Symmetry solutions and conservation laws of a new generalized 2D Bogoyavlensky-Konopelchenko equation of plasma physics

: In physics as well as mathematics, nonlinear partial di ﬀ erential equations are known as veritable tools in describing many diverse physical systems, ranging from gravitation, mechanics, ﬂuid dynamics to plasma physics. In consequence, we analytically examine a two-dimensional generalized Bogoyavlensky-Konopelchenko equation in plasma physics in this paper. Firstly, the technique of Lie symmetry analysis of di ﬀ erential equations is used to ﬁnd its symmetries and perform symmetry reductions to obtain ordinary di ﬀ erential equations which are solved to secure possible analytic solutions of the underlying equation. Then we use Kudryashov’s and ( G (cid:48) / G )-expansion methods to acquire analytic solutions of the equation. As a result, solutions found in the process include exponential, elliptic, algebraic, hyperbolic and trigonometric functions which are highly important due to their various applications in mathematic and theoretical physics. Moreover, the obtained solutions are represented in diagrams. Conclusively, we construct conservation laws of the underlying equation through the use of multiplier approach. We state here that the results secured for the equation understudy are new and highly useful.


Introduction
Plasma physics simply refers to the study of a state of matter consisting of charged particles. Plasmas are usually created by heating a gas until the electrons become detached from their parent atom or molecule. In addition, plasma can be generated artificially when a neutral gas is heated or subjected to a strong electromagnetic field. The presence of free charged particles makes plasma electrically conductive with the dynamics of individual particles and macroscopic plasma motion governed by collective electromagnetic fields [1].
Nonlinear partial differential equations (NPDE) in the fields of mathematics and physics play numerous important roles in theoretical sciences. They are the most fundamental models essential in studying nonlinear phenomena. Such phenomena occur in plasma physics, oceanography, aerospace industry, meteorology, nonlinear mechanics, biology, population ecology, fluid mechanics to mention a few. We have seen in [2] that the authors studied a generalized advection-diffusion equation which is a NPDE in fluid mechanics, characterizing the motion of buoyancy propelled plume in a bent-on absorptive medium. Moreover, in [3], a generalized Korteweg-de Vries-Zakharov-Kuznetsov equation was studied. This equation delineates mixtures of warm adiabatic fluid, hot isothermal as well as cold immobile background species applicable in fluid dynamics. Furthermore, the authors in [4] considered a NPDE where they explored important inclined magneto-hydrodynamic flow of an upper-convected Maxwell liquid through a leaky stretched plate. In addition, heat transfer phenomenon was studied with heat generation and absorption effect. The reader can access more examples of NPDEs in [5][6][7][8][9][10][11][12][13][14][15][16].
The (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation given as u tx + 6αu x u xx + 3βu x u xy + 3βu y u xx + αu xxxx + βu xxxy = 0, (1.1) where parameters α and β are constants, is a special case of the KdV equation in [34] which was introduced as a (2+1)-dimensional version of the KdV and it is described as an interaction of a long wave propagation along x-axis and a Riemann wave propagation along the y-axis [35]. In addition to that, few particular properties of the equation have been explored. The authors in [36] provided a Darboux transformation for the BK equation and the obtained transformation was used to construct a family of solutions of this equation. In [37], with 3β replaced by 4β and u y = v x in (1.1), the authors integrated the result once to get Further, they utilized Lie group theoretic approach to obtain solutions of the system of Eq (1.2). They also engaged the concept of nonlinear self-adjointness of differential equations in conjunction with formal Lagrangian of (1.2) for constructing nonlocal conservation laws of the system. In addition, various applications of BK equation (1.1) were highlighted in [37]. Further investigations on certain particular cases of (1.1) were also carried out in [38,39].
We notice that if one takes α = β = 1 in Eq (1.1) with the introduction of two new terms u xx and u yy , the new generalized version (1.7) is achieved. In consequence, we investigate explicit solutions of the new two-dimensional generalized Bogoyavlensky-Konopelchenko equation (1.7) of plasma physics in this study. In order to achieve that, we present the paper in the subsequent format. In Section 2, we employ Lie symmetry analysis to carry out the symmetry reductions of the equation. In addition, direct integration method will be employed in order to gain some analytic solutions of the equation by solving the resulting ordinary differential equations (ODEs) from the reduction process. We achieve more analytic solutions of (1.7) via the conventional (G /G)-expansion method as well as Kudryashov's technique. In addition, by choosing suitable parametric values, we depict the dynamics of the solutions via 3-D, 2-D as well as contour plots. Section 3 presents the conservation laws for 2D-gBK equation (1.7) through the multiplier method and in Section 4, we give the concluding remarks.

