On optimal system, exact solutions and conservation laws of the modified equal-width equation

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Abstract

In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

Abstract

In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

1 Introduction

In this paper we study the third-order modified equal-width (MEW) equation

ut+3αu2uxβutxx=0,α0,β0,

where α and β are non-zero real parameters. Equation (1) is used in handling the simulation of a single dimensional wave propagation in nonlinear media with dispersion processes [1]. Some researchers have used different techniques and methods to construct travelling wave solutions of (1). Recently MEW Equation (1) was investigated in [1], where the researchers employed extended simple equation method and also the exp((ξ)) expansion method to generate travelling wave solutions of the equation. In [2], dynamical system technique for integer order was used and travelling wave solutions of the MEW equation were found, which comprised of solitary, periodic waves and also kink and anti-kink wave solutions. Homotopy perturbation method was applied to (1) and numerical solution of the MEW equation was obtained in [3].

In our study we use an entirely different approach to obtain new exact travelling wave solutions, namely cnoidal and snoidal wave solutions of MEW Equation (1). Moreover, for the first time we derive conservation laws of the MEW equation by employing both the Noether approach as well as the multiplier approach.

2 Exact solutions of (1) constructed on optimal system

In this section, we first compute Lie point symmetries of (1) and then use them to construct an optimal system of one-dimensional subalgebras. Subsequently, we utilise this optimal system of one-dimensional subalgebras to obtain symmetry reductions and group-invariant solutions of (1) [4, 5, 6, 7, 8].

2.1 Lie point symmetries of (1)

The vector field

X=τ(t,x,u)t+ξ(t,x,u)x+η(t,x,u)u,

where τ, ξ and η depend on t, x and u is a Lie point symmetry of Equation (1) if

pr(3)XΔ|Δ=0=0,

where

Δut+3αu2uxβutxx

and pr(3)X is the third prolongation [6] of (2) defined as

pr(3)X=X+ζtut+ζxux+ζtxutx+ζtxxutxx.

Here ζt , ζx, ζtx and ζtxx are determined by

ζt=Dt(η)utDt(τ)uxDt(ξ),ζx=Dx(η)utDx(τ)uxDx(ξ),ζtx=Dx(ζt)uttDx(τ)utxDx(ξ),ζtxx=Dx(ζtx)uttxDx(τ)utxxDx(ξ),

where the total derivatives Dt and Dx are defined as

Dt=t+utu+uttut+utxux+,Dx=x+uxu+uxxux+uxtut+.

Expanding (3) and splitting on derivatives of u yields an overdetermined system of linear homogeneous partial differential equations (PDEs). Solving these equations we obtain the values of τ, ξ and η, which lead to three Lie point symmetries of (1) given by

X1=t,X2=x,X3=2ttuu.

The infinitesimal generator X3 represents scaling symmetry whereas the one-parameter groups generated by X1 and X2 demonstrate time and space-invariance of the MEW equation.

2.2 Optimal system of one-dimensional subalgebras

We now calculate an optimal system of one-dimensional subalgebras by using Lie point symmetries of (1) obtained in the previous subsection. We employ the method given in [6]. We first construct the commutator table. Thereafter we compute adjoint representation using the Lie series

Ad(exp(εXi))Xj=n=0εnn!(adXi)n(Xj)=Xjε[Xi,Xj]+ε22![Xi,[Xi,Xj]],

where ε is a real number and [Xi,Xj] denotes the commutator defined by

[Xi,Xj]=XiXjXjXi.

The table of commutators of Lie point symmetries of Equation (1) and adjoint representations of the symmetry group of (1) on its Lie algebra are presented in Table 1 and Table 2, respectively. Consequently, Table 1 and Table 2 are used to compute an optimal system of one-dimensional subalgebras for Equation (1).

Table 1

Lie brackets for equation (1)

[ , ]X1X2X3
X1002X1
X2000
X3-2X100
Table 2

Adjoint representation of subalgebras

AdX1X2X3
X1X1X2-2ɛX1 + X3
X2X1X2X3
X3e2ɛX1X2X3

Thus following [6] and utilising Tables 1 and 2 we can obtain an optimal system of one-dimensional subalgebras, which is given by {X1+cX2,X3+ aX2}, where c and a are arbitrary constants.

