In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.
In this paper we study the third-order modified equal-width (MEW) equation
where α and β are non-zero real parameters. Equation (1) is used in handling the simulation of a single dimensional wave propagation in nonlinear media with dispersion processes . Some researchers have used different techniques and methods to construct travelling wave solutions of (1). Recently MEW Equation (1) was investigated in , where the researchers employed extended simple equation method and also the exp(-φ(ξ)) expansion method to generate travelling wave solutions of the equation. In , dynamical system technique for integer order was used and travelling wave solutions of the MEW equation were found, which comprised of solitary, periodic waves and also kink and anti-kink wave solutions. Homotopy perturbation method was applied to (1) and numerical solution of the MEW equation was obtained in .
In our study we use an entirely different approach to obtain new exact travelling wave solutions, namely cnoidal and snoidal wave solutions of MEW Equation (1). Moreover, for the first time we derive conservation laws of the MEW equation by employing both the Noether approach as well as the multiplier approach.
2 Exact solutions of (1) constructed on optimal system
In this section, we first compute Lie point symmetries of (1) and then use them to construct an optimal system of one-dimensional subalgebras. Subsequently, we utilise this optimal system of one-dimensional subalgebras to obtain symmetry reductions and group-invariant solutions of (1) [4, 5, 6, 7, 8].
2.1 Lie point symmetries of (1)
The vector field
where τ, ξ and η depend on t, x and u is a Lie point symmetry of Equation (1) if
and pr(3)X is the third prolongation  of (2) defined as
Here ζt , ζx, ζtx and ζtxx are determined by
where the total derivatives Dt and Dx are defined as
Expanding (3) and splitting on derivatives of u yields an overdetermined system of linear homogeneous partial differential equations (PDEs). Solving these equations we obtain the values of τ, ξ and η, which lead to three Lie point symmetries of (1) given by
The infinitesimal generator X3 represents scaling symmetry whereas the one-parameter groups generated by X1 and X2 demonstrate time and space-invariance of the MEW equation.
2.2 Optimal system of one-dimensional subalgebras
We now calculate an optimal system of one-dimensional subalgebras by using Lie point symmetries of (1) obtained in the previous subsection. We employ the method given in . We first construct the commutator table. Thereafter we compute adjoint representation using the Lie series
where ε is a real number and [Xi,Xj] denotes the commutator defined by
The table of commutators of Lie point symmetries of Equation (1) and adjoint representations of the symmetry group of (1) on its Lie algebra are presented in Table 1 and Table 2, respectively. Consequently, Table 1 and Table 2 are used to compute an optimal system of one-dimensional subalgebras for Equation (1).
Lie brackets for equation (1)
|[ , ]||X1||X2||X3|
Adjoint representation of subalgebras
|X1||X1||X2||-2ɛX1 + X3|
2.3 Solutions and symmetry reductions
We now utilise the optimal system of one-dimensional subalgebras obtained above in the previous subsection and find group-invariant solutions and symmetry reductions for Equation (1).
Consider the first operator X1+ cX2 of the optimal system. This operator has two invariants
which give the group-invariant solution U = U (ξ). Using ξ as our new independent variable, Equation (1) is transformed into the nonlinear ordinary differential equation (ODE)
We now use the extended Jacobi elliptic function expansion method  to obtain travelling wave solutions of (1). We assume that solutions of the third-order nonlinear ODE (7) can be expressed in the form
where M is a positive integer obtained by the balancing procedure and Ai are constants to be determined. Here H (ξ) satisfies the nonlinear first-order ODE
We recall that the Jacobi cosine-amplitude function
is a solution to (9), whereas the Jacobi sine-amplitude function
is a solution to (10). Here ω is a parameter such that 0 ≤ ω ≤ 1 [9, 10].
We note that when ω → 1, then cn(ξ |w) → sech(ξ) and sn(ξ |w) → tanh(ξ). Also, when ω → 0, then cn(ξ |w)→cos(ξ) and sn(ξ|ω)→ sin(ξ).
