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

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 Bäcklund 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.