2.3 Solutions and symmetry reductions

We now utilise the optimal system of one-dimensional subalgebras obtained above in the previous subsection and find group-invariant solutions and symmetry reductions for Equation (1).

Consider the first operator X1+ cX2 of the optimal system. This operator has two invariants

ξ=xctandU=u,

which give the group-invariant solution U = U (ξ). Using ξ as our new independent variable, Equation (1) is transformed into the nonlinear ordinary differential equation (ODE)

cβU(ξ)+3αU2(ξ)U(ξ)cU(ξ)=0.

We now use the extended Jacobi elliptic function expansion method [9] to obtain travelling wave solutions of (1). We assume that solutions of the third-order nonlinear ODE (7) can be expressed in the form

U(ξ)=i=MMAiH(ξ)i,

where M is a positive integer obtained by the balancing procedure and Ai are constants to be determined. Here H (ξ) satisfies the nonlinear first-order ODE

H(ξ)=(1H2(ξ))(1ω+ωH2(ξ))

or

H(ξ)=(1H2(ξ))(1ωH2(ξ)).

We recall that the Jacobi cosine-amplitude function

H(ξ)=cn(ξ|ω)

is a solution to (9), whereas the Jacobi sine-amplitude function

H(ξ)=sn(ξ|ω)

is a solution to (10). Here ω is a parameter such that 0 ≤ ω ≤ 1 [9, 10].

We note that when ω → 1, then cn(ξ |w) sech(ξ) and sn(ξ |w) tanh(ξ). Also, when ω → 0, then cn(ξ |w)cos(ξ) and sn(ξ|ω) sin(ξ).

2.3.1 Cnoidal wave solutions

Considering the nonlinear ODE (7), the balancing procedure yields M = 1. Thus (8) takes the form

U(ξ)=A1H1(ξ)+A0+A1H(ξ).

Substitution of U from (13) into (7) and utilising (9) we obtain

H(ξ)4βcA1H(ξ)6βcA1+H(ξ)4βcA17H(ξ)2βcA112βcωA1+6βcω2A1+3H(ξ)10αωA136H(ξ)8αωA13H(ξ)8cωA1+6H(ξ)7αA0A12+3H(ξ)6αωA133αωA133H(ξ)6αA13+3H(ξ)8αA13H(ξ)6cA1+H(ξ)4cA1+H(ξ)4cA13H(ξ)2αA13H(ξ)2cA110H(ξ)6βcω2A12H(ξ)6βcω2A17H(ξ)8βcωA1+14H(ξ)8βcω2A16H(ξ)10βcω2A1+21H(ξ)2βcωA114H(ξ)2βcω2A13H(ξ)4βcωA110H(ξ)4βcωA1+2H(ξ)4βcω2A1+10H(ξ)4βcω2A1+10H(ξ)6βcωA1+H(ξ)6βcωA1+3H(ξ)8αωA02A1+3H(ξ)8αωA1A12+6H(ξ)9αωA0A126H(ξ)αωA12A03H(ξ)2αωA1A023H(ξ)2αωA12A1+12H(ξ)3αωA12A0+3H(ξ)4αωA02A1+3H(ξ)4αωA1A12+6H(ξ)4αωA1A02+6H(ξ)4αωA12A1+6H(ξ)5αωA0A126H(ξ)5αωA12A06H(ξ)6αωA02A16H(ξ)6αωA1A123H(ξ)6αωA1A023H(ξ)6αωA12A112H(ξ)7αωA0A12+6H(ξ)αA12A0+3H(ξ)2αA1A02+H(ξ)2cωA1+3H(ξ)2αA12A1+6H(ξ)2αωA13H(ξ)4cωA16H(ξ)3αA12A02H(ξ)4cωA13H(ξ)4αA1A123H(ξ)4αA02A13H(ξ)4αA12A13H(ξ)4αA1A02+2H(ξ)6cωA16H(ξ)5αA0A123H(ξ)4αωA13+3H(ξ)6αA1A12+3H(ξ)6αA02A1+H(ξ)6cωA1+6βcA1+3αA13=0.

