Symmetry Solutions and Conservation Laws for the 3D Generalized Potential Yu-Toda-Sasa-Fukuyama Equation of Mathematical Physics

In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are obtained. The obtained group invariant solutions include the solution in terms of incomplete elliptic integral. Furthermore, conservation laws for the gpYTSF equation are derived using both the multiplier and Noether’s methods. The multiplier method provides eight conservation laws, while the Noether’s theorem supplies seven conservation laws. These conservation laws include the conservation of energy and mass.


Introduction
Many natural phenomena of the real world are modelled using nonlinear partial differential equations (NPDEs). It is therefore important to find their exact solutions in order to understand the real world better. There have been several studies done on NPDEs and many researchers have suggested various techniques for finding exact solutions for such equations, since there is no general theory that can be applied to find exact solutions. These techniques include the Jacobi elliptic function expansion method [1], the homogeneous balance method [2], the Kudryashov's method [3], the ansatz method [4], the inverse scattering transform method [5], the Bäcklund transformation [6], the Darboux transformation [7], the Hirota bilinear method [8], the (G /G)−expansion method [9], and the Lie symmetry method [10][11][12][13][14][15], just to mention a few.
In the late 19th century, a powerful symmetry-based technique for solving differential equations (DEs), known today as Lie group analysis, was developed by the Norwegian mathematician Marius Sophus Lie (1844-1899). This technique is an efficient technique that can be used to compute exact solutions of DEs. It only became well-known in the early 1960s when the Russian mathematician L. V. Ovsyannikov  demonstrated the power of these methods for computing explicit solutions of complicated partial differential equations (PDEs) arising in mathematical physics. Since then, a robust amount of research based on Lie's work has been published by various researchers.
The German mathematician Emmy Noether (1882Noether ( -1935 in 1918 presented a procedure for deriving conservation laws for systems of DEs that are derived from the variational principle, and this procedure is referred to as Noether's theorem [16]. A given DE that is derived from the variational principle should have a Lagrangian. However, there are DEs that are not derived from a variational principle and as a result Noether's theorem cannot be invoked to determine conservation laws for such DEs. In such a case, multiplier method [12] or Ibragimov's theorem on conservation laws [17] can be employed to construct conservation laws. The computation of conservation laws are very important as they play a vital role in the study of DEs. They describe physical conserved quantities, e.g., conservation of mass, energy, momentum, charge, and other constants of motion. They are also important for the investigation of integrability and uniqueness of solutions. See for example [16][17][18][19][20][21][22][23][24][25]. For the connection between the Lie and Noether symmetries, the reader is referred to [26,27]. The three-dimensional potential Yu-Toda-Sasa-Fukuyama (3DYTSF) equation given by was introduced in [28] using the strong symmetry, and its travelling solitary wave solutions were presented. Yan [29] studied Equation (1) and obtained auto-Bäcklund transformations.
Using auto-Bäcklund transformations, some exact solutions of (1) were found. These included the non-travelling wave solutions, soliton-like solutions, and rational solutions. The authors of [30] investigated Equation (1) using homoclinic and extended homoclinic test techniques, the two-soliton method along with bilinear form method, and obtained some new exact wave solutions that included periodic kink-wave, periodic soliton, cross kink wave, and doubly periodic wave solutions. In [31], the exp-function method, with the aid of symbolic computation, was employed, and new generalized solitary solutions and periodic solutions with free parameters were obtained. Using a modification of extended homoclinic test approach, the authors of [32] obtained some analytic solutions of 3DYTSF Equation (1). In [33], using some 1D subalgebras, group invariant solutions were constructed for (1) that involve arbitrary functions. Additionally, some particular solutions were sketched. Exact solutions that included lump solutions and interaction solutions of (1) were obtained using the generalized Hirota bilinear method [34]. In [35] analytical solutions and conservation laws for the 2D form of (1) were presented. Also, 2D and 3D graphical representations of the some solutions were given. N-soliton solutions were derived for (1) by using bilinear transformation that included period soliton, line soliton, lump soliton, and their interaction. Moreover, for some solutions their images were drawn and their dynamic behavior was discussed in [36]. The authors of [37] invoked the extended homoclininc test and Hirota bilinear method and constructed a class of lump solutions of (1). Additionally, periodic lump-type solutions were obtained in [37]. In [38], Equation (1) was reduced to the potential YTSF equation, which is a 2D equation (see also the ref [35]). General lump solutions of this equation were established and its propagation path was discovered. By letting u = w x , the authors of [39] increased the order of the (3+1) YTSF equation to five and applied Lie symmetry methods and constructed dark, bright, topological, Peregrine, and multi-soliton.
In this paper, we shall work with the three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (3DgYTSF) equation, namely where α, β, and γ are real constants. We seek to derive its exact solutions by using symmetry analysis, along with various other methods. Moreover, conserved quantities of Equation (2) are established using two approaches: multiplier approach and Noether's approach.

