Conservation Laws and Exact Solutions for some Nonlinear Partial Differential Equations

An effective algorithmic method (Anco, S. C. and Bluman, G. (1996). Journal of Mathematical Physics 37, 2361; Anco, S. C. and Bluman, G. (1997). Physical Review Letters 78, 2869; Anco, S. C. and Bluman, G. (1998). European Journal of Applied Mathematics 9, 254; Anco, S. C. and Bluman, G. (2001). European Journal of Applied Mathematics 13, 547; Anco, S. C. and Bluman, G. (2002). European Journal of Applied Mathematics 13, 567 is used for finding the local conservation laws for some nonlinear partial differential equations. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that of finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. Different methods to construct new exact solution classes for the same nonlinear partial differential equations are also presented, which are named hyperbolic function method and the Bäcklund transformations. On the other hand, other methods and transformations are developed to obtain exact solutions for some nonlinear partial differential equations.