The above equation can be separated on like powers of H (ξ) to obtain an overdetermined system of eleven algebraic equations

A0A12=0,A12A0ωA12A0=0,2ωA12A0A12A0=0,A0A122ωA0A12=0,αA132βcωA1=0,ωA0A12ωA12A0A0A12=0,2βcω2A1αωA13+αA134βcωA1+2βcA1=0,3αωA1A12+3αωA02A16αωA13+14βcω2A1+3αA137βcωA1cωA1=0,6αωA133αωA12A13αωA1A0214βcω2A13αA13+3αA12A1+3αA1A02+21βcωA17βcA1+cωA1cA1=0,3αωA133αωA12A13αωA1A026αωA1A126αωA02A12βcω2A1+3αA1A1210βcω2A1+3αA02A13αA13+βcωA1+10βcωA1βcA1+cωA1+2cωA1cA1=0,6αωA12A13αωA13+6αωA1A02+3αωA1A12+3αωA02A1+10βcω2A1+2βcω2A13αA12A13αA1A023αA1A123αA02A110βcωA13βcωA1+βcA1+βcA12cωA1cωA1+cA1+cA1=0.

Solving the above system of equations we obtain

ω=8β+3k116β,A0=0,A1=±c(3k+8β1)8α,A1=3β±kβ+1A1

with k=8β2+1

Thus reverting to the original variables the solutions of (1) are

u(t,x)=±c(3k+8β1)8αcnξω3β±kβ+1ncξω,

where nc = 1/cn.

2.3.2 Snoidal wave solutions

We now obtain snoidal wave solutions for Equation (1). Here again M = 1. Substituting the value of U from (13) into (7) and making use of (10) we obtain

3H(ξ)6αA13H(ξ)4cA1+H(ξ)2cA1+3H(ξ)2αA13+3H(ξ)4αA12A1+3H(ξ)4αA1A02+H(ξ)6βcA1H(ξ)4βcA1H(ξ)4βcA1+7H(ξ)2βcA1+3H(ξ)4αA1A12+3H(ξ)4αA02A1+3H(ξ)10αωA133H(ξ)8αωA13H(ξ)8cωA16H(ξ)7αA0A123H(ξ)6αA1A123H(ξ)6αA02A1+H(ξ)6cωA1+H(ξ)6cωA1+6H(ξ)5αA0A123H(ξ)4αωA133αA136H(ξ)7αωA0A123H(ξ)6αωA12A13H(ξ)6αωA1A023H(ξ)6αωA1A123H(ξ)6αωA02A16H(ξ)5αωA12A0+3H(ξ)4αωA12A1+3H(ξ)4αωA1A02+6H(ξ)9αωA0A12+3H(ξ)8αωA1A12+3H(ξ)8αωA02A13H(ξ)8αA13+H(ξ)6cA1H(ξ)4cA16βcA1+7H(ξ)2βcωA1H(ξ)4βcωA18Hξ4βcωA1H(ξ)4βcω2A1+8H(ξ)6βcωA1+H(ξ)6βcωA1+H(ξ)6βcω2A1+H(ξ)6βcω2A17H(ξ)8βcωA17H(ξ)8βcω2A13H(ξ)2αA1A026HξαA12A0+3H(ξ)2αωA133H(ξ)2αA12A1H(ξ)4cωA1+6H(ξ)3αA12A0+6H(ξ)10βcω2A1+6H(ξ)3αωA12A0=0.

Splitting on powers of H (ξ) yields the following overdetermined system of algebraic equations:

A12A0=0,A0A12=0,αA13+2βcA1=0,ωA12A0+A12A0=0,ωA0A12+A0A12=0,αA13+2βcωA1=0,ωA12A0A0A12=0,3αωA1A12+3αωA02A13αωA137βcω2A13αA137βcωA1cωA1=0,3αωA13+3αA133αA12A13αA1A02+7βcωA1+7βcA1+cA1=0,βcω2A13αωA12A13αωA1A023αωA1A123αωA02A1+βcω2A1+3αA133αA1A123αA02A1+βcωA1+8βcωA1+βcA1+cωA1+cωA1+cA1=0,3αωA12A13αωA13+3αωA1A02βcω2A1+3αA12A1+3αA1A02+3αA1A12+3αA02A18βcωA1βcωA1βcA1βcA1cωA1cA1cA1=0.

Solving the above system of equations we get

β=11+ω,A1=A0=0,A1=±2c(β+1)α.