Symmetry analysis and analytic solutions of (1.7)
In this section we in the first place compute the Lie point symmetries of Eq (1.7) and thereafter engage them to generate analytic solutions.

Lie point symmetries of (1.7)
A one-parameter Lie group of symmetry transformations associated with the infinitesimal generators related to (gbk) can be presented as (2.1) We calculate symmetry group of 2D-gBK equation (1.7) using the vector field where ξ i , i = 1, 2, 3 and φ are functions depending on t, x, y and u. We recall that (2.2) is a Lie point symmetry of Eq (1.7) if where Q = u tx + 6u x u xx + u xxxx + u xxxy + 3 u x u y x + u xx + u yy . Here, R [4] denotes the fourth prolongation of R defined by where coefficient functions η t , η x , η y , η xt , η xx , η xy , η yy , η xxxx and η xxxy can be calculated from [22][23][24]. Writing out the expanded form of the determining equation (2.3), splitting over various derivatives of u and with the help of Mathematica, we achieve the system of linear partial differential equations (PDEs): The solution of the above system of PDEs is where A 1 -A 3 are arbitrary constants and F(t), G(t) are arbitrary functions of t. Consequently, we secure the Lie point symmetries of (1.7) given as (2.5)

Lie group transformations associated to (2.5)
We contemplate the exponentiation of the vector fields (2.5) by computing the flow or one parameter group generated by (2.5) via the Lie equations [22,23]: Therefore, by taking F(t) = G(t) = t in (2.5), one computes a one parameter transformation group of 2D-gBK (1.7). Thus, we present the result in the subsequent theorem.

Symmetry reduction of 2D-gBK equation (1.7)
In this subsection, we utilize symmetries (2.5) with a view to reduce Eq (1.7) to ordinary differential equations and thereafter obtain the analytic solutions of Eq (1.7) by solving the respective ODEs.
Case 1. Invariant solutions via R 1 -R 3 Taking F(t) = 1/3, we linearly combine translational symmetries R 1 -R 3 as R = bR 1 + cR 2 + aR 3 with nonzero constant parameters a, b and c. Subsequently utilizing the combination reduces 2D-gBK equation (1.7) to a PDE with two independent variables. Thus, solution to the characteristic equation associated with the symmetry R leaves us with invariants (2.6) Now treating θ above as the new dependent variable as well as r, s as new independent variables, (1.7) then transforms into the PDE: We now utilize the Lie point symmetries of (2.7) in a bid to transform it to an ODE. From (2.7), we achieve three translation symmetries: The linear combination Q = Q 1 + ωQ 2 (ω 0 being an arbitrary constant) leads to two invariants: that secures group-invariant solution Θ = Θ(z). Thus, on using these invariants, (2.7) is transformed into the fourth-order nonlinear ODE: which we rewrite in a simple structure as

Some analytic solutions of 2D-gBK equation (1.7)
In this section, we seek travelling wave solutions of the 2D-gBK equation (1.7).
A. Elliptic function solution of (1.7) On integrating equation (2.9) once, we accomplish a third-order ODE: where C 1 is a constant of integration. Multiplying Eq (2.10) by Θ (z), integrating once and simplifying the resulting equation, we have the second-order nonlinear ODE: where C 2 is a constant of integration. The above equation can be rewritten as Suppose that the cubic equation has real roots c 1 -c 3 such that c 1 > c 2 > c 3 , then Eq (2.12) can be written as whose solution with regards to Jacobi elliptic function [45,46] is with (cn) being the elliptic cosine function. Integration of (2.15) and reverting to the original variables secures a solution of 2D-gBK equation (1.7) as   However, contemplating a special case of (2.9) with B = 0, we integrate the equation twice and so we have where K 1 and K 2 are integration constants. Solving the second-order linear ODE (2.17) and reverting to the basic variables, we achieve the trigonometric solution of 2D-gBK equation (1.7) as  We further explore Weierstrass elliptic function solution of (1.7), which is a technique often involved in getting general exact solutions to NPDEs [47,48]. In order to accomplish this, we use the transformation and transform the nonlinear ordinary differential equation (NODE) (2.12) to with the invariants g 2 and g 3 given by Thus, we have the solution of NODE (2.12) as where ℘ denotes the Weierstrass elliptic function [46]. In consequence, integration of (2.21) and reverting to the basic variables gives the solution of 2D-gBK equation (1.7) as with arbitrary constant z 0 , z = cx + (aω − b)y − cωt and ζ being the Weierstrass zeta function [46]. We give wave profile of Weierstrass function solution (2.22) in Figure 3 with 3D, contour and 2D plots using parameter values a = 1, b = 0.2, c = −0.1, ω = 0.1, A = 10, B = −2, z 0 = 0, C = 1, C 1 = 1, C 2 = 10, where t = 2 and −10 ≤ x, y ≤ 10.

