Symmetry Analysis , Explicit Solutions , and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation

In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results.


Introduction
It is well-known that the majority of real-world physical phenomena are modeled by mathematical equations, especially partial differential equations (PDEs).These phenomena include the problems from fluid mechanics, elasticity, plasma physics and optical fibers, general relativity, gas dynamics, thermodynamics, and so on [1].In order to understand the understanding of such physical phenomena, it is vital to look for the exact solutions of the PDEs.In the last few decades, many scientists and mathematicians have extensively studied the dynamic behaviors of several PDEs  using different concepts.Symmetry analyses have been used to study the PDEs [2][3][4][5][6].It has been found that some new solutions to PDEs can be obtained from the old ones through symmetry transformations [2].On the other hand, conservation laws (CLs) are very important in the study of PDEs.CLs are important in determining the integrability of PDEs [2].
As an important integrable nonlinear model, the integrable sixth-order nonlinear Ramani equation [10][11][12][13][14][15][16] has attracted much attention in soliton theory in recent years.The equation was first proposed in reference [10].The equation was obtained as a five-reduction of the bilinear Kadomtsev-Petviashvili (BKP) equation [11].It has been shown that the Ramani equation possesses bilinear B äcklund transformation and CL [12].In reference [13], the truncated singular expansion scheme was applied to construct the B äcklund self-transformation and Lax pairs for Equation (1).In reference [14], the Lax pairs and B äcklund transformation were applied to study the equation.An extension of Equation ( 1), called the coupled Ramani equation, was extensively studied in references [15][16][17][18].
To our knowledge, a Lie symmetry analysis of Equation ( 1) has not been completed.The main aim of this work is to derive the Lie point symmetries [2,4], the optimal system of one-dimensional sub-algebras, invariant solutions, and the CL of the equation by invoking the conservation theorem due to Ibragimov [7,8].The invariant solutions are derived by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction process using the Riccati Bernoulli sub-ODE (RBSO) [19].

Lie Symmetry Analysis of Equation (1)
In this part, we construct the vector fields of Equation (1).The associated vector field of Equation ( 1) is given by where the coefficient functions ξ(x, t, ψ), η(x, t, ψ), φ(x, t, ψ) are the infinitesimals.The one-parameter Lie group is represented by where represents a group parameter.If the vector field in Equation ( 2) generates a point symmetry of Equation ( 1), then X should satisfy the invariance condition given by where P 6 r is the sixth-order prolongation of Γ [2], and Putting Equation (4) into Equation (3) together with the sixth-order prolongation, we obtain a set of determining equations of linear PDEs.From solving the system of linear PDEs, we obtain the infinitesimals ξ, φ, and η, given by where C 1 , C 2 , C 3 and C 4 are constants.The symmetries of Equation ( 1) are spanned by the vector fields given by

Optimal System of Algebras
In this part, we derive the optimal system of sub-algebras [4] of the vector fields in Equations ( 5)- (8).To do this, we begin by noting that each X i (i = 1, 2, 3, 4) yields an adjoint representation Ad(exp( X i ))X j , defined by [4] where [X i , X j ] represents the commutator There is a need to figure out the invariants of the adjoint, because they pose restrictions on the element The adjoint representation group is spanned by the Lie algebra g A , which is spanned by the following expression , where c k ij are constants obtained from Table 1.Thus, one can obtain Table 1.Commutator of vector fields in Equations ( 5)- (8).
In addition, we can have From Equations ( 9)-( 12) and Tables 1 and 2, we derive the following optimal system of Lie algebras: Table 2. Adjoint representation of the vector fields in Equations ( 5)- (8).

