The decomposition method and some integrable families of nonlinear PDEs

Abstract:
We propose a new (partly algorithmic) method for finding closed form solutions (general or particular) of some types of nonlinear PDEs of any order (n>1) with any number of independent variables. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. We demonstrate on examples that there exist extensive nontrivial families for different types of decomposable nonlinear PDEs. It is shown that for such PDEs the decomposition process can be done by iterative procedure, each step of which is reduced to solution of some auxiliary second-order PDEs system(s) for one dependent variable. Moreover, we find on this way the explicit expression of the first-order PDEs system(s) for first integral of decomposable PDE, so initial PDE is decomposable iff the corresponding auxiliary first-order PDEs system for first integral is consistent. Remarkably that this first-order PDEs system(s) is linear if initial PDE is linear in its highes! t derivatives. From practical standpoint the first integral method of attack is more preferred as it leads to more simple auxiliary problem and in many cases the first-order PDEs system can be solved automatically in present CAS (see Maple procedure prototype for finding first integrals at http://arxiv.org/abs/0704.0072v1). Examples of nonlinear PDEs of order 2-4 with calculated their general solutions demonstrate a potential of the method.