Solution of (1.7) by Kudrayshov's approach
This part of the study furnishes the solution of 2D-gBK equation (1.7) through the use of Kudryashov's approach [33]. This technique is one of the most prominent way to obtain closed-form solutions of NPDEs. Having reduced Eq (1.7) to the NODE (2.9), we assume the solution of (2.9) as with Q(z) satisfying the first-order NODE Q (z) = Q 2 (z) − Q(z).

(2.24)
We observe that the solution of (2.24) is . (2.25) The balancing procedure for NODE (2.9) produces N = 1. Hence, from (2.23), we have Now substituting (2.26) into (2.9) and using (2.24), we gain a long determining equation and splitting on powers of Q(z), we get algebraic equations for the coefficients B 0 and B 1 as The solution of the above system gives (2.28) Hence, the solution of 2D-gBK equation (1.7) associated with (2.28) is given as The wave profile of solution (2.29) is shown in Figure 4 with 3D, contour and 2D plots using parameter values a = 1, b = −0.2, c = 20, ω = 0.05, B 0 = 0 with t = 7 and −6 ≤ x, y ≤ 6.

Case 2. Group-invariant solutions via R 4
Lagrange system associated with the symmetry R 4 = 3t∂/∂y + (x − 2y)∂/∂u is , (2.36) which leads to the three invariants T = t, X = x, Q = u+(y 2 /3t)−(xy/3t). Using these three invariants, the 2D-gBK equation (1.7) is reduced to Case 3. Group-invariant solutions via R 1 , R 2 and R 5 We take G(t) = 1 and by combining the generators R 1 , R 2 as well as R 5 , we solve the characteristic equations corresponding to the combination and get the invariants X = x, Y = y−t with group-invariant u = Q(X, Y) + t. With these invariants, the 2D-gBK equation (1.7) transforms to the NPDE whose solution is given by with arbitrary constants A 1 -A 3 . Thus, we achieve the hyperbolic solution of (1.7) as The wave profile of solution (2.40) is shown in Figure 8 with 3D, contour and 2D plots using parameter values A 1 = 70.1, A 2 = −30, A 3 = 0, where t = 0.5 and −10 ≤ x, y ≤ 10. Besides, symmetries of (2.38) are found as Now, the symmetry P 1 furnishes the solution Q(X, Y) = f (z), z = Y. So, Eq (2.38) gives the ODE f (z) = 0. Hence, we have a solution of (1.7) as with A 0 , A 1 as constants. Further, the symmetry P 2 yields Q(X, Y) = f (z), z = X and so Eq (2.38) reduces to Integration of the above equation three times with respect to z gives and taking constants A 0 = A 1 = 0 and then integrating it results in the solution of (1.7) as The wave profile of solution (2.44) is shown in Figure 9 with 3D, contour and 2D plots using parameter values A 1 = 40, t = 3.5 and −10 ≤ x ≤ 10. On combining P 1 -P 3 as P = c 0 P 1 + c 1 P 2 + c 2 P 3 , we accomplish Using the newly acquired invariants (2.45), Eq (2.38) transforms to the NODE: Engaging the Lie point symmetry P 4 , we obtain and Eq (2.38) reduces to the NODE Case 4. Group-invariant solutions via R 6 Lie point symmetry R 6 dissociates to the Lagrange system which gives Substituting the expression of u in (1.7), we obtain the NPDE which has two symmetries: The symmetry P 2 gives Q(X, Y) = f (z) + (1/15)T X, z = T and hence (2.50) reduces to Solving the above ODE and reverting to the basic variables gives the solution of (1.7) as where A 1 and A 2 are integration constants. The wave profile of solution (2.51) is shown in Figure 10 with 3D, contour and 2D plots using parameter values A 1 = −0.3, A 2 = −50 with t = 1.1 and −10 ≤ x, y ≤ 10. Next, we invoke the symmetry P 1 + P 2 . This yields Q(X, Y) = f (z) + X + (1/15)T X, z = T . Consequently, we have the transformed version of (2.50) as Solving the above ODE and reverting to basic variables gives the solution of (1.7) as The wave profile of solution (2.52) is shown in Figure 11 with 3D, contour and 2D plots using parameter values A 1 = −3.6, A 2 = 50 with t = 1.1 and −10 ≤ x, y ≤ 10.  Figure 11. The wave profile of solution (2.52) at t = 1.1.

