Von mises- and crocco-type hydrodynamical transformations: Order reduction of nonlinear equations, construction of Bäcklund transformations and of new integrable equations

Abstract

Broad classes of nonlinear equations of mathematical physics are described that admit order reduction by applying the von Mises transformation (with the unknown function used as a new independent variable and with a suitable partial derivative used as a new dependent variable) and by applying the Crocco transformation (with the first and second partial derivatives used as new independent and dependent variables, respectively). Associated Backlund transformations are constructed that connect evolution equations of general form (their special cases include Burgers, Korteweg-de Vries, and Harry Dym type equations and many other nonlinear equations of mathematical physics). Transformations are indicated that reduce the order of hydrodynamic-type equations of higher orders. The generalized Calogero equation and a number of other new integrable nonlinear equations, reducible to linear equations, are considered.