Abdullahi Rashid Adem

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.

The generalized ( 1 + 1 )-dimensional and ( 2 + 1 )-dimensional Ito equations

The multiple exp-function algorithm, as a generalization of Hirotas perturbation scheme, is used to construct multiple wave solutions to the generalized ( 1 + 1 )-dimensional and ( 2 + 1 )-dimensional Ito equations. Some of the resulting solutions involve generic phase shifts and wave frequencies.

Solutions of Zakharov-Kuznetsov equation with power law nonlinearity in (1+3) dimensions

This paper studies the Zakharov-Kuznetsov equation in (1+3) dimensions with an arbitrary power law nonlinearity. The method of Lie symmetry analysis is used to carry out the integration of the Zakharov-Kuznetsov equation. The solutions obtained are cnoidal waves, periodic solutions, singular periodic solutions, and solitary wave solutions. Subsequently, the extended tanh-function method and the G′/G method are used to integrate the Zakharov-Kuznetsov equation. Finally, the nontopological soliton solution is obtained by the aid of ansatz method. There are numerical simulations throughout the paper to support the analytical development.

On the solutions and conservation laws of a coupled KdV system

Abstract In this paper Lie symmetry analysis is performed on the coupled KdV system which describes a resonant interaction of two wave modes in a shallow stratified liquid. The similarity reductions and exact solutions with the aid of simplest equation method and Jacobi elliptic function method are obtained based on the optimal systems of one-dimensional subalgebras for the KdV system. In addition, the conservation laws of the coupled KdV system are also derived using the multiplier (and homotopy) approach and a conservation theorem.

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.

Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method

Abstract A coupled Kadomtsev–Petviashvili equation is investigated by using Lie symmetry analysis. The similarity reductions and new exact solutions are obtained via the extended tanh method with symbolic computation. Exact solutions including solitons are shown. The solutions derived have dissimilar physical structures and depend on the real parameters.

Symbolic computation of conservation laws and exact solutions of a coupled variable-coefficient modified Korteweg–de Vries system

In this paper we study a generalized coupled variable-coefficient modified Korteweg–de Vries (CVCmKdV) system that models a two-layer fluid, which is applied to investigate the atmospheric and oceanic phenomena such as the atmospheric blockings, interactions between the atmosphere and ocean, oceanic circulations and hurricanes. The conservation laws of the CVCmKdV system are derived using the multiplier approach and a new conservation theorem. In addition to this, a similarity reduction and exact solutions with the aid of symbolic computation are computed.

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.

Exact Solutions and Conservation Laws of a Two-Dimensional Integrable Generalization of the Kaup-Kupershmidt Equation

We study a two-dimensional integrable generalization of the Kaup-Kupershmidt equation, which arises in various problems in mathematical physics. Exact solutions are obtained using the Lie symmetry method in conjunction with the extended tanh method and the extended Jacobi elliptic function method. In addition to exact solutions we also present conservation laws which are derived using the multiplier approach.

Exact Solutions and Conservation Laws of the Joseph-Egri Equation with Power Law Nonlinearity

In this chapter we obtain exact solutions of the Joseph-Egri equation with power law nonlinearity, which arises in various problems in many scientific applications. The Lie group analysis and simplest equation method are used to carry out the integration of this equation. The solutions obtained are travelling wave solutions. Moreover, the conservation laws for the Joseph-Egri equation with power law nonlinearity are constructed by using the multiplier method.