2.3.1 Cnoidal wave solutions
Considering the nonlinear ODE (7), the balancing procedure yields M = 1. Thus (8) takes the form
Substitution of U from (13) into (7) and utilising (9) we obtain
The above equation can be separated on like powers of H (ξ) to obtain an overdetermined system of eleven algebraic equations
Solving the above system of equations we obtain
Thus reverting to the original variables the solutions of (1) are
where nc = 1/cn.
2.3.2 Snoidal wave solutions
We now obtain snoidal wave solutions for Equation (1). Here again M = 1. Substituting the value of U from (13) into (7) and making use of (10) we obtain
Splitting on powers of H (ξ) yields the following overdetermined system of algebraic equations:
Solving the above system of equations we get
Reverting to original variables we obtain solutions of (1) as
We now consider the second operator X3 + aX2 of the optimal system. This symmetry operator yields two invariants J1 = ex t-a/2 and J2 = ω1/2. Thus J2 = f (J1) provides a group-invariant solution to (1). That is
Substituting the above value of u in (1), we obtain the third-order nonlinear ODE
3 Conservation laws of the modified equal width Equation (1)
In the section we derive conservation laws for (1) by employing two different techniques, namely the multiplier method and Noether approach.
Conservation laws have several important uses in the study of partial differential equations, especially for determining conserved quantities and constants of motion, detecting integrability and linearizations, finding potentials and nonlocally-related systems, as well as checking the accuracy of numerical solution methods [11, 12, 13, 14, 15, 16, 17].
3.1 Conservation laws of (1) using multiplier approach
We look for zeroth-order multiplier 𝛬=𝛬(t,x,u). Thus, the determining equation for this multiplier is stated as
where δ/δu is the Euler-Lagrange operator defined as
and the total derivatives Dt and Dx are defined as in (6). The above equation yields
which on expanding gives
Splitting the above equation on derivatives of u, we obtain
By solving the above equations we get two multipliers given by
Corresponding to these two multipliers, we obtain the following two conservation laws:
3.2 Conservation laws of (1) using Noether’s theorem
In this subsection we derive conservation laws for the modified equal-width Equation (1) using Noether’s theorem [18, 19]. This equation as it is does not have a Lagrangian. In order to apply Noether’s theorem we transform Equation (1) to a fourth-order equation which will have a Lagrangian. Thus using the transformation u =Vx, Equation (1) becomes
It can readily be verified that a Lagrangian for equation (17) is given by
because δℒ/δV = 0 on (17). Here δ/δV is the Euler-Lagrange operator defined as
Consider the vector field
where τ, ξ and η depend on t, x and V. To determine Noether point symmetries X of (17) we insert the value of L from (18) in the determining equation
where Bt = Bt(t,x,V) and Bx = Bx(t,x,V) are gauge terms and prX is the second prolongation of X defined as
with ζt , ζx, ζxx and ζtx as defined in (5). Expansion of equation (20) and separating with respect to derivatives of V yields an overdetermined system of linear PDEs. Thereafter solving these PDEs we obtain the following Noether point symmetries together with their gauge functions:
we obtain three conserved vectors associated with three Noether point symmetries X1, X2 and X f . Then reverting to the original variable u, we have
Remark: It should be noted that due to the presence of arbitrary function f (t) we have infinitely many nonlocal conservation laws.
In this paper we studied the modified equal-width Equation (1). For the first time, Lie point symmetries of (1) were computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter utilising this optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions of (1) were presented. The solutions obtained were cnoidal and snoidal waves. Again for the first time, we computed the conservation laws for (1) by employing two different methods, the multiplier method and Noether approach.
Communicated by Juan L.G. Guirao
The authors would like to thank T Motsepa for fruitful discussions. CMK thanks the North-West University, Mafikeng Campus for its continued support. ODA and IS thank the North-West University for financial aid through the post-graduate bursary scheme. Finally, we thank the anonymous reviewers for their useful comments, which helped improve the presentation of the paper.
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