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

  • In physics as well as mathematics, nonlinear partial differential 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 differential equations is used to find its symmetries and perform symmetry reductions to obtain ordinary differential equations which are solved to secure possible analytic solutions of the underlying equation. Then we use Kudryashov's and (G/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.

    Citation: Chaudry Masood Khalique, Oke Davies Adeyemo, Kentse Maefo. Symmetry solutions and conservation laws of a new generalized 2D Bogoyavlensky-Konopelchenko equation of plasma physics[J]. AIMS Mathematics, 2022, 7(6): 9767-9788. doi: 10.3934/math.2022544

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  • In physics as well as mathematics, nonlinear partial differential 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 differential equations is used to find its symmetries and perform symmetry reductions to obtain ordinary differential equations which are solved to secure possible analytic solutions of the underlying equation. Then we use Kudryashov's and (G/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.



    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].

    In order to really understand these physical phenomena it is of immense importance to solve NPDEs which govern these aforementioned phenomena. However, there is no general systematic theory that can be applied to NPDEs so that their analytic solutions can be obtained. Nevertheless, in recent times scientists have developed effective techniques to obtain viable analytical solutions to NPDEs, such as inverse scattering transform [16], simple equation method [17], Bäcklund transformation [18], F-expansion technique [19], extended simplest equation method[20], Hirota technique [21], Lie symmetry analysis [22,23,24,25,26,27], bifurcation technique [28,29], the (G/G)-expansion method [30], Darboux transformation [31], sine-Gordon equation expansion technique [32], Kudryashov's method [33], and so on.

    The (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation given as

    utx+6αuxuxx+3βuxuxy+3βuyuxx+αuxxxx+βuxxxy=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 uy=vx in (1.1), the authors integrated the result once to get

    ut+αuxxx+βvxxx+3αu2x+4βuxvx=0,uyvx=0. (1.2)

    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].

    In [40], the 2D generalized BK equation that reads

    utx+k1uxxxx+k2uxxxy+2k1k3k2uxuxx+k3(uxuy)x+γ1uxx+γ2uxy+γ3uyy=0 (1.3)

    was studied and lump-type and lump solutions were constructed by invoking the Hirota bilinear method. Liu et al. [41] applied the Lie group analysis together with (G/G)-expansion and power series methods and obtained some analytic solutions of (1.3).

    Yang et al. [42] recently examined a generalized combined fourth-order soliton equation expressed as

    α(6uxuxx+uxxxx)+β[3(uxut)x+uxxxt]+γ[3(uxuy)x+uxxxy]+δ1uyt+δ2uxx+δ3uxt+δ4uxy+δ5uyy+δ6utt=0, (1.4)

    with constant parameters α,β and γ which are not all zero, whereas all constant coefficients δi,1i6, are arbitrary. It was observed that Eq (1.4) comprises three fourth-order terms and second-order terms that consequently generalizes the standard Kadomtsev-Petviashvili equation. Soliton equations are known to have applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories.

    Assuming α=0, β=1, γ=0 and δ1=δ2=1, δ3=δ4=δ5=δ6=0, the authors gain an integrable (1+2)-dimensional extension of the Hirota-Satsuma equation commonly referred to as the Hirota-Satsuma-Ito equation in two dimensions[43] given as

    uty+uxx+3(uxut)x+uxxxt=0 (1.5)

    that satisfies the Hirota three-soliton condition and also admits a Hirota bilinear structure under logarithmic transformation presented in the form

    u=2(lnf)x,where(D3xDt+DyDt+D2x)ff=0, (1.6)

    whose lump solutions have been calculated in [44]. On taking parameters α=1, β=0, γ=0 along with δ1=δ4=δ6=0 whereas δ2=δ3=δ5=1, they eventually came up with a two dimensional equation[42]:

    utx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy=0, (1.7)

    which is called a two-dimensional generalized Bogoyavlensky-Konopelchenko (2D-gBK) equation. We notice that if one takes α=β=1 in Eq (1.1) with the introduction of two new terms uxx and uyy, 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.

    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.

    A one-parameter Lie group of symmetry transformations associated with the infinitesimal generators related to (\real{gbk}) can be presented as

    ˉt=t+ϵξ1(t,x,y,u)+O(ϵ2),ˉx=x+ϵξ2(t,x,y,u)+O(ϵ2),ˉy=y+ϵξ3(t,x,y,u)+O(ϵ2),ˉu=u+ϵϕ(t,x,y,u)+O(ϵ2). (2.1)

    We calculate symmetry group of 2D-gBK equation (1.7) using the vector field

    R=ξ1(t,x,y,u)t+ξ2(t,x,y,u)x+ξ3(t,x,y,u)y+ϕ(t,x,y,u)u, (2.2)

    where ξi,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

    R[4](utx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy)|Q=0=0, (2.3)

    where Q=utx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy. Here, R[4] denotes the fourth prolongation of R defined by