Similarity Reductions and Exact Solutions
In this part, we use the derived optimal system of Lie algebras in Equation ( 13) to investigate the solutions of Equation (1).To achieve this, one needs to solve the characteristic equations denoted by For the reduction by the vector field X 1 = ∂ ∂x , we acquire the similarity variables ψ(x, t) = F (t) with F (t) satisfying the ODE: The solution of Equation (1) after solving Equation ( 14) is given by 1.3.2.Symmetry Reduction with the Vector Field X 2 For the reduction by the vector field 1.3.3.Symmetry Reduction with the Vector Field X 3 For the reduction by the vector field X 3 = ∂ ∂t , we get the similarity variables ψ(x, t) = F (x), where F (x) satisfies the following ODE: 1.3.4.Symmetry Reduction with the Vector Field X 4 For the reduction by the vector field x , where ζ = t x 3 .F (ζ) thus satisfies the following ODE: 1.3.5.Invariant Solutions of Equation ( 15) Equation ( 15) is a sixth-order nonlinear ODE.We apply the RBSO technique [19] to derive its solutions.In what follows, we provide the description of the RBSO method.
Consider the PDE given by where ψ = ψ(x, t).
Step 1: By introducing the transformation Equation ( 16) is transformed to the following ODE with F (ζ) = du dζ .
Step 2: Suppose the solution of Equation ( 17) is the solution of the RBE with a, b, c, and m being arbitrary constants.By integrating Equation ( 18), we acquire
Step 3: By putting the derivatives of F into Equation ( 17), one can obtain algebraic expressions involving F and other parameters.By choosing the value of m according to the steps described above, comparing the coefficients of F i , i = (1, 2, . . ., 0, performing all the necessary algebraic computations, and utilizing Equations ( 19)-( 26), the solutions of Equation ( 16) may be derived.
To solve Equation ( 15) using the RBSO technique, we substitute Equation ( 18) along with the 2nd, 3rd, 4th and 6th derivatives into Equation (15) and set m = 0 in the resulting algebraic expression to get By collecting the terms of F i (i = 0, 1..., 7) in Equation ( 27) and performing all the necessary algebraic computations, we get Constants : F 1 : −315a 3 bα 4 1 − 6aα + 8a 2 α 2 = 0, (34) From solving Equations ( 28)-( 35), we acquire the following family of parameter values: we have cases and solutions given by Case A: If α > 0, we acquire the kink-type solution given by and the singular solution Case B: If α < 0, we acquire the following periodic traveling wave solutions Case C: From substituting the parameters in Equation (36) into Equation ( 21), we obtain the following exact solution of Equation (1): Family 2: When we acquire the periodic traveling wave solutions represented by and

Conservation Laws
In this part, we derive the nonlinear self-adjointness of Equation ( 1) for the purpose of constructing the CL.We begin by considering the following theorem from references [7,8]: Theorem 1.The system of m differential equations , with m dependent variables ψ = (ψ 1 , . . ., ψ m ) has an adjoint equation where The formal Lagrangian L for Equation ( 42) is given by where υ β = υ β (x, t) is a nonlocal dependent variable.
The formal Lagrangian Equation (44) for Equation ( 1) is written as where υ = υ(x, t) is a nonlocal variable.Subsequently, we derive the adjoint of Equation (1) as Theorem 2. Equation ( 42) is nonlinear self-adjoint if it becomes equivalent to its adjoint Equation (43) upon the substitution Equation (47) means that not all components of φ α (x, ψ) of φ vanish.
Theorem 3. Equation ( 1) is nonlinear self-adjoint if υ in Equation ( 46) is given by By putting Equation (49) into the adjoint Equation (46), we acquire Using the Mathematica package called SYM [24], we obtain the solution of Equation (50) as Theorem 4. Any infinitesimal symmetry (Lie point, B äcklund, nonlocal) of Equation (42) leads to a CL D i (T i ) = 0, constructed by the formula where W α = η α − ξ j ψ α j and T i are the conserved vectors.

Figure 1 .
Figure 1.(a) The 3D surface of the kink-type solution Equation (37) and (b) the singular solution to Equation (38) by setting α = 0.7, C = 1.These solutions have several physical applications in a Bloch wall between two magnetic domains in a ferromagnetic.The solitary waves propagate without change in the dynamics of the amplitude and width.It can be observed that the solitary waves move along the 3D axis with positive phase velocity and a constant period.The amplitude and phase of the solitary waves do not change during the evolution.