The von Mises transformation: order reduction and construction of Backlund transformations and new integrable equations

TL;DR

This paper introduces the von Mises transformation as a method for order reduction and constructing Bäcklund transformations for various nonlinear equations in mathematical physics, leading to new integrable equations and exact solutions.

Contribution

It develops a systematic approach using the von Mises transformation for order reduction and Bäcklund transformations, resulting in new integrable equations and solutions in mathematical physics.

Findings

Constructed Bäcklund transformations for general evolution equations.

Derived new integrable nonlinear equations including a generalized Calogero equation.

Applied the method to obtain exact solutions in hydrodynamics.

Abstract

Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the von Mises transformation, with the unknown function taken as the new independent variable and an appropriate partial derivative taken as the new dependent variable. RF-pairs and associated B\"acklund transformations are constructed for evolution equations of general form (special cases of which are Burgers, Korteweg--de Vries, and Harry Dym type equations as well as many other nonlinear equations of mathematical physics). The results obtained are used for order reduction and constructing exact solutions of hydrodynamics equations. A generalized Calogero equation and a number of other new integrable nonlinear equations are considered.