    R[4]=R+ηtut+ηxux+ηyuy+ηtxutx+ηxxuxx+ηyyuyy+ηxxxxuxxxx+ηxxxyuxxxy, (2.4)

    where coefficient functions ηt, ηx, ηy, ηxt, ηxx, ηxy, ηyy, ηxxxx and ηxxxy 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):

    ξ3x=0,ξ1x=0,ξ1y=0,ξ2u=0,ξ1u=0,ξ3u=0,ξ1tt+5ξ2yy=0,ξ1t+5ϕu=0,5ξ2xξ1t=0,5ξ2y2ξ1t=0,5ξ3y3ξ1t=0,3ϕxxξ3yy=0,ξ3t3ϕx+2ξ2y=0,4ξ1t5ξ2t+30ϕx+15ϕy=0,ϕtx+ϕxx+ϕxxxx+ϕxxxy+ϕyy=0.

    The solution of the above system of PDEs is

    ξ1=A1+A2t,ξ2=F(t)+15A2(x+2y),ξ3=A445A2t+3A3t+35A2y,η=G(t)15A2u+A3x415A2y2A3y+13yF(t),

    where A1A3 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

    R1=t,R2=y,R3=3F(t)x+yF(t)u,R4=3ty+(x2y)u,R5=G(t)u,R6=15tt+(3x+6y)x+(9y12t)y(4y+3u)u. (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]:

    dˉtdϵ=ξ1(ˉt,ˉx,ˉy,ˉu),ˉt|ϵ=0=t,dˉxdϵ=ξ2(ˉt,ˉx,ˉy,ˉu),ˉx|ϵ=0=x,dˉydϵ=ξ3(ˉt,ˉx,ˉy,ˉu),ˉy|ϵ=0=y,dˉudϵ=ϕ(ˉt,ˉx,ˉy,ˉu),ˉu|ϵ=0=u.

    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.

    Theorem 2.1. Let Tiϵ(t,x,y,u),i=1,2,3,,6 be transformations group of one parameter generated by vectors R1,R2,R3,R6 in (2.5), then, for each of the vectors, we have accordingly

    T1ϵ:(˜t,˜x,˜y,˜u)(t+ϵ1,x,y,u),T2ϵ:(˜t,˜x,˜y,˜u)(t,x,y+ϵ2,u),T3ϵ:(˜t,˜x,˜y,˜u)(t,3ϵ3t+x,y,ϵ3y+u),T4ϵ:(˜t,˜x,˜y,˜u)(t,x,3ϵ4t+y,u+(x2y)ϵ43ϵ24t),T5ϵ:(˜t,˜x,˜y,˜u)(t,x,y,ϵ5t+u),T6ϵ:˜t,˜x,˜y,˜u)(te15ϵ6,(2e9ϵ6e3ϵ6e15ϵ6)t+xe3ϵ6+(e9ϵ6e3ϵ6)y,(2e9ϵ62e15ϵ6)t+ye9ϵ6,19[(4e18ϵ66e12ϵ6+2)t+(33e12ϵ6)y+9u]e3ϵ6),

    where ϵR is regarded as the group parameter.

    Theorem 2.2. Hence, suppose u(t,x,y)=Θ(t,x,y) satisfies the 2D-gBK (1.7), in the same vein, the functions given in the structure

    u1(t,x,y)=Θ(tϵ1,x,y,z),u2(t,x,y)=Θ(t,x,yϵ2,z,u),u3(t,x,y)=Θ(t,x3ϵ3t,y)ϵ3y,u4(t,x,y)=Θ(t,x,3ϵ4t+y)(x2y)ϵ4+3ϵ24t,u5(t,x,y)=Θ(t,x,y)ϵ5t,u6(t,x,y)=Θ[te15ϵ6,(2e9ϵ6e3ϵ6e15ϵ6)t+xe3ϵ6+(e9ϵ6e3ϵ6)y,(2e9ϵ62e15ϵ6)t+ye9ϵ6]23te12ϵ6+49te18ϵ6+29t13ye12ϵ6+13y

    will do, where ui(t,x,y)=TϵiΘ(t,x,y),i=1,2,3,,6 with ϵ<<1 regarded as any positive real number.

    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 R1R3

    Taking F(t)=1/3, we linearly combine translational symmetries R1R3 as R=bR1+cR2+aR3 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

    r=ctay,s=cxby,θ=u. (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:

    c2θrs+6c3θsθss+c4θssss3ac2θsθsr6bc2θsθss3ac2θrθssac3θsssrbc3θssss+c2θss+a2θrr+2abθsr+b2θss=0. (2.7)

    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:

    Q1=r,Q2=s,Q3=θ.

    The linear combination Q=Q1+ωQ2 (ω0 being an arbitrary constant) leads to two invariants:

    z=sωr,θ=Θ, (2.8)

    that secures group-invariant solution Θ=Θ(z). Thus, on using these invariants, (2.7) is transformed into the fourth-order nonlinear ODE:

    (c2ωc2+a2ω22baω+b2)Θ(z)6(βbc2βc2aωc3)Θ(z)Θ(z)+(c3aω+c4bc3)Θ(z)=0,

    which we rewrite in a simple structure as

    AΘ(z)BΘ(z)Θ(z)+CΘ(z)=0, (2.9)

    where A=c2ωc2+a2ω22baω+b2, B=6(bc2c2aωc3), C=c3aω+c4bc3 and z=cx+(aωb)ycωt.

