# Conservation Laws and Exact Solutions of Generalized Nonlinear System and Nizhink-Novikov-Veselov Equation

Authors: A. A. Zaidi; M. D. Khan; I. Naeem
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The Lie symmetries, conservation laws, and exact solutions of a generalized nonlinear system and a (2+1)-dimensional generalized Nizhink-Novikov-Veselov (NNV) equation, arising in the study of hydrodynamics, are investigated. The multiplier approach is employed to compute the conservation laws for systems under consideration. The Lie point symmetries are derived and the association between symmetries and conserved vectors are established using symmetries conservation laws relationship. The double reduction theory is utilized which results in the reduction and exact solutions of models under investigation. All cases are discussed in detail and new solutions are determined.


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## Body

## 1. Introduction

Noether [1] formulated a general approach for constructing conservation laws for Euler-Lagrange equations when Lie-Bäcklund symmetries came to the fore. Noether’s approach not only established a connection between conservation laws and symmetry properties of a physical system, but also provided a basis for the application of double reduction method. However, its applicability remained limited to those physical systems which have a Lagrangian formulation. Kara and Mahomed [2] relaxed this condition by introducing the concept of partial Lagrangian and associated Noether type symmetries with conservation laws. However, a connection between other types of symmetries and conservation laws remained elusive. Sjöberg [3, 4] developed the theory of double reduction for two independent variables. Bokhari et al. [5] then generalized this idea of reductions using conservation laws for k independent variables.Symmetries play an important role in many areas of physical sciences. In particular, hydrodynamics has remained an area of great interest owing to its applications in physics. Most of the studies have focused on providing an underlying mathematical theory by establishing empirical and semiempirical laws derived from the flow measurement of fluids and using them to solve real world problems. Mathematical models thus obtained provide vital information about various hydrological processes including the designing of dams, harbours, and bridges. These models usually entail systems of nonlinear partial differential equations (PDEs). Consequently, developing techniques for exact solutions is crucial to the analysis of such systems.A system involving two nonlinear PDEs,(1)vtt,x+avt,xvxt,x+bux=0,utt,x+aut,xvt,xx+cvxxxt,x=0,where a, b, and c are constants, arises in various physical phenomena in physics and engineering. As noted by Congy et al. [6], for a=1, b=1, and c=-1/4, the corresponding Kaup-Boussinesq (KB) system models the physics of shallow water waves. In this context, u denotes the local height of the water layer and v denotes the local mean flow velocity. The corresponding KB system is in fact a “positive dispersion” system, which also models the propagation of capillary waves on top of a thin fluid layer [7]. System (1) also serves as an approximation to various equations arising in physics and engineering. For instance, (1) provides an approximation to the Landau-Lifshitz equation, which models the propagation of magnetization waves in easy plane magnets. Similarly, the choice of coefficients a=1, b=1, and c=1/3 models a (1+1)-dimensional dispersive long wave equation [8], and the values a=-2, b=-1, c=-1 yield the Kaup-Boussinesq system [9]. System (1) also models the long wave phenomena in shallow water, where the depth of water is considered to be uniform and u(t,x) and v(t,x) denote the elevation of water waves and the surface velocity of water along x-direction, respectively. Further applications of (1) can be found in [10–13] and some more recent works are given in [14] where soliton solutions are determined using Hirota’s direct method and in [15] where symmetric analysis is used.Another PDE that arises in the study of fluid dynamics is(2)ut+kuxxx+ruyyy+sux+quy=3ku∂y-1uxx+3ru∂x-1uyy,where ∂-1 denotes the integral with respect to the subscripts, u=u(t,x,y) represents a physical field, and k,s,q, and r are specified constants.The integro-partial differential equation (2) is the well-known generalized Nizhnik-Novikov-Veselov (GNNV) equation [16, 17] and is essentially a symmetric generalization of the famous KdV equation to the(2+1)dimensions [18], which has numerous applications in engineering and physics including the electrical transmission lines, surface gravity waves, internal solitons in the ocean, nonlinear acoustics of bubbly liquids, voidage slugs in fluidized beds, and the “Great Red spot” in Jupiter. For a detailed account of these applications, we refer the reader to [19]. Eq. (2) can be simplified using the transformations ux=vy and uy=wx, where v and w are functions of t,x, and y. This yields the system(3)ut+kuxxx+ruyyy+sux+quy=3kuvx+3ruwy,ux=vy,uy=wx.System (3) also arises in applications in plasma physics, optics, and condense matter physics and has been studied extensively. For instance, Boitiet al. [20] used inverse scattering method to procure solutions to (2), and Radha et al. [21] employed Painleve analysis and bilinear method to find multi-dromion solutions to (2). Some recent works on the GNNV equation can be found in [22, 23].In this paper, we provide a double reduction analysis of the system of PDEs (1) and (3). The double reduction theory not only reduces the order and the number of independent variables involved but also makes use of symmetry, if it exists, to reduce a PDE further. Some of the solutions to (1) and (3) are of a “travelling wave” type and are based on a well-known substitution, which is possible only if translational symmetries exist. In double reduction, not only are travelling wave type solutions obtained by making use of translational symmetries but also solutions are obtained by forming an association of the symmetry present with conserved vectors of the equation. This association is not only limited to the translational symmetries and it may exist for a wide range of symmetries like rotations, dilatations, inversions. Here, we exploit the dilation symmetry of the systems to extract new exact solutions. The interested reader is referred to some recent works in the field of symmetries, conservation laws, and the double reduction theory (see, e.g., [24, 25]).In Section2, we provide a double reduction analysis of (1). We procure exact solutions to (1) by finding Lie symmetries and conservation laws. In Section 3, we extend this work to system (3). The results are discussed in Section 4. The conclusions are presented in Section 5.


