Lie symmetry methods play a vital role in the study and finding solutions for nonlinear partial differential equations (NLPDE) [4,5, 6,7,8,9,10,11,12,13,14,15,16]. Different techniques are used in the literature for the construction of Cls for different system of equation and these Cls are important for the investigation of a physical system [4,5, 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Moreover, Authors made rigorous attempts for construction of the construction of one-dimensional and higher-dimensional optimal systems optimal system of Lie algebra [22,23,24,25,26]. Cls and symmetries have many application in science, physics and engineering [32,33,34,35,36,37].
In this work, we obtain one-dimensional optimal system, exact solutions and Cls for the GSWW equation given by
where a ≠ 0, b ≠ 0 are arbitrary constants. Eq. (1) have been studied by different authors using a variety of techniques. For example  introduced exact solutions for Eq. (1) by the general projective Ricatti equations method. Periodic wave solution for Eq. (1) by the improved Jacobi elliptic function method was investigated by  . Homogeneous balance method  was applied to investigate some solutions for Eq. (1) and some new solution of Eq. (1) with extended elliptic function method was proposed in  and many more.
2 One-dimensional optimal system of subalgebras of GSWW
In this section, we establish the optimal system of one-dimensional subalgebras of L4 and their corresponding exact solutions. Consider one parameter Lie group of the infinitesimal transformation below
where ϵ is the group parameter. The corresponding Lie algebra of the infinitesimal symmetries is the set of vector field of the form
Considering the fourth order prolongation Pr(4) of the vector field X such that
where Δ = (1), whenever Δ = 0. Using SYM package introduced in , the determining equations for Eq. (1) are obtained. Solving for η(x, t, u), ξ2(x, t, u), and ξ1(x, t, u) from the obtained determining equations, we get
where c1, c2, c3 and c4 are arbitrary constants and F(t) is an arbitrary function of t. The Lie symmetry algebra admitted by Eq. (1) is spanned by four infinitesimals generators below
The corresponding commutator table of the infinitesimal generators is given by
2.1 Construction of one-dimensional optimal system of subalgebras
The Lie algebra L4 spanned by the given generators X1, X2, X3, and X4 can guarantee a possibility to obtain invariant solutions of Eq. (1). This will be based mainly on one-dimensional subalgebra of L4. There may be an infinite number of one-dimensional subalgebras of L4. Therefore, one can write an arbitrary generators from L4 as
which depend on the four arbitrary constants l1, l2, l3, and l4. We construct the one-dimensional optimal system of subalgebras using the method introduced in [22,23,24,25,26]. After the transformation of L4, we can get a 4-parameter group of linear transformations of the generators as
where l1, l2, l3, and l4 are the coefficients in Eq. (13).
2.2 Linear transformation
Here, we investigate the linear transformations by using their generators which is given as
and the structure constants of the Lie algebra L4 defined by is given as
Consider the following cases:
Case 1: For μ = 1, v = 4, and λ = 1, 2 in Table 1. [X1, X4] = X1 X2 and the non vanishing structure constants are ( ) are = a, = 2.
Case 2: For μ = 2, v = 4, and λ = 2 in Table 1. [X2, X4] = X2 and the non vanishing structure constants ( ) are = −a.
Case 3: For μ = 3, v = 4, and λ = 3 in Table 1. [X3, X4] = X3 and the non vanishing structure constants ( ) are = ∗a.
Case 4: For μ = 4, v = 1, and λ = 1, 2 in Table 1. [X4, X1] = X1 X2 and the non vanishing structure constants ( ) are = −a, = −2. Setting v = 2, λ = 2 row four column two, we get [X4, X2] = X2 and = a, Setting v = 3, λ = 3 row four column three, we get [X4, X3] = X3 and = a.
Now, Substituting the values of the non-vanishing structure constants in Eq. (15) for μ = 1, 2, …, 4, we obtain
2.3 Lie equation
To obtain the Lie equation, we integrate the generators E1, E2, E3, E4 in Eq. (17) using the initial condition l|ϵ=0 = l.
For the generator E1, the Lie equation with the parameter ϵ are given by Integrating and using the initial condition we obtain l1 = aϵ1 l4 + l1, l2 = 2ϵ1 l4 + l2, l3 = l3, l4 = l4.
similarly for the other generators by following the same approach we get
For E2, we obtain l1 = l1, l2 = −aϵ2 l4 + l2, l3 = l3, l4 = l4.