    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:

    AΘ(z)12BΘ2(z)+CΘ(z)+C1=0, (2.10)

    where C1 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:

    12AΘ(z)216BΘ(z)3+12CΘ(z)2+C1Θ(z)+C2=0,

    where C2 is a constant of integration. The above equation can be rewritten as

    Θ(z)2=B3CΘ(z)3ACΘ(z)22C1CΘ(z)2C2C. (2.11)

    Letting U(z)=Θ(z), Eq (2.11) becomes

    U(z)2=B3CU(z)3ACU(z)22C1CU(z)2C2C. (2.12)

    Suppose that the cubic equation

    U(z)33ABU(z)26C1BU(z)6C2B=0 (2.13)

    has real roots c1c3 such that c1>c2>c3, then Eq (2.12) can be written as

    U(z)2=B3C(U(z)c1)(U(z)c2)(U(z)c3), (2.14)

    whose solution with regards to Jacobi elliptic function [45,46] is

    U(z)=c2+(c1c2)cn2{B(c1c2)12Cz,Δ2},Δ2=c1c2c1c3, (2.15)

    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

    u(t,x,y)=12C(c1c2)2B(c1c3)Δ8{EllipticE[sn(B(c1c3)12Cz,Δ2),Δ2]}+{c2(c1c2)1Δ4Δ4}z+C3, (2.16)

    with z=cx+(aωb)ycωt and C3 a constant of integration. We note that (2.16) is a general solution of (1.7), where EllipticE[p;q] is the incomplete elliptic integral [46,47] expressed as

    EllipticE[p;q]=p01q2r21r2dr.

    We present wave profile of periodic solution (2.16) in Figure 1 with 3D, contour and 2D plots with parametric values a=4, b=0.2, c=0.1, ω=0.1, c1=100, c2=50.05, c3=60, B=10, C=70, where t=1 and 10x,y10.

    Figure 1.  Elliptic solution wave profile of (2.16) at t=1.

    However, contemplating a special case of (2.9) with B=0, we integrate the equation twice and so we have

    CΘ(z)+AΘ(z)+K1z+K2=0, (2.17)

    where K1 and K2 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

    u(t,x,y)=A1sin(a2ω2ω(2ab+c2)+b2+c2zc3(aωb+c))+A2cos(a2ω2ω(2ab+c2)+b2+c2zc3(aωb+c))K1z+K2a2ω2ω(2ab+c2)+b2+c2, (2.18)

    with A1 and A2 as the integration constants as well as z=cx+(aωb)ycωt. We depict the wave dynamics of periodic solution (2.18) in Figure 2 via 3D, contour and 2D plots with dissimilar parametric values a=1, b=0.2, c=0.1, ω=0.1, A1=20, A2=2, K1=1, K2=10, where t=2 and 10x,y10.

    Figure 2.  Wave profile of the trigonometric function solution (2.18) at t=2.

    B. Weierstrass elliptic solution of 2D-gBK equation (1.7)

    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

    U(z)=W(z)+AB (2.19)

    and transform the nonlinear ordinary differential equation (NODE) (2.12) to

    W2ξ=4W3g2Wg3,ξ=B12Cz, (2.20)

    with the invariants g2 and g3 given by

    g2=12A2B224C1Bandg3=8A3B324AC1B224C2B.

    Thus, we have the solution of NODE (2.12) as

    U(z)=AB+(112C(zz0);g2;g3), (2.21)

    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

    u(t,x,y)=AB(zz0)12BCζ[B12C(zz0);g2,g3], (2.22)

    with arbitrary constant z0, z=cx+(aωb)ycωt 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, z0=0, C=1, C1=1, C2=10, where t=2 and 10x,y10.

    Figure 3.  Wave profile of (2.22) at t=2 of the Weierstrass zeta function solution.

    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

    Θ(z)=Nn=0BnQn(z), (2.23)

    with Q(z) satisfying the first-order NODE

    Q(z)=Q2(z)Q(z). (2.24)

    We observe that the solution of (2.24) is

    Q(z)=11+exp(z). (2.25)

    The balancing procedure for NODE (2.9) produces N=1. Hence, from (2.23), we have

    Θ(z)=B0+B1Q(z). (2.26)

    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 B0 and B1 as

    Q(z)5:2aB1c3ω+aB21c2ω2bB1c3bB21c2+2B1c4+B21c3=0,Q(z)4:2bB1c32aB1c3ωaB21c2ω+bB21c22B1c4B21c3=0,Q(z)3:a2B1ω22abB1ω+25aB1c3ω+12aB21c2ω+b2B125bB1c312bB21c2+25B1c4+12B21c3B1c2ω+B1c2=0,Q(z)2:2abB1ωa2B1ω25aB1c3ω2aB21c2ωb2B1+5bB1c3+2bB21c25B1c42B21c3+B1c2ωB1c2=0,Q(z):a2B1ω22abB1ω+aB1c3ω+b2B1bB1c3+B1c4B1c2ω+B1c2=0. (2.27)

    The solution of the above system gives

    B0=0,B1=2c,a=2bωc3ωc2ω2(c44c2+4ω4)2ω2. (2.28)

    Hence, the solution of 2D-gBK equation (1.7) associated with (2.28) is given as

    u(t,x,y)=2c1+exp{cx+(aωb)ycωt}. (2.29)

    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, B0=0 with t=7 and 6x,y6.