## 2. Generalized Nonlinear System

We use the theory of double reduction to compute the exact solutions to (1) by first finding the conservation laws using the variational derivative approach. Many software packages are available to compute Lie symmetries of differential equations. For instance, the mathematical software Maple can be employed to obtain the Lie symmetries to (1). This yields(4)X1=∂∂t,X2=∂∂x,X3=at∂∂x+∂∂v,X4=-2t∂∂t-x∂∂x+2u∂∂u+v∂∂v.We investigate the conservation laws of (1) using the multiplier approach. The multiplier Λα=Λα(x,u,u(1),…,u(σ)) satisfies (see [26, 27])(5)ΛαΔα=DiTi,where T is the conserved vector and(6)Di=∂∂xi+uiθ∂∂uθ+uijθ∂∂ujθ+⋯,i=1,⋯n,θ=1,⋯m,is the total derivative operator. The repeated indices throughout indicate the summation convention. The total derivative operator for two independent variables (t,x) and two dependent variables (u,v) takes the form(7)Dt=∂∂t+ut∂∂u+vt∂∂v+utt∂∂ut+utx∂∂ux+vtt∂∂vt+vtx∂∂vx+…,Dx=∂∂x+ux∂∂u+vx∂∂v+uxt∂∂ut+uxx∂∂ux+vxt∂∂vt+vxx∂∂vx+…The variational derivative of (5) yields(8)EuΛαΔα=0,where Eu is the Euler operator and is given by(9)Eu=∂∂uα+∑s≥1-1sDi1…Dis∂∂ui1…isα.α=1,2,…,m.The Euler operatorEu for two independent variables (t,x) and two dependent variables (u,v) leads to(10)Eu=∂∂u-Dt∂∂ut-Dx∂∂ux+Dt2∂∂utt+DtDx∂∂utx+Dx2∂∂uxx-…,where total derivative operators Dt and Dx are defined above.Relation (8) gives(11)EuΛ1vt+avvx+bux+Λ2ut+auvx+cvxxx=0EvΛ1vt+avvx+bux+Λ2ut+auvx+cvxxx=0,where Λ1=Λ1(t,x,u,v) and Λ2=Λ2(t,x,u,v). Equation (11) results in(12)Λ1uvt+avvx+bux+Λ2uut+auvx+cvxxx+aΛ2vx-DxbΛ1+aΛ2v-DtΛ2=0,Λ1vvt+avvx+bux+Λ2vut+auvx+cvxxx+aΛ1vx+Λ2aux-DxavΛ1+auΛ2-DtΛ1+Dx3Λ2=0.The separation of (12) with respect to the derivatives of u and v yields the system(13)Λ1x=0,Λ1v=0,Λ2u=0,Λ2xx=0,Λ2xv=0,Λ2vv=0,Λ1u=Λ2v,Λ1t=-auΛ2x,Λ2t=-avΛ2x,so that(14)Λ1t,x,u,v=-atuc1+uc2+c4,Λ2t,x,u,v=x-atvc1+c2v+c3,where c1, c2, c3, and c4 are arbitrary constants. Four sets of multipliers can be obtained by setting one of the constants equal to 1 and the rest to zero. This yields(15)Λ11t,x,u,v=-atu,Λ21t,x,u,v=x-atv,Λ12t,x,u,v=u,Λ22t,x,u,v=v,Λ13t,x,u,v=0,Λ23t,x,u,v=1,Λ14t,x,u,v=1,Λ24t,x,u,v=0.The conserved vectors corresponding to these multipliers are(16)T1t=-atuv+xu,T1x=-a2tuv2+axuv-12abtu2-actvvxx+12actvx2+cxvxx-cvx,T2t=uv,T2x=auv2+12bu2+cvvxx-12cvx2,T3t=u,T3x=auv+cvxx,T4t=v,T4x=12av2+bu.The above analysis shows that system (1) admits four symmetries and four conserved vectors.We now compute the exact solutions of (1) by utilizing the double reduction theory. A Lie-Bäcklund symmetry generator X is associated with a conserved vector T of system (1) if it satisfies(17)XTi-TjDjξi+TiDjξj=0,i=1,2,…,n.The association between symmetry and conservation laws of system (1) is presented in Table 1.Table 1
Association between symmetries and conserved vectors.
Symmetries T 1 T 2 T 3 T 4 X 1 NA A A A X 2 NA A A A X 3 A NA A NA X 4 A NA NA A A=associated; NA=not associated.Case 1 (solution of (1) using X3).
LetX be any symmetry generator and T be a conserved vector associated with X. Under a similarity transformation of X, there exists T~ such that D~jT~j=0, where T~ can be calculated (see [5]) from the system(18)T~1T~2⋮T~n=JA-1TT1T2⋮Tn,where(19)A=D~1x1D~1x2…D~1xnD~2x1D~2x2…D~2xn⋮⋮⋮⋮D~nx1D~nx2…D~nxn,A-1=D1x~1D1x~2…D1x~nD2x~1D2x~2…D2x~n⋮⋮⋮⋮Dnx~1Dnx~2…Dnx~n,J=A.Relation (17) shows that the Lie symmetry X3 is associated with T1 and T3. The change of variables(20)s=xat,r=t,Ar=-x+atvat,Br=u.reduces the generator X3=at∂/∂x+∂/∂v to the canonical form ∂/∂s. For two independent variables, (18) reduces to the conserved form(21)TrTs=JA-1TTtTx,where(22)A-1=DtrDtsDxrDxs,J=det⁡A.The r-components of T1 and T3 are obtained by exploiting relations (21) and (22). This gives(23)T1r=-a2r2AB,T3r=arB,so that(24)-a2r2AB=g7,(25)arB=g8,where g7 and g8 are constants. Equations (24) and (25) yield(26)Ar=-g7g8ar,Br=g8ar.Consequently, the solution to (1) is(27)u=g8at,v=-g7+xg8atg8.Case 2 (solution of (1) using X1 and X2).
