Abstract formulation of the Miura transform

关于演化方程的连续类

On continual classes of evolution equations

本文指出Miura变换可应用于连续演化方程类，并揭示其与规范不变的零曲率表示的联系。

On the r-th dispersionless Toda hierarchy I: Factorization problem, symmetries and some solutions

For a family of Poisson algebras, parametrized by by an integer number r, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed f…

通过深度神经网络学习方案进行 Bäcklund 变换与孤子演化方程的数据驱动发现

Data-driven discoveries of Bäcklund transforms and soliton evolution equations via deep neural network learning schemes

提出两种深度学习方案，分别用于数据驱动学习 Bäcklund 与 Miura 变换，并借助高阶孤子信息提升 mKdV 和 sine-Gordon 方程发现精度。

KdV Preserves White Noise

It is shown that white noise is an invariant measure for the Korteweg-deVries equation on $\mathbb T$. This is a consequence of recent results of Kappeler and Topalov establishing the well-posedness of the equation on appropriate negative Sobolev spaces, together with a result of Cambronero and McKean that white noise is the image under the Miura transform (Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation, proven to be invariant for that equation by Bourgain.

Solutions to the complex Korteweg-de Vries equation: Blow-up solutions and non-singular solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, including blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Korteweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the complex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.

Abstract formulation of the Cole-Hopf transform

Operator representation of Cole-Hopf transform is obtained based on the logarithmic representation of infinitesimal generators. For this purpose the relativistic formulation of abstract evolution equation is introduced. Even independent of the spatial dimension, the Cole-Hopf transform is generalized to a transform between linear and nonlinear equations defined in Banach spaces. In conclusion a role of transform between the evolution operator and its infinitesimal generator is understood in the context of generating nonlinear semigroup.

A Paley-Wiener Theorem for Periodic Scattering with Applications to the Korteweg-de Vries Equation

Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg-de Vries equation is also obtained.

关于 $\mathbb R^2$ 上 KP-II 方程的 Bäcklund 变换与线孤子的稳定性

On the Bäcklund transform and the stability of the line soliton of the KP-II equation on $\mathbb R^2$

本文通过 Miura 映射构造 Bäcklund 变换，研究 $\mathbb R^2$ 上 KP-II 方程线孤子的 $L^2$ 稳定性，证明余维数 1 流形上的尖锐正则性稳定性，并推广到多孤子加法映射。

The Korteweg-de-Vries equation at H^{-1} regularity

In this paper we will prove the existence of weak solutions to the Korteweg-de Vries initial value problem on the real line with H^{-1} initial data; moreover, we will study the problem of orbital and asymptotic H^{s} stability of solitons for integers s\ge -1; finally, we will also prove new a priori H^{-1} bounds for solutions to the Korteweg-de Vries equation. The paper will utilise the Miura transformation to link the Korteweg-de Vries equation to the modified Korteweg-de Vries equation.