For E3, we have l1 = l1, l2 = l2, l3 = −aϵ3 l4 + l3 and l4 = l4.
For E4, we have and l4 = l4.
using SYM package , we can get the optimal system raw data in matrix form as:
and the number of the functionally invariants, which is found to be l4. The corresponding one-dimensional optimal system of subalgebras are found to be the following:
αX2 + X3,
X1 + X4,
αX3 + X4,
αX1 + βX2 + X3,
where α, β ∈ ℝ. In the following, we list the corresponding similarity variables, similarity solutions as well as the reduced PDEs obtained from the generators of optimal system and their exact solutions.
Similarity variable related to X3 is u(x, t) = F(x) and F(x) satisfies Fxx = 0 two times integration implies that F(x) = c1 + xc2 and we have the exact solution
Similarity variable related to αX2 + X3 is u(x, t) = αt + F(x) and F(x) satisfies Fxx = 0 which after integrating twice gives F(x) = c1 + xc2 and we have the exact solution
Similarity variable related to X4 is u(x, t) = ζ = tx and F(ζ) satisfies
thrice integration of Eq. (20) and letting c1 = 0 yields
solving for F(ζ) in Eq. (21) we obtain
Hence by back substituting the similarity variables we get the exact solution as
L1, L2 are as stated above and ζ = tx.
Similarity variable related to X1 + X4 is u(x, t) = ζ = t(1 + ax) and F(ζ) satisfies the following
three times integration and letting c1 = c2 = c3 = 0 in the above equation, gives
solving for F(ζ) we have the following equation in Bessel function form
and the exact solution is
where ζ = t(1 + ax).
Similarity variable related to βX3 + X4 is u(x, t) = and F(ζ) satisfies the following
here the reduced PDE is the same as that in reduction 3, the only different is the variable ζ. Therefore, we get
and L1, L2, Q0, Q1, Q2, and Q3 are as stated in Eq. (23) and ζ =
Similarity variable related to αX1 + βX2 + X3 is u(x, t) = and F(ζ) satisfies the following
three times integration of Eq. (30) with c1 = 0 leads
solving for Fζ, we obtain
Thus, by back substituting the similarity variables we get
4 Nonlinear self-adjointness
α = 1, …, m, with m dependent variables u = (u1,…,um) is said to be NSA if the adjoint equations
α = 1, …,m, are satisfied for all solutions u of the original system Eq. (34) upon a substitution
α = 1, …, m, such that
On the other hand, the equation below holds:
α = 1, …, m, where are undetermined coefficients, and φ(σ) are derivatives of Eq. (36),
,σ = 1, …, s. Here v and φ are the m-dimensional vectors v = (v1, …, v(m)), φ = (φ1, …, φm), and also, it is worth noting that not all components φα(x, u) of φ vanish simultaneously from Eq. (37).
4.1 Test for self-adjointness for GSWW
Let v = ϕ(x, t, u), after some calculations and equating the coefficients of the derivatives ut, ux, uxt, uxx, uxxt and uxxx to zero, we have
The solution for ϕ(t, x, u) from the above equation is simply found to be the following
5 Conservation laws for GSWW
The reality that Eq. (1) is NSA with the obtained differential substitution in Eq. (40), we can use the Noether operator 𝓝 to obtain its conserved vectors (C1, C2) [17,18,19,20]. The obtained conserved vectors will satisfy the conservation equation DxC1 + DtC2 = 0. Moreover, the non local variables appearing in that formula must be substituted according to equation (40).
The following are the conserved vectors obtained from the four infinitesimals say X1, X2, X3, and X4 when C1 = 1 respectively.
Similarly, one can verify and see that DxC1 + DtC2 = 0 which is a trivial conservation laws.
In this study, Lie symmetry analysis, one dimensional optimal system and Cls for GSWW equation were studied. Some reductions and their solutions were reported from the obtained one dimensional optimal system. We presented some figures for some of the obtained exact solutions. The exact solutions include an expression with a Bessel function and an Airy function. We verified the authenticity of the solution by substitution into the original equation. The GSWW equation is a NSA with the obtained differential substitution, we obtained Cls using the new conservation theorem presented by Ibragmov.
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About the article
Published Online: 2018-06-26
Citation Information: Open Physics, Volume 16, Issue 1, Pages 364–370, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0049.
© 2018 Dumitru Baleanu et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0