On order reduction of non-linear equations of mechanics and mathematical physics, new integrable equations and exact solutions

Some classes of non-linear equations of mechanics and mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where a first-order partial derivative is taken as a new independent variable and a second-order partial derivative is taken as the new dependent variable. The results obtained are used for order reduction of hydrodynamic equations (Navier–Stokes, Euler, and boundary layer) and deriving exact solutions to these equations. Associated Bäcklund transformations are constructed for evolution equations of general form (special cases include Burgers, Korteweg-de Vries, and many other non-linear equations of mathematical physics). A number of new integrable non-linear equations, inclusive of the generalized Calogero equation, are considered.

► Some classes of non-linear equations of mechanics and mathematical physics admit order reduction with the Crocco transformation.
► The order of the Navier–Stokes, Euler and boundary-layer equations is reduced with this approach.
► New exact solutions are obtained for several other classes of equations.
► Associated Bäcklund transformations are constructed for the Burgers, Korteweg-de Vries, and many other non-linear equations of mathematical physics.