Von Mises- and Crocco-type hydrodynamical transformations: order reduction of nonlinear equations, construction of Bäcklund transformations and of new integrable equations
The von Mises transformation is a transformation in which the unknown function is used as a new independent variable and a partial derivative with respect to the coordinate directed along the normal to the streamlined surface is the new dependent variable. The Crocco transformation is a transformation in which the first and second derivatives are used as new independent and dependent variables, respectively. These transformations, which are used in hydrodynamics in the study of boundary layer equations, are employed in this paper to reduce the order of a broad class of nonlinear equations with mixed derivatives. Several examples of nonlinear equations admitting a reduction of order by using these two types of transformations are presented. Also, by using von Mises and Crocco transformations in a wide class of evolution equations, the authors construct Bäcklund transformations and use them to construct new integrable equations. Some examples of equations with three independent variables are also considered.
scientific article; zbMATH DE number 6097090