On multisoliton solutions of the constant astigmatism equation

More exact solutions of the constant astigmatism equation

By using Bäcklund transformation for the sine-Gordon equation, new periodic exact solutions of the constant astigmatism equation $ z_{yy} + ({1}/{z})_{xx} + 2 = 0 $ are generated from a seed which corresponds to Lipschitz surfaces of constant astigmatism.

Some results concerning the constant astigmatism equation

In this paper we continue investigation of the constant astigmatism equation z_{yy} + (1/z)_{xx} + 2 = 0. We newly interpret its solutions as describing spherical orthogonal equiareal patterns, with relevance to two-dimensional plasticity. We show how the classical Bianchi superposition principle for the sine-Gordon equation can be extended to generate an arbitrary number of solutions of the constant astigmatism equation by algebraic manipulations. As a by-product, we show that sine-Gordon solutions give slip line fields on the sphere. Finally, we compute the solutions corresponding to classic…

A Reciprocal Transformation for the Constant Astigmatism Equation

We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter $λ= \pm1$. It is also a nonlocal symmetry.

The Constant Astigmatism Equation. New Exact Solution

In this paper we present a new solution for the Constant Astigmatism equation. This solution is parameterized by an arbitrary function of a single variable.

常像散方程的非局部守恒律

Nonlocal conservation laws of the constant astigmatism equation

该研究为常像散方程构造了在倒数变换下封闭的非局部守恒律系统，并给出了模去朗斯基型关系的函数独立势。

Lagrangian and Hamiltonian Structures for the Constant Astigmatism Equation

In this paper we found a Lagrangian representation and corresponding Hamiltonian structure for the constant astigmatism equation. Utilizing this Hamiltonian structure and extra conservation law densities we construct a first evolution commuting flow of the third order. Also, we apply the recursion operator and present a second Hamiltonian structure. This bi-Hamiltonian structure allows to replicate infinitely many local commuting flows and corresponding local conservation law densities.