Reverting to original variables we obtain solutions of (1) as

u(t,x)=±2c(β+1)αsnξω.

We now consider the second operator X3 + aX2 of the optimal system. This symmetry operator yields two invariants J1 = ex t-a/2 and J2 = ω1/2. Thus J2 = f (J1) provides a group-invariant solution to (1). That is

u=t1/2f(exta/2).

Substituting the above value of u in (1), we obtain the third-order nonlinear ODE

aβz3f(z)+β3a+1z2f(z)+(aβa+β)zf(z)+6αzf(z)2f(z)f(z)=0,

where z=exta/2.

3 Conservation laws of the modified equal width Equation (1)

In the section we derive conservation laws for (1) by employing two different techniques, namely the multiplier method and Noether approach.

Conservation laws have several important uses in the study of partial differential equations, especially for determining conserved quantities and constants of motion, detecting integrability and linearizations, finding potentials and nonlocally-related systems, as well as checking the accuracy of numerical solution methods [11, 12, 13, 14, 15, 16, 17].

3.1 Conservation laws of (1) using multiplier approach

We look for zeroth-order multiplier 𝛬=𝛬(t,x,u). Thus, the determining equation for this multiplier is stated as

δδu{Λ(t,x,u)(ut+3αu2uxβutxx)}=0,

where δ/δu is the Euler-Lagrange operator defined as

δδu=uDtutDxuxDtDx2utxx

and the total derivatives Dt and Dx are defined as in (6). The above equation yields

utΛu+3αu2uxΛuβutxxΛu+6αuuxΛDt(Λ)Dx(3αu2Λ)+βDx2Dt(Λ)=0,

which on expanding gives

utΛu+3αu2uxΛuβuxxtΛu+6αuuxΛΛtutΛu3αu2Λx6αuuxΛ3αu2uxΛu+βΛtxx+βutΛuxx+βuxΛtux+βutuxΛuux+βutxΛux+βuxΛtux+βutuxΛuux+βux2Λtuu+βutux2Λuuu+βutxuxΛuu+βutxΛux+βutxuxΛuu+βuxxΛtu+βuxxutΛuu+βuxxtΛu=0.

Splitting the above equation on derivatives of u, we obtain

Λuu=0,Λux=0,Λtu=0,βΛtxx3αu2ΛxΛt=0.

By solving the above equations we get two multipliers given by

Λ1(t,x,u)=u

and

Λ2(t,x,u)=1.

Corresponding to these two multipliers, we obtain the following two conservation laws:

T1t=12u2+12βux2,T1x=34αu4βuutx

and

T2t=u,T2x=αu3βutx.

3.2 Conservation laws of (1) using Noether’s theorem

In this subsection we derive conservation laws for the modified equal-width Equation (1) using Noether’s theorem [18, 19]. This equation as it is does not have a Lagrangian. In order to apply Noether’s theorem we transform Equation (1) to a fourth-order equation which will have a Lagrangian. Thus using the transformation u =Vx, Equation (1) becomes

Vtx+3αVx2VxxβVtxxx=0.

It can readily be verified that a Lagrangian for equation (17) is given by

L=12VxVt14αVx412βVxxVtx

because δℒ/δV = 0 on (17). Here δ/δV is the Euler-Lagrange operator defined as

δδV=VDtVtDxVx+Dx2Vxx+DtDxVtx.

Consider the vector field

X=τ(t,x,V)t+ξ(t,x,V)x+η(t,x,V)V,

where τ, ξ and η depend on t, x and V. To determine Noether point symmetries X of (17) we insert the value of L from (18) in the determining equation

pr[2]X(L)+LDtτ+Dxξ=DtBt+DxBx,

where Bt = Bt(t,x,V) and Bx = Bx(t,x,V) are gauge terms and pr[2]X is the second prolongation of X defined as

pr[2]X=X+ζtVt+ζxVx+ζxxVxx+ζtxVtx

with ζt , ζx, ζxx and ζtx as defined in (5). Expansion of equation (20) and separating with respect to derivatives of V yields an overdetermined system of linear PDEs. Thereafter solving these PDEs we obtain the following Noether point symmetries together with their gauge functions:

X1=t,Bt=0,Bx=0,X2=x,Bt=0,Bx=0,Xf=f(t)V,Bt=0,Bx=12f(t)V.

