Variational approach for a coupled Zakharov-Kuznetsov System and the (2+1)-dimensional breaking soliton equation

Abstract

In this dissertation the generalized coupled Zakharov-Kuznetsov system and the (2+ 1 )-dimensional breaking soliton equation will be studied. Exact solutions for the coupled Zakharov-Kuznetsov equations are obtained using the Kudryashov method and the Jacobi elliptic function method while the exact solutions for the (2+1)dimensional breaking soliton equation are derived using the double reduction theory. Furthermore, N oether theorem is employed to construct conservation laws for the above mentioned partial differential equations. Since the coupled system is of thirdorder, it does not have a Lagrangian. Therefore, we use the transformations u = Ux and v = Vx to increase a third-order system to a fourth-order coupled system in U and V variables and let a = l. Thus, the new system of equations have a Lagrangian. However the (2+1)-dimensional breaking soliton equation has a Lagrangian in its natural form. Finally, the conservation laws are expressed in 'U and v variables for the generalized coupled Zakharov-Kuznetsov system. Some local and infinitely many nonlocal conserved quantities are found and the Kudryashov method and Jacobi elliptic function method are used to obtain the exact solutions for the coupled Zakharov-Kuznetsov system. The (2+1)-dimensional breaking soliton equation possesses only local conserved quantities and the double reduction theory is applied to obtain some exact solutions.