Generalized Laplace invariants and the method of Darboux (Q1366696)

In rough but suggestive terms, the authors transparently demonstrate the important idea that certain constructions as yet well-known only for the linear differential equations can be carried over to the nonlinear case if appropriate differential forms undertake the role of functions. In this sense, the familiar Laplace method which permits to resolve some linear differential equations \(u_{xy}+ a(x,y)u_x+ b(x,y)u_y+ c(x,y)u= 0\) can be adapted in such a manner to give the classical Darboux method for the hyperbolic equations \(F(x,y,u, u_x,u_y, u_{xx}, u_{xy}, u_{yy})= 0\). The rather involved calculations are based upon the so-called adapted Laplace coframe (the counterpart of the Laplace series of transformations in the linear subcase) which is thoroughly analyzed and illustrated by the examples of equations \(u_{xx} u_{yy}= u_y\), \(u_{xy}+ uu_{xx}= f(x,u_x)\), \(u_{xx}= f(u_{yy})\).