Exact solutions and conservation laws of a generalized ITO -type coupled KdV system, boussinesq system of KdV-KdV type and coupled KdV system

Abstract

In this dissertation, three systems of nonlinear partial differential equations, namely, the generalized Ito-type coupled KdV system, Boussinesq system of KdV-KdV type equation and coupled KdV system are studied. The exact solutions of the Boussinesq system of KdV-KdV type equations and the coupled KdV system are obtained. The Lie symmetry approach along with the simplest equation method is used to obtain these solutions. Moreover, conservation laws for all the three systems are derived using the Noether's theorem. The three systems are third-order systems of PDEs and do not have Lagrangian. The transformation u -+ Ux, v -+ Vx are used to convert the above mentioned systems to fourth-order systems in U, V variables, which have Lagrangians. Noether's approach is then used to derive the conservation laws. Finally, the conservation laws are expressed in the variables u, v and they constitute the conservation laws for the third-order systems. Infinitely many non-local conserved quantities are obtained for these three systems.