Ben Muatjetjeja

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Abstract We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y) = 0 with respect to the standard Lagrangian L = xny′2/2 — xn ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αyr , where r ≠ 0,1. The case r = 5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r = 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.

Conservation laws and exact solutions for a 2D Zakharov–Kuznetsov equation

Abstract This paper aims to compute conservation laws for the 2D Zakharov–Kuznetsov equation using Noether’s approach through an interesting method of increasing the order of the 2D Zakharov–Kuznetsov equation. Moreover, exact solutions for the 2D Zakharov–Kuznetsov equation are obtained using the Kudryashov method and Jacobi elliptic function method.

Conservation Laws for a Variable Coefficient Variant Boussinesq System

We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.

Variational approach and exact solutions for a generalized coupled Zakharov–Kuznetsov system

Abstract In the present paper, we obtain a variational principle for a generalized coupled Zakharov–Kuznetsov system, which does not admit any Lagrangian formulation in its present form. The eminent Noether‘s theorem will then be employed to compensate for this approach. In addition, exact solutions will be constructed for the generalized coupled Zakharov–Kuznetsov system using the Kudryashov method and the Jacobi elliptic function method.

Group analysis of a hyperbolic Lane-Emden system

In this paper we carry out a complete Noether and Lie group classification of the radial form of a coupled system of hyperbolic equations. From the Noether symmetries we establish the corresponding conserved vectors. We also determine constraints that the nonlinearities should satisfy in order for the scaling symmetries to be Noetherian. This led us to a critical hyperbola for the systems under consideration. An explicit solution is also obtained for a particular choice of the parameters.

Rosenau-KdV Equation Coupling with the Rosenau-RLW Equation: Conservation Laws and Exact Solutions

Abstract We compute the conservation laws for the Rosenau-Kortweg de Vries equation coupling with the Regularized Long-Wave equation using Noether’s approach through a remarkable method of increasing the order of the Rosenau-KdV-RLW equation. Furthermore, exact solutions for the Rosenau- KdV-RLW equation are acquired by employing the Kudryashov method.

Benjamin-Bona-Mahony Equation with Variable Coefficients: Conservation Laws

This paper aims to construct conservation laws for a Benjamin–Bona–Mahony equation with variable coefficients, which is a third-order partial differential equation. This equation does not have a Lagrangian and so we transform it to a fourth-order partial differential equation, which has a Lagrangian. The Noether approach is then employed to construct the conservation laws. It so happens that the derived conserved quantities fail to satisfy the divergence criterion and so one needs to make adjustments to the derived conserved quantities in order to satisfy the divergence condition. The conservation laws are then expressed in the original variable. Finally, a conservation law is used to obtain exact solution of a special case of the Benjamin–Bona–Mahony equation.

A variational formulation approach to a generalized coulped inhomogenous emden–flowler system

Abstract We study a generalized coupled inhomogeneous Emden–Fowler system from the Lagrangian formulation standpoint. A special case of this system was considered in the literature, and necessary and sufficient conditions for the existence of multiple positive solutions were obtained. Here we perform preliminary Noether classification of the generalized system by the direct method. We obtain nine cases for which the system has Noether point symmetries. First integrals are then obtained for the cases which admit Noether point symmetries.

Group Classification of a Generalized Lane-Emden System

We perform the group classification of the generalized Lane-Emden system , which occurs in many applications of physical phenomena such as pattern formation, population evolution, and chemical reactions. We obtain four cases depending on the values of n.

Conservation laws for a generalized Ito-type coupled KdV system

In this paper, the conservation laws for a generalized Ito-type coupled Korteweg-de Vries (KdV) system are constructed by increasing the order of the partial differential equations. The generalized Ito-type coupled KdV system is a third-order system of two partial differential equations and does not have a Lagrangian. The transformation u=Ux, v=Vx converts the generalized Ito-type coupled KdV system into a system of fourth-order partial differential equations in U and V variables, which has a Lagrangian. Noether’s approach is then used to construct the conservation laws. Finally, the conservation laws are expressed in the original variables u and v. Some local and infinitely many nonlocal conserved quantities are found for the generalized Ito-typed coupled KdV system.