The decomposition method and Maple procedure for finding first integrals of nonlinear PDEs.pdf

a r X i v : 0 7 0 4 .0 0 7 2 v 1 [ m a t h - p h ] 1 A p r 2 0 0 7 The decomposition method and Maple procedure for finding first integrals of nonlinear PDEs of any order with any number of independent variables Yu. N. Kosovtsov Lviv Radio Engineering Research Institute, Ukraine email: kosovtsov@ Abstract In present paper we propose seemingly new method for finding solu- tions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that decomposition process can be done by iterative procedure(s), each step of which is reduced to solution of some auxiliary PDEs system(s) for one dependent variable. Moreover, we find on this way the explicit expression of the first-order PDE(s) for first inte- gral of decomposable initial PDE. Remarkably that this first-order PDE is linear if initial PDE is linear in its highest derivatives. The developed method is implemented in Maple procedure, which can really solve many of different order PDEs with different number of in- dependent variables. Examples of PDEs with calculated their general solutions demonstrate a potential of the method for automatic solving of nonlinear PDEs. 1 Introduction Nonlinear partial differential equations (PDEs) play very important role in many fields of mathematics, physics, chemistry, and biology, and numerous applica- tions. If for nonlinear ordinary differential equations (ODEs) one can observe incontestable progress in their automatic solving, the situation for nonlinear PDEs seems as nearly hopeless one. Despite the fact that various methods for solving nonlinear PDEs have been developed in 19-20 centuries as the suitable groups of transformations, such as point or contact transformations, differential substitutions, and Backlund trans- formations etc., the most powerful method for explicit integration of second- order nonlinear PDEs in two independent variables remains the method of Dar