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

TL;DR

This paper introduces the Crocco transformation for reducing the order of nonlinear equations in mathematical physics, constructs associated Backlund transformations, and derives new integrable equations, aiding in solving complex hydrodynamics problems.

Contribution

It develops a systematic approach using Crocco transformation for order reduction and constructs Backlund transformations for a broad class of nonlinear equations, including new integrable cases.

Findings

Derived new integrable nonlinear equations including generalized Calogero equation

Constructed Backlund transformations for evolution equations like Burgers and KdV

Applied methods to obtain exact solutions for hydrodynamics equations

Abstract

Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the Crocco transformation, with a first-order partial derivative taken as a new independent variable and a second-order partial derivative taken as the new dependent variable. Associated Backlund transformations are constructed for evolution equations of general form (special cases of which are Burgers, Korteweg-de Vries, and many other nonlinear equations of mathematical physics). The results obtained are used for order reduction and constructing exact solutions of hydrodynamics equations (Navier-Stokes, Euler, and boundary layer). A number of new integrable nonlinear equations, inclusive of the generalized Calogero equation, are considered.