Solutions of the 3DgYTSF Equation
In this section, we firstly present Lie point symmetries and symmetry reductions of 3DgYTSF Equation (2). Moreover, we obtain travelling wave solution of (2) by employing Kudryashov's method.

Group Invariant Solution under X 7
Here we consider the Lie symmetry X 7 , which is given by Solving the associated characteristic equations for X 7 , we accomplish the invariants The use of these invariants reduces the 3DgYTSF Equation (2) to Equation (15) possesses six Lie symmetries The symmetry Γ 4 gives three invariants g, h, U = Φ − 2 f h and consequently, the group-invariant solution is which reduces Equation (15) to whose solution is given by where M and K are functions of h. Thus, the invariant solution of the 3DYTSF Equation (2) under the symmetry X 7 is In Figure 2, we depict the solution (16) with M z/t 3 = cos z/t 3 , K z/t 3 = sech z/t 3 and parametric values β = 2, α = 2, x = 0, t = 1, z = 15.

Group Invariant Solution under X 8
For the symmetry X 8 , we get the group-invariant solution as where f , g, h, F are the invariants given by Substituting (17) into the 3DgYTSF Equation (2) gives the NODE whose symmetry includes Γ 1 = ∂/∂ f . Using the symmetry Γ 1 , we get the invariants j 1 = g, j 2 = h, F = Ψ, which reduces Equation (18) to the NPDE with two independent variables. Thus, we have reduced the number of independent variables of 3DgYTSF Equation (2) by two.

Conservation Laws of (2)
In this section we construct conservation laws of the 3DgYTSF Equation (2) by using two different approaches, namely, the multiplier method and Noether's approach.

Conservation Laws Using the Multiplier Approach
We seek first-order multiplier Q = Q(t, x, y, z, u, u t , u x , u y ) by applying the determining equation for the multipliers where the Euler-Lagrange operator δ/δu in our case is defined as Expansion of Equation (19) yields which, on applying the total derivatives D t , D x , D y , D z and splitting over the derivatives of u, yields the following simplified determining equations: Q x = 0, Q u = 0, Q yy = 0, Q zz = 0, Q tu t = 0, Q yu t = 0, Q yu y = 0, Q zu t = 0, Q zu x = 0, Q zu y = 0, Q u t u t = 0, Q u t u x = 0, Q u t u y = 0, 2Q tu x − Q z = 0, Q u x u x = 0, Q u y u y = 0, The solution of the above system of overdetermined equations is where F(t), G(t), H(t), and J(t) are functions of t, whereas C 1 , C 2 , C 3 , and C 4 are arbitrary constants. The conservation laws are now obtained by using the divergence identity where T t is the conserved density, and T x , T y , T z are spatial fluxes. Thus, after some calculations, conservation laws corresponding to the eight multipliers are given below. Case 1. For the first multiplier Q 1 = yu x + (βtu y )/α, the corresponding conservation law is given by Case 2. For the second multiplier Q 2 = u t , we obtain the corresponding conservation law as Case 3. For the second multiplier Q 3 = u x , we obtain the corresponding conservation law as Case 4. For the multiplier Q 4 = u y , we obtain the corresponding conservation law as Case 5. For multiplier Q 5 = F (t)z + 1 2 F (t) y + 1 2α βF(t)u y , we obtain the corresponding conservation law as βF(t)u z u xxy βF(t)uu xxxy βF(t)u xx u xy . Case 6. For the multiplier Q 6 = G(t)y, we obtain the corresponding conservation law as Case 7. For the multiplier Q 7 = H (t)z + 1 2 H(t)u x , we obtain the corresponding conservation law as T t 7 = − αzu x H (t) + Case 8. For the last multiplier Q 8 = J(t), we obtain the corresponding conservation law as

Conservation Laws Using Noether's Approach
In this subsection, we utilize Noether's approach to derive consevation laws for the 3DgYTSF Equation (2). This equation is of fourth-order and it has a Lagrangian. It can be verified that Equation (2) has a second-order Lagrangian L given by L = 1 2 u xx u xz − βu 2 y − αu 2 x u z + αu t u x as δL/δu = 0, on the Equation (2). Here, the Euler-Lagrange operator δ/δu is given as in (20). The determining equation for Noether point symmetries is where gauge terms B 1 , B 2 , B 3 , and B 4 depend on t, x, y, z, and u. Here, X [2] is the second prolongation of the infinitesimal generator X and is defined by Institutional Review Board Statement: Not applicable.

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