Exact Solutions and Painleve Analysis of some Nonlinear Partial Differential Equations

Abstract

The integrability of nonlinear partial differential equations has been addressed in this thesis. The various approaches for complete integrability have been comprehensively exploited for important equations in mathematical physics. Along with usual the Lie symmetry analysis, a detailed discussion on various group classification techniques has been given. More general symmetries for variable coefficient coupled KdV equations have been reported. Based on Lie algebra isomorphism, an obscured technique for Lie algebra classification has been re-introduced to improve existing Lie algebra classifications. Motivated by the powerful Hirota's bilinear method, a new test function has been proposed which generalizes the existing test functions for finding exact solutions of PDEs. The Hirota's bilinear method has also been used to find Backlund transformations, Lax system, an infinite number of nontrivial conservation laws. Using direct method and new conservation theorem, nontrivial conservation laws have been constructed for some important PDEs of physical relevance.