The symmetriesX1 and X2 are associated with T2, T3, and T4 (see Table 1). We derive the reduced conserved form by considering a combination of X1 and X2, namely, X=αX1+X2. The generator X has a canonical form ∂/∂s when(28)dtα=dx1=du0=dv0=dr0=ds1=dw0=dk0.The characteristic equation (28) yields(29)r=αx-t,s=x,wr=u,kr=v.The conserved vectors T2,T3, and T4, using (29) and (21)-(22), reduce to(30)T2r=wk-αawk2-12αbw2-α3ckkrr+12cα3kr2,T2s=-awk2-12bw2-α2ckkrr-12cα2kr2,(31)T3r=w-αawk-α3ckrrT3s=-awk+cα2krr,(32)T4r=k-12αak2-αbwT4s=-12ak2-bw.The conserved vectors T2r, T3r, and T4r satisfy the reduced conserved form(33)DrTr=0.Consequently (30), (31), and (32) give(34)wk-αawk2-12αbw2-α3ckkrr+12cα3kr2=d1,(35)w-αawk-α3ckrr=d2,(36)k-12αak2-αbw=d3,where d1, d2, and d3 are arbitrary constants. Any pair of equations in the reduced conserved form (34)-(36) can be used to construct a solution to (1) with the help of Maple. For instance, (34) and (35) give(37)wr=d2+α3ckrr1-αak,and(38)±∫2bcaα3e2aαc1bc1+8bα3a2d1e4aαc1±4aαke2aαc1∓2a2α2k2e2aαc1+8a2bd2α3ke4aαc1+a4α4k4e4aαc1-4aαke4aαc1+6a2α2k2e4aαc1-4a3α3k3e4aαc1+e4aαc1∓2e2aαc1-1/2dk-r-g2=0.Another exact solution to (1) can be gleaned using (34) and (36). This gives(39)k1r=1aα1+tan⁡2aαd3-12α2bcr+c12aαd3-1,w1r=-1α2baaαd3-12sec22aαd3-12α2bcr+c1,and(40)k2r=1aα1-tan⁡2aαd3-12α2bcr+c12aαd3-1,w2r=1-2aαd32α2basec22aαd3-12α2bcr+c1,d1=0.It can be verified directly that solutions (39) and (40) satisfy (1). The same solution is obtained from (35) and (36) by setting d2=0 and solving the equations simultaneously.The transformationr=αx-t introduced in (29) describes one-dimensional travelling wave, where α is the wave number and 1/α is the wave speed. If we choose a=1,b=1, and c=1/3, then system (1) is identified as dispersive long wave system [8]. Here a is a nonlinearity parameter and b is dispersive parameter. If we set 1/α>2ad3, then the exact solution (39) and (40) corresponds for internal waves. If the wave speed 1/α is chosen to be 1.01 and d3=-0.51, then the elevation in (39) of waves is shown in Figure 1.Figure 1If1/α<2ad3, then the solution (39) represents the rogue waves. This situation is depicted in Figure 2, where we assume α=1 and d3=1.Figure 2The elevation functionu(t,x) of (39) is the same as (40).Case 3 (solution of (1) using X4).
Relation (17) shows that the symmetry X4 is associated with T1 and T4. Canonical coordinates of X4 are (41)r=xt,s=-12ln⁡t,Ar=vt,Br=tu.Equations (21) and (22) along with (41) can be used to procure the reduced conserved form of vectors T1 and T4. This yields(42)3arAB-r2B-2a2A2B-abB2-2acAArr+acAr2+2crArr-2cAr=g9,and(43)-rA+aA2+2bB=g10.Since (42) and (43) do not admit any symmetry, the double reduction theory can not be applied to find the exact solution of (1). However, numerical methods can be employed to obtain approximate solutions (Figure 3).Figure 3
Solution profile ofM1 where k=1, r=1, s=-1, q=1, γ=-1, α=-1, β=-1. This sequence of frames is constructed from time t=0 to t=10.