    Figure 4.  The wave profile of solution (2.29) at t=7.

    We reckon the (G/G)-expansion technique [30] in the construction of analytic solutions of 2D-gBK equation (1.7) and so we contemplate a solution structured as

    Θ(z)=Mj=0Bj(Q(z)Q(z))j, (2.30)

    where Q(z) satisfies

    Q(z)+λQ(z)+μQ(z)=0 (2.31)

    with λ and μ taken as constants. Here, B0,,BM are parameters to be determined. Utilization of balancing procedure for (2.9) produces M=1 and as a result, the solution of (1.7) assumes the form

    Θ(z)=B0+B1(Q(z)Q(z)). (2.32)

    Substituting the value of Θ(z) from (2.32) into (2.9) and using (2.31) and following the steps earlier adopted, leads to an algebraic equation in B0 and B1, which splits over various powers of Q(z) to give the system of algebraic equations whose solution is secured as

    B0=0,B1=2c,a=16bωB31λ2ω±Ω0+64B21ω364B21ω2+4B31μω16ω2,

    where Ω0=B61λ4ω28B61λ2μω216B41λ2ω2+16B61μ2ω2+64B41μω2. Thus, we have three types of solutions of the 2D-gBK equation (1.7) given as follows:

    When λ24μ>0, we gain the hyperbolic function solution

    u(t,x,y)=B0+B1(Δ1A1sinh(Δ1z)+A2cosh(Δ1z)A1cosh(Δ1z)+A2sinh(Δ1z)λ2), (2.33)

    with z=cx+(aωb)ycωt, Δ1=12λ24μ together with A1, A2 being arbitrary constants. The wave profile of solution (2.33) is shown in Figure 5 with 3D, contour and 2D plots using parameter values a=3, b=0.5, c=10, ω=0.1, B0=0, λ=0.971, μ=10, A1=5, A2=1, where t=10 and 10x,y10.

    Figure 5.  The wave profile of solution (2.33) at t=10.

    When λ24μ<0, we achieve the trigonometric function solution

    u(t,x,y)=B0+B1(Δ2A2cos(Δ2z)A1sin(Δ2z)A1cos(Δ2z)+A2sin(Δ2z)λ2), (2.34)

    with z=cx+(aωb)ycωt, Δ2=124μλ2 together with A1 and A2 are arbitrary constants. The wave profile of solution (2.34) is shown in Figure 6 with 3D, contour and 2D plots using parameter values a=1, b=0.5, c=0.3, ω=0.3, B0=0, λ=0.971, μ=2, A1=5, A2=1 with t=10 and 10x,y10.

    Figure 6.  The wave profile of solution (2.34) at t=10.

    When λ24μ=0, we gain the rational function solution

    u(t,x,y)=B0+B1(A2A1+A2zλ2), (2.35)

    with z=cx+(aωb)ycωt and A1, A2 being arbitrary constants. We plot the graph of solution (2.35) in Figure 7 via 3D, contour and 2D plots using parametric values a=1, b=1.01, c=100, ω=0.1, B0=10, λ=10, A1=3, A2=10, where t=2.4 and 5x,y5.

    Figure 7.  The wave profile of solution (2.35) at t=2.4.

    Case 2. Group-invariant solutions via R4

    Lagrange system associated with the symmetry R4=3t/y+(x2y)/u is

    dt0=dx0=dy3t=du(x2y), (2.36)

    which leads to the three invariants T=t,X=x,Q=u+(y2/3t)(xy/3t). Using these three invariants, the 2D-gBK equation (1.7) is reduced to

    18TQXQXX+3TQTX+3TQXX+3XQXX+3QX+3TQXXXX2=0. (2.37)

    Case 3. Group-invariant solutions via R1,R2andR5

    We take G(t)=1 and by combining the generators R1,R2 as well as R5, we solve the characteristic equations corresponding to the combination and get the invariants X=x, Y=yt with group-invariant u=Q(X,Y)+t. With these invariants, the 2D-gBK equation (1.7) transforms to the NPDE

    QXX+QYYQXY+3QXQXY+3QYQXX+6QXQXX+QXXXX+QXXXY=0, (2.38)

    whose solution is given by

    Q(X,Y)=2A2tanh[A2X+A2(121216A4224A2232A22)Y+A1]+A3, (2.39)

    with arbitrary constants A1A3. Thus, we achieve the hyperbolic solution of (1.7) as

    u(t,x,y)=t+2A2tanh[12A2(ty)16A4224A223+12(4t4y)A32+12(y+2xt)A2+A1]+A3. (2.40)

    The wave profile of solution (2.40) is shown in Figure 8 with 3D, contour and 2D plots using parameter values A1=70.1, A2=30, A3=0, where t=0.5 and 10x,y10.