Next, we use the above results to compute conserved vectors of the fourth-order equation (17). Using formulae for the conserved vector (Tt ,Tx) [20]

Fk=Lτk+(ξαψxjατj)(Lψxkαl=1kDxl(Lψxlxkα))+l=kn(ηlαψxlxjατj)Lψxkxlαfk

we obtain three conserved vectors associated with three Noether point symmetries X1, X2 and X f . Then reverting to the original variable u, we have

T1t=14αu412βuxut12βuxxutdx,T1x=12utdx2+αu3utdx12βuxtutdx+12βut2+12βuxuttdx;T2t=12u212βuuxx,T2x=34αu412βuuxt+12βuxut;Tft=12f(t)u+12βf(t)uxx,Tfx=12f(t)utdxαf(t)u3+12βf(t)uxt12βf(t)ux+12f(t)udx.

Remark: It should be noted that due to the presence of arbitrary function f (t) we have infinitely many nonlocal conservation laws.

4 Conclusions

In this paper we studied the modified equal-width Equation (1). For the first time, Lie point symmetries of (1) were computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter utilising this optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions of (1) were presented. The solutions obtained were cnoidal and snoidal waves. Again for the first time, we computed the conservation laws for (1) by employing two different methods, the multiplier method and Noether approach.

Communicated by Juan L.G. Guirao

Acknowledgments

The authors would like to thank T Motsepa for fruitful discussions. CMK thanks the North-West University, Mafikeng Campus for its continued support. ODA and IS thank the North-West University for financial aid through the post-graduate bursary scheme. Finally, we thank the anonymous reviewers for their useful comments, which helped improve the presentation of the paper.

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[1]

D. Lu, A. R. Seadawy, A. Ali. Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathematical methods and its applications. Results in Physics 9 (2018): 313-320.

[2]

H. Wang, L. Chen, H. Wang, Exact travelling wave solutions of the modified equal width equation via the dynamical system method, Nonlinear Analysis and Differential Equations, 4 (2016) 9–15.

[3]

S.T. Mohyud-Din, A. Yildirim, M.E. Berberler, M. M. Hosseini, Numerical Solution of Modified Equal Width Wave Equation, World Applied Sciences Journal 8 (2010) 792–798

[4]

L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.

[5]

G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989.

[6]

P.J. Olver, Applications of Lie Groups to Differential Equations, second ed., Springer-Verlag, Berlin, 1993.

[7]

N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Vols 1–3, CRC Press, Boca Raton, Florida, 1994–1996.

[8]

N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, Chichester, NY, 1999.

[9]

T. Motsepa, C.M. Khalique, Cnoidal and snoidal waves solutions and conservation laws of a generalized (2+1)-dimensional KdV equation, Proceedings of the 14th Regional Conference on Mathematical Physics, 2018, Pages: 253–263 10.1142/9789813224971-0027

[10]

I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 7th edn. Academic Press, New York, 2007.

[11]

S. Anco and G. Bluman. Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications. European J. Appl. Math., 13:545–566, 2002.

[12]

I. Simbanefayi and C.M. Khalique. Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation. Results in Physics, 8:57–63, 2018.

[13]

M. Rosa and M.L. Gandarias. Multiplier method and exact solutions for a density dependent reaction-diffusion equation. Applied Mathematics and Nonlinear Sciences, 1(2):311–320, 2016.

[14]

R. de la Rosa and M.S. Bruzón. On the classical and nonclassical symmetries of a generalized Gardner equation. Applied Mathematics and Nonlinear Sciences, 1(1):263–272, 2016.

[15]

M.L. Gandarias and M.S. Bruzón. Conservation laws for a Boussinesq equation. Applied Mathematics and Nonlinear Sciences, 2(2):465–472, 2017.

[16]

T. Motsepa and C.M. Khalique. On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics. Open Phys., 16:211–214, 2018.

[17]

C.M. Khalique, I.E. Mhlanga. Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation. Applied Mathematics and Nonlinear Sciences, 2018, in press.

[18]

E. Noether, Invariante variationsprobleme, Nachr. König. Gesell. Wissen., Göttingen. Math Phys Kl Heft, 2 (1918) 235–257. English translation in Transp. Theor. Stat. Phys. 1(3) (1971) 186–207.

[19]

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