## 3. (2+1)-Dimensional GNNV Equation

In this section, we compute exact solutions to the (2+1)-dimensional system (3) by following a pattern similar to that used in Section 2. This gives the Lie point symmetries admitted by (3) (44)X1=∂∂t+G1t∂∂x-13kG1′t∂∂vX2=∂∂y+G2t∂∂x-13kG2′t∂∂vX3=t∂∂t+x3+G3t∂∂x+y3∂∂y-2u3∂∂u+19k-3G3′t-6kv+2s∂∂v+19r2q-6rw∂∂w,and the following set of determining equations for multipliers Λk=Λk(t,x,u,v,w), where k=1,2,3(45)Λ2xx=0,Λ3yy=0,Λ1t=-Λ2x-Λ3y,Λ1x=0,Λ3x=0,Λ1y=0,Λ2y=0,Λ1ux=0,Λ1u=0,Λ2u=0,Λ3u=0,Λ1v=0,Λ2v=0,Λ3v=0,Λ1w=0,Λ2w=0,Λ3w=0,Λ2ux=0,Λ3ux=0,Λ1vx=0,Λ2vx=0,Λ3vx=0,Λ1wx=0,Λ2wx=0,Λ3wx=0.The last set of equations give(46)Λ1=-F1t-F3t+c1,Λ2=F1tx+F2t,Λ3=F3ty+F4t,so that the conserved vectors corresponding to these multipliers are(47)T1t=u,T1x=-3ukv+us+uxxk,T1y=-3urw+uq+uyyr,T2t=-uF1t,T2x=3uF1tkv-uF1ts-uxxF1tk+uF1′tx,T2y=3uF1trw-uF1tq-uyyF1tr-vF1′tx,T3t=0,T3x=uF2t,T3y=-vF2t,T4t=-uF3t,T4x=3uF3tkv-uF3ts-wF3′ty-uxxF3tk,T4y=3uF3trw-uF3tq+uF3′ty-uyyF3tr,T5t=0,T5x=-wF4t,T5y=uF4t.The association between symmetries and conservation laws is shown in Table 2.Table 2
Association between symmetries and conservation laws.
T 1 T 2 T 3 T 4 T5 X1 associated associated if associated if associated if associated if F 1 ( t ) constant F 2 ( t ) and G1(t) constant F 3 ( t ) constant F 4 ( t ) constant  X 2 associated associated if associated if associated if associated F 1 ( t ) constant F 2 ( t )=0, G2(t)≠0 F 3 ( t ) constant orG2(t)=0 orF2(t)≠0, G2(t) constant  X 3 associated associated if associated if associated if not associated F 1 ( t ) constant G 3 ( t ) = 2 s t / 3, F2(t)=t-2/3 F 3 ( t ) constantCase 1 (solutions of (3) using X1 and X2).
The Lie symmetriesX1 and X2 are associated with the conserved vectors T1, T2, T3, T4, and T5 if Fi(t) and Gj(t) are constants. For simplicity, we assume these constants to be equal to 1. Consequently, the Lie symmetries reduce to(48)X1=∂∂t+∂∂x,X2=∂∂x+∂∂y,and the associated conserved vectors take the form(49)T1t=u,T1x=-3kuv+su+kuxx,T1y=-3ruw+qu+ruyy,T2t=-u,T2x=3kuv-su-kuxx,T2y=3ruw-qu-ruyy,T3t=0,T3x=u,T3y=-v,T4t=-u,T4x=3kuv-su-kuxx,T4y=3ruw-qu-ruyy,T5t=0,T5x=-w,T5y=u.We note that the conserved vectors T1,T2, and T4 are the same, so we only use T1 to construct the solution of (3). A linear combination of X1 and X2, given in (48), can be expressed as(50)X=∂∂t+α∂∂x+β∂∂y,where α and β are constants. The generator X in the last equation reduces to ∂/∂c with the help of similarity transformations(51)a=y-βt,b=x-αt,c=t,ut,x,y=Aa,b,vt,x,y=Ba,b,wt,x,y=Ca,b.Equation (18) gives(52)TaTbTc=JA-1TTtTxTy,where(53)J=-1andA-1T=β0-1α-10-100.This yields(54)Ta=βTt-Ty,Tb=αTt-Tx,Tc=-Tt,so