Conservation laws of (1.7)
In this section, we construct the conservation laws for 2D-gBK equation (1.7) by making use of the multiplier approach [22,49,50], but first we give some basic background of the method that we are adopting.

Construction of conservation laws for (1.7)
Conservation laws of 2D-gBK equation (1.7) are derived by utilizing second-order multiplier Ω(t, x, y, u, u t , u x , u y , u xx , u xy ), in Eq (3.7), where G is given as G ≡ u tx + 6u x u xx + u xxxx + u xxxy + 3 u x u y x + u xx + u yy , and the Euler operator δ/δu is expressed in this case as Expansion of Eq (3.7) and splitting on diverse derivatives of dependent variable u gives Ω u = 0, Ω x = 0, Ω yy = 0, Ω yu x = 0, Ω u x u x = 0, Ω tu x = 0, Ω u t = 0, Ω u xx = 0, Ω u xy = 0, Ω u y = 0. (3.8) Solution to the above system of equations gives Ω(t, x, y, u, u t , u x , u y , u xx , u xy ) as Ω(t, x, y, u, u t , u x , u y , u xx , u xy ) = C 1 u x + f 1 (t)y + f 2 (t), (3.9) with C 1 being an arbitrary constant and f 1 (t), f 2 (t) being arbitrary functions of t. Using Eq (3.6), one obtains the following three conserved vectors of Eq (1.7) that correspond to the three multipliers u x , f 1 (t) and f 2 (t): Case 1. For the first multiplier Q 1 = u x , the corresponding conserved vector (T t 1 , T x 1 , T y 1 ) is given by u xx u xy + 1 2 u x u xxy + u xxx u x + 1 2 u xxx u y + 1 2 uu xxxy + 1 2 uu yy + uu x u xy + 2u y u 2 x , T y 1 = 1 2 u y u x − uu x u xx − 1 2 uu xy − 1 2 uu xxxx .
Case 2. For the second multiplier Q 2 = f 1 (t), we obtain the corresponding conserved vector (T t 2 , T x 2 , T y 2 ) as T t 2 = u x f 1 (t)y, T x 2 = 3y f 1 (t)u 2 x + 3y f 1 (t)u x u y − y f 1 (t)u + y f 1 (t)u x + y f 1 (t)u xxx + y f 1 (t)u xxy , T y 2 = u y f 1 (t)y − u f 1 (t). Case 3. Finally, for the third multiplier Q 3 = f 2 (t), the corresponding conserved vector (T t 3 , T x 3 , T y 3 ) is T t 3 = u x f 2 (t), T x 3 = 3u 2 x f 2 (t) + 3u x u y f 2 (t) − u f 2 (t) + u x f 2 (t) + u xxx f 2 (t) + u xxy f 2 (t), T y 3 = u y f 2 (t). Remark 3.1. We notice that this method assists in the construction of conservation laws of (1.7) despite the fact that it possesses no variational principle [51]. Moreover, the presence of arbitrary functions in the multiplier indicates that 2D-gBK (1.7) has infinite number of conserved vectors.

Conclusions
In this paper, we carried out a study on two-dimensional generalized Bogoyavlensky-Konopelchenko equation (1.7). We obtained solutions for Eq (1.7) with the use of Lie symmetry reductions, direct integration, Kudryashov's and (G /G)-expansion techniques. We obtained solutions of (1.7) in the form of algebraic, rational, periodic, hyperbolic as well as trigonometric functions. Furthermore, we derived conservation laws of (1.7) by engaging the multiplier method were we obtained three local conserved vectors. We note here that the presence of the arbitrary functions f 1 (t) and f 2 (t) in the multipliers, tells us that one can generate unlimited number of conservation laws for the underlying equation.