    Figure 8.  The wave profile of solution (2.40) at t=0.5.

    Besides, symmetries of (2.38) are found as

    P1=X,P2=Y,P3=Q,P4=(13X+23Y)X+YY+(23X2Y13Q)Q.

    Now, the symmetry P1 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

    u(t,x,y)=t+A0(yt)+A1, (2.41)

    with A0, A1 as constants. Further, the symmetry P2 yields Q(X,Y)=f(z), z=X and so Eq (2.38) reduces to

    f(z)+6f(z)f(z)+f(z)=0. (2.42)

    Integration of the above equation three times with respect to z gives

    f(z)2+2f(z)3+f(z)2+2A0f(z)+2A1=0, (2.43)

    and taking constants A0=A1=0 and then integrating it results in the solution of (1.7) as

    u(t,x,y)=t12{1+tan(12A112x)2}. (2.44)

    The wave profile of solution (2.44) is shown in Figure 9 with 3D, contour and 2D plots using parameter values A1=40, t=3.5 and 10x10.

    Figure 9.  The wave profile of solution (2.44) at t=3.5.

    On combining P1P3 as P=c0P1+c1P2+c2P3, we accomplish

    Q(X,Y)=c2c0X+f(z),wherez=c0Yc1X. (2.45)

    Using the newly acquired invariants (2.45), Eq (2.38) transforms to the NODE:

    c0c21f(z)+6c21c2f(z)3c0c1c2f(z)+c20c1f(z)+c30f(z)+6c20c21f(z)f(z)6c0c31f(z)f(z)+c0c41f(z)c20c31f(z)=0. (2.46)

    Engaging the Lie point symmetry P4, we obtain

    Q(X,Y)=X2Y+Y1/3f(z)withz=Y1/3(XY), (2.47)

    and Eq (2.38) reduces to the NODE

    6zf(z)+z2f(z)18f(z)29f(z)f(z)+4f(z)18zf(z)f(z)12f(z)3zf(z)=0. (2.48)

    Case 4. Group-invariant solutions via R6

    Lie point symmetry R6 dissociates to the Lagrange system

    dt15t=dx3x+6y=dy9y12t=du(4y+3u),

    which gives

    u=t1/5Q(T,X)29t13y,withT=(2t+y)t3/5andX=(xty)t1/5. (2.49)

    Substituting the expression of u in (1.7), we obtain the NPDE

    5QTT3TQTXXQXX2QX+15QXQTX+15QTQXX+5QTXXX=0, (2.50)

    which has two symmetries:

    P1=Q,P2=X+115TQ.

    The symmetry P2 gives Q(X,Y)=f(z)+(1/15)TX, z=T and hence (2.50) reduces to

    75f(z)4z=0.

    Solving the above ODE and reverting to the basic variables gives the solution of (1.7) as

    u(t,x,y)=15t{(2t+y)(xty)15t4/5+2(2t+y)3225t9/5+(2t+y)t3/5A1+A2}2t9y3, (2.51)

    where A1 and A2 are integration constants. The wave profile of solution (2.51) is shown in Figure 10 with 3D, contour and 2D plots using parameter values A1=0.3, A2=50 with t=1.1 and 10x,y10.

    Figure 10.  The wave profile of solution (2.51) at t=1.1.

    Next, we invoke the symmetry P1+P2. This yields Q(X,Y)=f(z)+X+(1/15)TX,z=T. Consequently, we have the transformed version of (2.50) as

    75f(z)4z15=0.

    Solving the above ODE and reverting to basic variables gives the solution of (1.7) as

    u(t,x,y)=15t{(2t+y)(xty)15t4/5+2(2t+y)3225t9/5+(2t+y)210t6/5+(2t+y)t3/5A1+A2}2t9y3. (2.52)

    The wave profile of solution (2.52) is shown in Figure 11 with 3D, contour and 2D plots using parameter values A1=3.6, A2=50 with t=1.1 and 10x,y10.

    Figure 11.  The wave profile of solution (2.52) at t=1.1.

    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.

    Consider the n independent variables x=(x1,x2,,xn) and m dependent variables u=(u1,u2,,um). The derivatives of u with respect to x are defined as

    uαi=Di(uα),uαij=DjDi(ui),, (3.1)

    where

    Di=xi+uαiuα+uαijuαj+,i=1,...,n, (3.2)

    is the operator of total differentiation. The collection of all first derivatives uαi is denoted by u(1), i.e., u(1)={uαi},α=1,...,m,i=1,...,n. In the same vein u(2)={uαij},α=1,...,m,i,j=1,...,n and u(3)={uαijk} and likewise u(4) etc. Since uαij=uαji, u(2) contains only uαij for ij.