that the reduced conserved vectors are(55)T1a=βA+3rAC-qA-rAaa,T1b=αA+3kAB-sA-kAbb,(56)T3a=B,T3b=-A,and(57)T5a=-A,T5b=C.We note that the new conservation law satisfies(58)DaTa+DbTb=0.The transformed vectors T1, T3, and T5 in (55)-(57) inherit symmetries ∂/∂a and ∂/∂b which are helpful in further reduction. The combination of these symmetries X~=∂/∂a+γ∂/∂b yields the similarity variables(59)m=a,n=-γa+b,Ln=Ca,b,Mn=Aa,b,Nn=Ba,b,and (19) can be used to procure(60)Tn=γTa-Tb,Tm=-Ta,which satisfies(61)DnTn=0.Integrating the last equation yields(62)Tn=d,where d is the constant of integration. The conserved vectors (55)-(57) using (60) and (62) can be expressed as(63)γβM+3rγML-qγM-rγ3Mnn-αM-3kMN+sM+kMnn=d1,γN+M=d2,γM+L=d3,where d1, d2, and d3 are arbitrary constants. The solution of system (63) is(64)N=d2-Mγ,L=d3-γM,±∫γrγ3-kγ2βM2+3γ2rd3M2-2γ3rM3-γ2qM2-αγM2-3kd2M2+2kM3+sγM2+c1rγ4-c1γk-2d1γM-1/2dM-n-c2=0,where(65)Mn=ut,x,y,Nn=vt,x,y,Ln=wt,x,y,n=βγ-αt-γy+x.It can be verified directly that these solutions satisfy (3). Explicit solutions to (63) can be procured by setting constants d1, d2, and d3 equal to zero and then employing the sine-cosine method [28]. This gives(66)M1n=-12γα+qγ-γβ-srγ3-ksec212-γβ-s+qγ+αrγ3-kn,N1n=12α+qγ-γβ-srγ3-ksec212-γβ-s+qγ+αrγ3-kn,L1n=12γ2α+qγ-γβ-srγ3-ksec212-γβ-s+qγ+αrγ3-kn,-γβ-s+qγ+αrγ3-k>0.(67)M2n=-12γα+qγ-γβ-srγ3-kcsc212-γβ-s+qγ+αrγ3-kn,N2n=12α+qγ-γβ-srγ3-kcsc212-γβ-s+qγ+αrγ3-kn,L2n=12γ2α+qγ-γβ-srγ3-kcsc212-γβ-s+qγ+αrγ3-kn,-γβ-s+qγ+αrγ3-k>0.(68)M3n=-12γα+qγ-γβ-srγ3-ksech212γβ+s-qγ-αrγ3-kn,N3n=12α+qγ-γβ-srγ3-ksech212γβ+s-qγ-αrγ3-kn,L3n=12γ2α+qγ-γβ-srγ3-ksech212γβ+s-qγ-αrγ3-kn,-γβ-s+qγ+αrγ3-k<0.(69)M4n=12γα+qγ-γβ-srγ3-kcsch212γβ+s-qγ-αrγ3-kn,N4n=-12α+qγ-γβ-srγ3-kcsch212γβ+s-qγ-αrγ3-kn,L4n=-12γ2α+qγ-γβ-srγ3-kcsch212γβ+s-qγ-αrγ3-kn,-γβ-s+qγ+αrγ3-k<0.
In system (3), u represents physical field and v and w describe potentials. We recover solution (M3,N3,L3) in (68) as reported in [17]. This solution is identified as dromion solution. Here, we obtain three additional solutions (M1,N1,L1), (M2,N2,L2), and (M4,N4,L4) of system (3) which were not reported before and their profiles are shown below.
Figure4 represents the breather wave.Figure 4
Solution profileM2 where k=1, r=1, s=-1, q=1, γ=-1, α=-1, β=-1. This sequence of frames is constructed from time t=0 to t=10.Case 2 (solution of (3) using X3).
