1. Introduction
Many physical phenomena of the real world are governed by nonlinear partial differential equations (NLPDEs). It is therefore absolutely necessary to analyse these equations from the point of view of their integrability and finding exact closed form solutions. Although this is not an easy task, many researchers have developed various methods to find exact solutions of NLPDEs. These methods include the sine-cosine method [
1], the extended tanh method [
2], the inverse scattering transform method [
3], the Hirota’s bilinear method [
4], the multiple exp-function method [
5], the simplest equation method [
6,
7], non-classical method [
8], method of generalized conditional symmetries [
9], and the Lie symmetry method [
10,
11].
This paper aims to study one NLPDE; namely, the (2 + 1)-dimensional Zoomeron equation [
12]
which has attracted some attention in recent years. Many authors have found closed-form solutions of this equation. For example, the
expansion method [
12,
13], the extended tanh method [
14], the tanh-coth method [
15], the sine-cosine method [
16,
17], and the modified simple equation method [
18] have been used to find closed-form solutions of (
1). The (2 + 1)-dimensional Zoomeron equation with power-law nonlinearity was studied in [
19] from a Lie point symmetries point of view and symmetry reductions, and some solutions were obtained. Additionally, in [
19], the authors have given a brief history of the (1 + 1)-dimensional Zoomeron equation. See also [
20,
21,
22].
In this paper we first use the classical Lie point symmetries admitted by Equation (
1) to find an optimal system of one-dimensional subalgebras. These are then used to perform symmetry reductions and determine new group-invariant solutions of (
1). It should be noted that such approach was previously used for examination of a wide range of nonlinear PDEs [
23,
24,
25,
26,
27,
28,
29,
30,
31]. Furthermore, we derive the conservation laws of (
1) using the multiplier method [
32,
33].
The paper is organized as follows: in
Section 2, we compute the Lie point symmetries of (
1) and use them to construct the optimal system of one-dimensional subalgebras. These are then used to perform symmetry reductions and determine new group-invariant solutions of (
1). In
Section 3, we derive conservation laws of (
1) by employing the multiplier method. Finally, concluding remarks are presented in
Section 4.
3. Conservation Laws of (1)
Conservation laws describe physical conserved quantities, such as mass, energy, momentum and angular momentum, electric charge, and other constants of motion [
32]. They are very important in the study of differential equations. Conservation laws can be used in investigating the existence, uniqueness, and stability of the solutions of nonlinear partial differential equations. They have also been used in the development of numerical methods and in obtaining exact solutions for some partial differential equations.
A local conservation law for the (2 + 1)-dimensional Zoomeron Equation (
1) is a continuity equation
holding for all solutions of Equation (
1), where the conserved density
T and the spatial fluxes
X and
Y are functions of
t,
x,
y,
u. The results in [
11] show that all non-trivial conservation laws arise from multipliers. Specifically, when we move off of the set of solutions of Equation (
1), every non-trivial local conservation law (
12) is equivalent to one that can be expressed in the characteristic form
holding off of the set of solutions of Equation (
1) where
is the multiplier, and where
differs from
by a trivial conserved current. On the set of solutions
of Equation (
1), the characteristic form (
13) reduces to the conservation law (
12).
In general, a function
is a multiplier if it is non-singular on the set of solutions
of Equation (
1), and if its product with Equation (
1) is a divergence expression with respect to
. There is a one-to-one correspondence between non-trivial multipliers and non-trivial conservation laws in characteristic form.
The determining equation to obtain all multipliers is
where
is the Euler–Lagrange operator given by
Equation (
14) must hold off of the set of solutions of Equation (
1). Once the multipliers are found, the corresponding non-trivial conservation laws are obtained by integrating the characteristic Equation (
13) [
11].
We will now find all multipliers
and obtain corresponding non-trivial (new) conservation laws. The determining Equation (
14) splits with respect to the variables
. This yields a linear determining system for
which can be solved by the same algorithmic method used to solve the determining equation for infinitesimal symmetries. By applying this method, for Equation (
1), we obtain the following linear determining equations for the multipliers:
It is straightforward using Maple to set up and solve this determining system (
15)–(
18), and we get the four multipliers given by
For each solution
Q, a corresponding conserved density and flux can be derived (up to local equivalence) by integration of the divergence identity (
13) [
11,
36]. We obtain the following results.
Corresponding to these multipliers, we obtain four conservation laws. Thus, the multiplier (
19) gives the conservation law with the following conserved vector:
Likewise, the multiplier (
20) yields
as conserved vector.
Similarly, the multiplier (
21) results in the following conserved vector
Lastly, the multiplier (
22) gives the conserved vector whose components are