    Now consider a kth-order system of PDEs:

    Gα(x,u,u(1),,u(k))=0,α=1,2,,m. (3.3)

    The {Euler-Lagrange operator}, for every α, is presented as

    δδuα=uα+s1(1)sDi1Disuαi1i2is,α=1,,m. (3.4)

    An n-tuple T=(T1,T2,,Tn), such that

    DiTi=0 (3.5)

    holds for all solutions of (3.3) is known as the conserved vector of system (3.3).

    The multiplier Ωα(x,u,u(1),) of system (3.3) has the property that

    DiTi=ΩαGα (3.6)

    holds identically [22]. The determining equations for multipliers are obtained by taking the variational derivative of (3.6), namely

    δδuα(ΩαGα)=0. (3.7)

    The moment multipliers are generated from (3.7), the conserved vectors can be derived systematically using (3.6) as the determining equation.

    Conservation laws of 2D-gBK equation (1.7) are derived by utilizing second-order multiplier Ω(t,x,y,u,ut,ux,uy,uxx,uxy), in Eq (3.7), where G is given as

    Gutx+6uxuxx+uxxxx+uxxxy+3(uxuy)x+uxx+uyy,

    and the Euler operator δ/δu is expressed in this case as\newpage

    δδu=uDtutDxuxDyuy+DtDxutx+DxDyuxy+D2xuxx+D2yuyy+D4xuxxxx+D3xDyuxxxy.

    Expansion of Eq (3.7) and splitting on diverse derivatives of dependent variable u gives

    Ωu=0,Ωx=0,Ωyy=0,Ωyux=0,Ωuxux=0,Ωtux=0,Ωut=0,Ωuxx=0,Ωuxy=0,Ωuy=0. (3.8)

    Solution to the above system of equations gives Ω(t,x,y,u,ut,ux,uy,uxx,uxy) as

    Ω(t,x,y,u,ut,ux,uy,uxx,uxy)=C1ux+f1(t)y+f2(t), (3.9)

    with C1 being an arbitrary constant and f1(t), f2(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 ux,f1(t) and f2(t):

    Case 1. For the first multiplier Q1=ux, the corresponding conserved vector (Tt1,Tx1,Ty1) is given by

    Tt1=12u2x,Tx1=12u2x+2u3x12u2xx12uxxuxy+12uxuxxy+uxxxux+12uxxxuy+12uuxxxy+12uuyy+uuxuxy+2uyu2x,Ty1=12uyuxuuxuxx12uuxy12uuxxxx.

    Case 2. For the second multiplier Q2=f1(t), we obtain the corresponding conserved vector (Tt2,Tx2,Ty2) as

    Tt2=uxf1(t)y,Tx2=3yf1(t)u2x+3yf1(t)uxuyyf1(t)u+yf1(t)ux+yf1(t)uxxx+yf1(t)uxxy,Ty2=uyf1(t)yuf1(t).

    Case 3. Finally, for the third multiplier Q3=f2(t), the corresponding conserved vector (Tt3,Tx3,Ty3) is

    Tt3=uxf2(t),Tx3=3u2xf2(t)+3uxuyf2(t)uf2(t)+uxf2(t)+uxxxf2(t)+uxxyf2(t),Ty3=uyf2(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.

    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 f1(t) and f2(t) in the multipliers, tells us that one can generate unlimited number of conservation laws for the underlying equation.

    The authors state no conflicts of interest.



