemann-Hilbert problems and use it to study the semiclassical limit of the focusing nonlinear Schroedinger equation with real analytic, even, bell-shaped initial data. We provide explicit strong locally uniform asymptotics for a sequence of exact solutions corresponding to initial data that has been modified in an asymptotically small sense. We call this sequence of exact solutions a semiclassical soliton ensemble. Our asymptotics are valid in regions of the (x,t) plane where a certain scala…
The Baker-Akhiezer (BA) function theory was successfully developed in the mid 1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and theory of completely integrable nonlinear equations such as Korteweg-de Vries equation, nonlinear Schrödinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation. Subsequently the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch (MB) system or for the Karpman-Kaup equations, which conta…
On zero-background solitons of the sharp-line Maxwell-Bloch equations
本文系统研究零背景锐线Maxwell-Bloch方程的一般N-孤子解，提出N-DSG显式公式，证明其局域性并计算长时间渐近，推广至其他可积系统。
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