基于守恒律的偏微分方程的部分线性化和完全线性化

Similarity transformations and linearization for a family of dispersionless integrable PDEs

We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this study is that from the application of the similarity transformations provided by the Lie point symmetries all the members of the family of the partial differential equations are reduced to second-order differential equations which are maximal symmetric and can be linearized.

孤立子周围可积偏微分方程线性化产生无反射散射

Emergence of Reflectionless Scattering from Linearizations of Integrable PDEs around Solitons

我们提出了四个数学物理的可积偏微分方程（PDE）的例子-当线性化周围的一个固定的孤子-表现出散射无反射在{ \it所有}的能量。从最著名的和最相关的经验现象的透明性的一维明亮玻色孤子Bogoliubov激发，我们进行到正弦戈登，Korteweg-de Vries和刘维方程的定态孤子也支持我们的断言。可积性和无反射散射之间的联系似乎跨越了至少两种不同的可积性范式：前三种情况下的S-可积性和最后一种情况下的C-可积性。我们认为，线性化可积偏微分方程的透明性是必要的，以确保他们可以支持原来的可积偏微分方程中的固定孤子的透明性。作为对比，分析进一步扩展到两个不可可积系统：锯齿Gordon和$ϕ^4$模型。

Nonlocal conservation laws of PDEs possessing differential coverings

In his 1892 paper [L. Bianchi, Sulla trasformazione di Bäcklund per le superfici pseudosferiche, Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, 3--12], L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the Bäcklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential coverings [I.S. Krasil'shchik, A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Mat…

非线性偏微分方程在弹性力学与声学中的守恒律

Conservation laws of nonlinear PDEs arising in elasticity and acoustics in Cartesian, cylindrical, and spherical geometries

本文通过尺度齐次性和乘子法，研究了弹性力学与声学中非线性偏微分方程的守恒律，包括剪切波传播模型和参数化本构方程组。

Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

Nonlinear self-adjointness method for constructing conservation laws of partial differential equations (PDEs) is further studied. We show that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and vice versa. Consequently, each symmetry of PDEs corresponds to a conservation law via a formula if the system of PDEs is nonlinearly self-adjoint with differential substitution. As a byproduct, we find that the set of differential substitutions includes the set of conservation law multipliers as a subset. The results are illustrated by three typical examples.

平坦双微分理想与半Hamilton偏微分方程

Flat bidifferential ideals and semihamiltonian PDEs

本文考虑一类具有特殊守恒定律的半Hamilton系统。这个守恒定律的密度和流满足一个二阶偏微分方程组，这个方程组在平坦双曲理想理论中有一个自然的解释。我们考虑的系统类包含重要的半哈密顿系统的著名例子。其他的例子，如KdV的1类Whitham调制方程，通过互逆变换与该类相关。