    [1] A. I. Morozov, Introduction to plasma dynamics, Boca Raton, Florida: CRC Press, 2012.
    [2] O. D. Adeyemo, T. Motsepa, C. M. Khalique, A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences, Alex. Eng. J., 61 (2022), 185–194. https://doi.org/10.1016/j.aej.2021.04.066 doi: 10.1016/j.aej.2021.04.066
    [3] C. M. Khalique, O. D. Adeyemo, A study of (3+1)-dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation via Lie symmetry approach, Results Phys., 18 (2020), 103197. https://doi.org/10.1016/j.rinp.2020.103197 doi: 10.1016/j.rinp.2020.103197
    [4] A. Shafiq, C. M. Khalique, Lie group analysis of upper convected Maxwell fluid flow along stretching surface, Alex. Eng. J., 59 (2020), 2533–2541. https://doi.org/10.1016/j.aej.2020.04.017 doi: 10.1016/j.aej.2020.04.017
    [5] N. Benoudina, Y. Zhang, C. M. Khalique, Lie symmetry analysis, optimal system, new solitary wave solutions and conservation laws of the Pavlov equation, Commun. Nonlinear Sci. Numer. Simul., 94 (2021), 105560. https://doi.org/10.1016/j.cnsns.2020.105560 doi: 10.1016/j.cnsns.2020.105560
    [6] J. J. Li, G. Singh, O. A. İlhan, J. Manafian, Y. S. Gasimov, Modulational instability, multiple exp-function method, SIVP, solitary and cross-kink solutions for the generalized KP equation, AIMS Math., 6 (2021), 7555–7584. https://doi.org/10.3934/math.2021441 doi: 10.3934/math.2021441
    [7] E. Alimirzaluo, M. Nadjafikhah, J. Manafian, Some new exact solutions of (3+1)-dimensional Burgers system via Lie symmetry analysis, Adv. Differ. Equ., 2021 (2021), 1–17. https://doi.org/10.1186/s13662-021-03220-3 doi: 10.1186/s13662-021-03220-3
    [8] P. G. Estévez, J. D. Lejarreta, C. Sardón, Symmetry computation and reduction of a wave model in (2+1)-dimensions, Nonlinear Dyn., 87 (2017), 13–23. https://doi.org/10.1007/s11071-016-2997-5 doi: 10.1007/s11071-016-2997-5
    [9] B. Muatjetjeja, D. M. Mothibi, C. M. Khalique, Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2803–2812. https://doi.org/10.3934/dcdss.2020219 doi: 10.3934/dcdss.2020219
    [10] C. M. Khalique, I. Simbanefayi, Conserved quantities, optimal system and explicit solutions of a (1+1)-dimensional generalised coupled mKdV-type system, J. Adv. Res., 29 (2020), 159–166. https://doi.org/10.1016/j.jare.2020.10.002 doi: 10.1016/j.jare.2020.10.002
    [11] R. J. Leveque, Numerical methods for conservation laws, Basel: Birkhäuser, 1992. https://doi.org/10.1007/978-3-0348-8629-1
    [12] W. Sarlet, Comment on 'Conservation laws of higher order nonlinear PDEs and the variational conservation laws in the class with mixed derivatives', J. Phys. A Math. Theor., 43 (2010), 458001. https://doi.org/10.1088/1751-8113/43/45/458001 doi: 10.1088/1751-8113/43/45/458001
    [13] T. Motsepa, M. Abudiab, C. M. Khalique, A study of an extended generalized (2+1)-dimensional Jaulent-Miodek equation, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 391–395. https://doi.org/10.1515/ijnsns-2017-0147 doi: 10.1515/ijnsns-2017-0147
    [14] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311–328. https://doi.org/10.1016/j.jmaa.2006.10.078 doi: 10.1016/j.jmaa.2006.10.078
    [15] C. M. Khalique, K. Maefo, A study on the (2+1)-dimensional first extended Calogero-Bogoyavlenskii-Schiff equation, Math. Biosci. Eng., 18 (2021), 5816–5835. https://doi.org/10.3934/mbe.2021293 doi: 10.3934/mbe.2021293
    [16] M. J. Ablowitz, P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, 1991.
    [17] N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Soliton. Fract., 24 (2005), 1217–1231. https://doi.org/10.1016/j.chaos.2004.09.109 doi: 10.1016/j.chaos.2004.09.109
    [18] C. H. Gu, Soliton theory and its application, Zhejiang Science and Technology Press, 1990.
    [19] Y. B. Zhou, M. L. Wang, Y. M. Wang, Periodic wave solutions to a coupled KdV equations with variable coefficients, Phys. Lett. A, 308 (2003), 31–36. https://doi.org/10.1016/S0375-9601(02)01775-9 doi: 10.1016/S0375-9601(02)01775-9
    [20] N. A. Kudryashov, N. B. Loguinova, Extended simplest equation method for nonlinear differential equations, Appl. Math. Comput., 205 (2008), 396–402. https://doi.org/10.1016/j.amc.2008.08.019 doi: 10.1016/j.amc.2008.08.019
    [21] R. Hirota, The direct method in soliton theory, Cambridge University Press, 2004.
    [22] P. J. Olver, Applications of Lie groups to differential equations, New York: Springer, 1993.
    [23] N. H. Ibragimov, CRC handbook of Lie group analysis of differential equations, Boca Raton, Florida: CRC Press, 1995.
    [24] N. H. Ibragimov, Elementary Lie group analysis and ordinary differential equations, New York: Wiley, 1999.
    [25] G. W. Wang, X. Q. Liu, Y. Y. Zhang, Symmetry reduction, exact solutions and conservation laws of a new fifth-order nonlinear integrable equation, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2313–2320. https://doi.org/10.1016/j.cnsns.2012.12.003 doi: 10.1016/j.cnsns.2012.12.003
