On RF-pairs, Bäcklund transformations and linearization of nonlinear equations

Some classes of nonlinear equations of mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where the unknown function is taken as a new independent variable and an appropriate partial derivative is taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form. The results obtained are used for order reduction of hydrodynamic equations (Navier–Stokes and boundary layer) and constructing exact solutions to these equations. A generalized Calogero equation and a number of other new linearizable nonlinear differential equations of the second, third and forth orders are considered. Some integro-differential equations are analyzed.

► Many nonlinear equations admit order reduction with the von Mises transformation.
► RF-pairs and Bäcklund transformations are obtained for general evolution equations.
► The order of the Navier–Stokes and boundary layer equations is reduced.
► New exact solutions are obtained for several other classes of equations.
► Integro-differential equations can be treated in a similar manner.