As shown in Table2, the Lie symmetry X3 is associated with the conserved vectors T1 and T3 if F2(t)=t-2/3 and G3(t)=2/3st. This leads to the similarity transformations of X3, which are given by(70)a=x-stt1/3,b=yt1/3,c=ln⁡t,Aa,b=vt2/3,Ba,b=-13rt2/3-3rw+q,Ca,b=ut2/3.The conserved vectors T1 and T3 in new variables can be expressed as(71)T1a=-13aC-3kAC+kCaa,T1b=-13bC-3rBC+rCbb,T3a=C,T3b=-A.The above vectors are invariant under Lie symmetry 9r∂/∂b-∂/∂B. Consequently, the similarity variables are(72)m=b9r,n=a,Ln=Ca,b,Mn=Aa,b,Nn=19rb+9rB,so that(73)Tn=9rTa,Tm=Tb.The reduced conserved form thus gives(74)-3rnL-27rkML+kLnn=d7,9rL=d8,where we have replaced Tn by d7 and Tm by d8. We note that the above system does not have any contribution from the third equation of (3), namely, uy-wx=0(Figures 5 and 6). This equation is invariant under X3, and the change of variables (70) yields(75)Cb=Ba.Since (75) is also invariant under the symmetry 9r∂/∂b-∂/∂B, its further reduction leads to(76)Nn=d9.Equations (76) and (74) in original variables give the solution of (3) (see Figures 7–9)(77)u=d89rt2/3,v=13kd8t2/3d8st-x3t1/3-d7,w=19rt2/39rd9+3qt2/3-yt-1/3.
We note from Table2 that T2 and T4 are also associated with X3 if F1(t) and F3(t) are constants but in that case T2 and T4 reduce to T1 which has already been discussed above.Figure 5
Solution profileM3 where k=1, r=1, s=-1, q=1, γ=2, α=-1, β=4. This sequence of frames is constructed for time t=0 to t=10.Figure 6
Solution profileM4 where k=1, r=1, s=-1, q=1, γ=2, α=-1, β=4. This sequence of frames is constructed for time t=0 to t=2.Figure 7
Solutionu given by (77) with d8=1 and r=1.Figure 8
Solutionv given by (77) with k=1,d8=1,d7=1, and s=1.Figure 9
Solutionw given by (77) with r=1, d9=1, and q=1.Case 3 (solution of (3) using X1, X2, T1, and T5).
ConsiderX=X1+αX2. As shown in Table 2, the symmetries X1 and X2 are associated with T1 and T5. The symmetries are also associated with T2 and T4 if F1(t) and F3(t) are constants, but in that case they coincide with T1. Consequently, considering T1 would be sufficient. The symmetry X can then be expressed as(78)∂∂t+G1t+αG2t∂∂x+α∂∂y-13kG1′t+αG2′t∂∂v.In similarity variables, we have(79)a=-∫G1t+αG2tdt+x,b=-αt+y,c=t,Aa,b=G1t+αG2t+3vk3k,Ba,b=w,Ca,b=u,so that (18) gives(80)Ta=-G1t+αG2tTt+Tx,Tb=-αTt+Ty,Tc=Tt.The reduced conserved vectors thus obtained are(81)T1a=-kAC+sC+kCaa,T1b=q-αC-3rCB+rCbb,(82)T5a=-B,T5b=C.The vectors T1 and T5 admit symmetries ∂/∂a and ∂/∂b, from which further reduction is possible. Using combination of symmetries X~=∂/∂a+γ∂/∂b and transforming it into new similarity variables yield(83)m=a,n=-γa+b,Ln=Ca,b,Mn=Aa,b,Nn=Ba,b.Again, from (18) it is straightforward to show that(84)Tn=γTa-Tb,Tm=-Ta,and system (3) satisfies(85)DnTn=0.Equations (84) and (85) give (86)-kγML+sγL+kγ3-rLnn-q-αL+3rLN=d4,γN+L=d5,M=-γL+d6,where d4, d5, and d6 are constants. System (86) reduces to Case 2.