    [26] H. Z. Liu, J. B. Li, Lie symmetry analysis and exact solutions for the short pulse equation, Nonlinear Anal., 71 (2009), 2126–2133. https://doi.org/10.1016/j.na.2009.01.075 doi: 10.1016/j.na.2009.01.075
    [27] H. Z. Liu, J. B. Li, Q. X. Zhang, Lie symmetry analysis and exact explicit solutions for general Burgers' equation, J. Comput. Appl. Math., 228 (2009), 1–9. https://doi.org/10.1016/j.cam.2008.06.009 doi: 10.1016/j.cam.2008.06.009
    [28] S. N. Chow, J. K. Hale, Methods of bifurcation theory, New York: Springer, 1982. https://doi.org/10.1007/978-1-4613-8159-4
    [29] L. J. Zhang, C. M. Khalique, Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 759–772. https://doi.org/10.3934/dcdss.2018048 doi: 10.3934/dcdss.2018048
    [30] M. L. Wang, X. Z. Li, J. L. Zhang, The (G/G)-expansion method and travelling wave solutions for linear evolution equations in mathematical physics, Phys. Lett. A, 24 (2005), 1257–1268.
    [31] V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Berlin: Springer, 1991.
    [32] Y. Chen, Z. Y. Yan, New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method, Chaos Soliton. Fract., 26 (2005), 399–406. https://doi.org/10.1016/j.chaos.2005.01.004 doi: 10.1016/j.chaos.2005.01.004
    [33] N. A. Kudryashov, One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2248–2253. https://doi.org/10.1016/j.cnsns.2011.10.016 doi: 10.1016/j.cnsns.2011.10.016
    [34] O. I. Bogoyavlenskiĭ, Overturning solitons in new two-dimensional integrable equations, Math. USSR Izv., 34 (1990), 245–259.
    [35] B. G. Konopelchenko, Solitons in multidimensions: Inverse spectral transform method, Singapore: World Scientific, 1993.
    [36] M. V. Prabhakar, H. Bhate, Exact solutions of the Bogoyavlensky-Konoplechenco equation, Lett. Math. Phys., 64 (2003), 1–6. https://doi.org/10.1023/A:1024909327151 doi: 10.1023/A:1024909327151
    [37] S. S. Ray, On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky-Konopelchenko equation in wave propagation, Comput. Math. Appl., 74 (2017), 1158–1165. https://doi.org/10.1016/j.camwa.2017.06.007 doi: 10.1016/j.camwa.2017.06.007
    [38] F. Calogero, A method to generate solvable nonlinear evolution equations, Lett. Nuovo Cimento, 14 (1975), 443–447. https://doi.org/10.1007/BF02763113 doi: 10.1007/BF02763113
    [39] K. Toda, S. J. Yu, A study of the construction of equations in (2+1) dimensions, Inverse Probl., 17 (2001), 1053.
    [40] Q. Li, T. Chaolu, Y. H. Wang, Lump-type solutions and lump solutions for the (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation, Comput. Math. Appl., 77 (2019), 2077–2085. https://doi.org/10.1016/j.camwa.2018.12.011 doi: 10.1016/j.camwa.2018.12.011
    [41] F. Y. Liu, Y. T. Gao, X. Yu, L. Q. Li, C. C. Ding, D. Wang, Lie group analysis and analytic solutions for a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation in fluid mechanics and plasma physics, Eur. Phys. J. Plus, 136 (2021), 1–14. https://doi.org/10.1140/epjp/s13360-021-01469-x doi: 10.1140/epjp/s13360-021-01469-x
    [42] J. Y. Yang, W. X. Ma, C. M. Khalique, Determining lump solutions for a combined soliton equation in (2+1)-dimensions, Eur. Phys. J. Plus, 135 (2020), 1–13. https://doi.org/10.1140/epjp/s13360-020-00463-z doi: 10.1140/epjp/s13360-020-00463-z
    [43] Y. Kosmann-Schwarzbach, B. Grammaticos, K. M. Tamizhmani, Integrability of nonlinear systems, Berlin, Heidelberg: Springer, 2004. https://doi.org/10.1007/b94605
    [44] S. T. Chen, W. X. Ma, Lump solutions of a generalized Calogero-Bogoyavlenskii-Schiff equation, Comput. Math. Appl., 76 (2018), 1680–1685. https://doi.org/10.1016/j.camwa.2018.07.019 doi: 10.1016/j.camwa.2018.07.019
    [45] N. A. Kudryashov, Analytical theory of nonlinear differential equations, Moskow-Igevsk, Institute of Computer Investigations, 2004.
    [46] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, New York: Academic Press, 2007.
    [47] N. I. Akhiezer, Elements of the theory of elliptic functions, Providence, Rhode Island: American Mathematical Society, 1990.
    [48] N. A. Kudryashov, First integrals and general solution of the Fokas-Lenells equation, Optik, 195 (2019), 163135. https://doi.org/10.1016/j.ijleo.2019.163135 doi: 10.1016/j.ijleo.2019.163135
    [49] S. C. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications, Eur. J. Appl. Math., 13 (2002), 545–566. https://doi.org/10.1017/S095679250100465X doi: 10.1017/S095679250100465X
    [50] S. C. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: General treatment, Eur. J. Appl. Math., 13 (2002), 567–585. https://doi.org/10.1017/S0956792501004661 doi: 10.1017/S0956792501004661
    [51] E. Noether, Invariante variationsprobleme, Nachr. Ges. Wiss. Göttingen Math. Phys. Kl., 1918 (1918), 235–257.
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