关于封闭式KdV敷料链的整合

Dressing the Dressing Chain

The dressing chain is derived by applying Darboux transformations to the spectral problem of the Korteweg-de Vries (KdV) equation. It is also an auto-Bäcklund transformation for the modified KdV equation. We show that by applying Darboux transformations to the spectral problem of the dressing chain one obtains the lattice KdV equation as the dressing chain of the dressing chain and, that the lattice KdV equation also arises as an auto-Bäcklund transformation for a modified dressing chain. In analogy to the results obtained for the dressing chain (Veselov and Shabat proved complete integrabilit…

Darboux dressing and undressing for the ultradiscrete KdV equation

We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We then show how to reconstruct the potential in the scattering problem at any time, using an ultradiscrete analogue of a Darboux transformation. This is achieved by obtaining data uniquely characterising the soliton content and the `background' from the initial potential by Darboux transformation.

Dressing for a vector modified KdV hierarchy

We present the hierarchy and soliton solutions associated to a multi-component generalisation of the modified Korteweg-de Vries equation. A recursive formula for obtaining the Lax operators associated to the higher flows of the hierarchy is provided. Using the method of rational dressing and the symmetries of the Lax operators we obtain the one-soliton solution. We also derive the general rank one-breather solution and express it in terms of determinants. We present the simplest case of the one-breather solution, which is parametrised by two constant unit vectors normal to each other. Finally,…

一个用于KdV方程解非线性叠加的数值方法

A numerical dressing method for the nonlinear superposition of solutions of the KdV equation

本文统一了两种现有数值方法，用于计算Korteweg-de Vries方程的解，通过组合方法得到精确的渐近(准)周期有限隙解，属于非线性叠加解。

Discrete Hirota reductions associated with the lattice KdV equation

We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to waves moving with rational speed N/M on the lattice, where N,M are coprime integers, we prove the Liouville integrability of the maps when N + M is odd, and prove various properties of the general case. There are two main ingredients to our construction: the cluster algebra associated with each of the Hirota bilinear equations, which provides invariant (pre)sym…

格点势KdV方程的连续对称性

Continuous Symmetries of the Lattice Potential KdV Equation

本文给出了格点势Korteweg-de Vries（lpKdV）方程的积分和对称性的一组结果。利用其相关的谱问题，我们构造的孤子解决方案和拉克斯技术使我们能够提供无限序列的广义对称性。最后，利用lpKdV方程的离散对称性，我们构造了一大类非自治广义对称。