In this work, we study the (2 + 1)-dimensional Zoomeron equation which is an extension of the famous (1 + 1)-dimensional Zoomeron equation that has been studied extensively in the literature. Using classical Lie point symmetries admitted by the equation, for the first time we develop an optimal system of one-dimensional subalgebras. Based on this optimal system, we obtain symmetry reductions and new group-invariant solutions. Again for the first time, we construct the conservation laws of the underlying equation using the multiplier method.

In this paper we study a nonlinear multi-dimensional partial differential equation, namely, a generalized second extended (3+1)-dimensional Jimbo-Miwa equation. We perform symmetry reductions of this equation until it reduces to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is obtained in terms of the Weierstrass zeta function. Also travelling wave solutions are derived using the simplest equation method. Finally, the conservation laws of the underlying equation are computed by employing the conservation theorem due to Ibragimov, which include conservation of energy and conservation of momentum laws.

In this work we study a Kaup-Boussinesq system, which is used in the analysis of long waves in shallow water. Travelling wave solutions are obtained by using direct integration. Secondly, conservation laws are derived by using the multiplier method.

In this paper a (2+1)-dimensional generalized breaking soliton system, which describes the (2+1)-dimensional interaction of a Riemann wave propagating along the y−axis with a long wave along the x−axis is studied. Exact explicit solutions of this system are obtained using its three translation symmetries along with the (G^′/G)−expansion method. In addition to this we employ the multiplier method to construct conservation laws for the underlying system.

In this paper we study a nonlinear generalized improved Boussinesq equation, which describes nonlinear dispersive wave phenomena. Exact solutions are derived by using the Lie symmetry analysis and the simplest equation methods. Moreover, conservation laws are constructed by using the multiplier method.