## 4. Results and Discussion

Whereas the symmetry method only reduces the number of independent variables in the PDE involved, the double reduction method, in addition to the reduction of independent variables, reduces the order of the PDEs. The association of symmetries and conservation laws reduces the number of independent variables and the order of PDEs simultaneously. The association of a symmetry with more than one conservation law yields the same solutions (Case 2 of Sections2 and 3), which are either implicit or in closed form. The association of scaling symmetry leads to double reduction whereas inversion symmetry yields a distinct solution to (1). The association of Lie symmetries and conserved vectors obtained in Section 3 involves arbitrary functions. The association of symmetries and conservation laws puts restriction on the form of these functions (as listed in Table 2). Under these restrictions, group invariant solutions are constructed. One solution is obtained using the relation between scaling symmetry and conservation law, while the other solutions are of travelling wave type.Solutions (66), (67), (68), (69), (70), (74), (75), (76), and (77) are procured in terms of trigonometric functions using the sine-cosine method. An equivalent formulation to these solutions is given in [21] where the method of ansatz is employed to obtain dromion solutions in terms of exponential functions. In addition to the existing explicit solutions to (3), some new solutions including an implicit solution (64) and an explicit solution (77) to (3) have also been obtained. Moreover, we recuperated solution (M3,N3,L3) as reported in [17] using the combination of symmetries and conservation laws. This solution is identified as a dromion solution. In this article, we obtained three additional solutions (M1,N1,L1), (M2,N2,L2), and (M4,N4,L4) of (3) which were not reported before.Since the double reduction approach reduces the order of the PDE and the number of independent variables involved, it makes the numerical computations of solutions easier as compared to the application of numerical techniques to the original GNNV equation. This study provides a new way of constructing variety of exact solutions for partial differential equations by forming an association of the symmetry present with conserved vectors of the equation. This association is not only limited to the translational symmetries or transformation of the formx-αt and it may exist for a wide range of symmetries like rotations, dilatations, and inversions. In this article, we exploited the dilation symmetry of the systems to extract new exact solutions as stated above.


## 5. Conclusion

Double reduction of a PDE is a well-known technique which could be used when a conservation law is associated with Noether symmetry of Euler-Lagrange equation. However, in many cases, the PDE system does not admit the Lagrangian. Consequently, the existence of Noether symmetry remains out of question. In this work we considered systems of evolution equations which do not admit the Lagrangian formulation. We used the variational approach to construct conservation laws and their Lie symmetries. Double reduction was then employed by investigating the association between symmetry and conservation law. An association of scaling symmetry with a conserved vector was established. This had the consequence of integrating the differential equation further, which is significant since in many cases further integration of a differential equation is not possible if it is reduced by scaling or inversion symmetry. Several explicit solutions were also obtained which is a contribution of this paper.



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