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formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation

Dumitru Baleanu
  • Corresponding author
  • Cankaya University, Department of Mathematics, Ögretmenler Cad. 1406530, Ankara, Turkey
  • Institute of Space Sciences, Magurele, Bucharest, Romania
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Mustafa Inc / Abdullahi Yusuf / Aliyu Isa Aliyu
Published Online: 2018-06-26 | DOI: https://doi.org/10.1515/phys-2018-0049

Abstract

In this article, the generalized shallow water wave (GSWW) equation is studied from the perspective of one dimensional optimal systems and their conservation laws (Cls). Some reduction and a new exact solution are obtained from known solutions to one dimensional optimal systems. Some of the solutions obtained involve expressions with Bessel function and Airy function [1,2,3]. The GSWW is a nonlinear self-adjoint (NSA) with the suitable differential substitution, Cls are constructed using the new conservation theorem.

Keywords: GSWW; optimal system; Cls; infinitesimal generators; NSA

PACS: 02.20.Qs; 02.20.Sv; 02.30.Hq; 02.30.Jr

1 Introduction

Lie symmetry methods play a vital role in the study and finding solutions for nonlinear partial differential equations (NLPDE) [4,5, 6,7,8,9,10,11,12,13,14,15,16]. Different techniques are used in the literature for the construction of Cls for different system of equation and these Cls are important for the investigation of a physical system [4,5, 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Moreover, Authors made rigorous attempts for construction of the construction of one-dimensional and higher-dimensional optimal systems optimal system of Lie algebra [22,23,24,25,26]. Cls and symmetries have many application in science, physics and engineering [32,33,34,35,36,37].

In this work, we obtain one-dimensional optimal system, exact solutions and Cls for the GSWW equation given by

Δ=uxxxt+auxuxt+butuxxuxtuxx=0,(1)

where a ≠ 0, b ≠ 0 are arbitrary constants. Eq. (1) have been studied by different authors using a variety of techniques. For example [27] introduced exact solutions for Eq. (1) by the general projective Ricatti equations method. Periodic wave solution for Eq. (1) by the improved Jacobi elliptic function method was investigated by [28] . Homogeneous balance method [29] was applied to investigate some solutions for Eq. (1) and some new solution of Eq. (1) with extended elliptic function method was proposed in [30] and many more.

2 One-dimensional optimal system of subalgebras of GSWW

In this section, we establish the optimal system of one-dimensional subalgebras of L4 and their corresponding exact solutions. Consider one parameter Lie group of the infinitesimal transformation below

x¯=x+ϵξ1(x,t,u)+O(ϵ2),(2)

t¯=t+ϵξ2(x,t,u)+O(ϵ2),(3)

u¯=u+ϵη(x,t,u)+O(ϵ2),(4)

where ϵ is the group parameter. The corresponding Lie algebra of the infinitesimal symmetries is the set of vector field of the form

X=ξ1(x,t,u)t+ξ2(x,t,u)x+η(x,t,u)u.(5)

Considering the fourth order prolongation Pr(4) of the vector field X such that

Pr(4)X(Δ)=0,

where Δ = (1), whenever Δ = 0. Using SYM package introduced in [31], the determining equations for Eq. (1) are obtained. Solving for η(x, t, u), ξ2(x, t, u), and ξ1(x, t, u) from the obtained determining equations, we get

ξ1=c1+12axc3,(6)

ξ2=12atc3+c4+bF(t),(7)

η=c212auc3+xc3+F(t).(8)

where c1, c2, c3 and c4 are arbitrary constants and F(t) is an arbitrary function of t. The Lie symmetry algebra admitted by Eq. (1) is spanned by four infinitesimals generators below

X1=x,(9)

X2=u,(10)

X3=t,(11)

X4=att(au2x)u+axx.(12)

The corresponding commutator table of the infinitesimal generators is given by

Table 1

Commutator table of the Lie algebra for GSWWE

2.1 Construction of one-dimensional optimal system of subalgebras

The Lie algebra L4 spanned by the given generators X1, X2, X3, and X4 can guarantee a possibility to obtain invariant solutions of Eq. (1). This will be based mainly on one-dimensional subalgebra of L4. There may be an infinite number of one-dimensional subalgebras of L4. Therefore, one can write an arbitrary generators from L4 as

X=l1X1+l2X2+l3X3+l4X4,(13)

which depend on the four arbitrary constants l1, l2, l3, and l4. We construct the one-dimensional optimal system of subalgebras using the method introduced in [22,23,24,25,26]. After the transformation of L4, we can get a 4-parameter group of linear transformations of the generators as

l=(l1,l2,l3,l4).(14)

where l1, l2, l3, and l4 are the coefficients in Eq. (13).

2.2 Linear transformation

Here, we investigate the linear transformations by using their generators which is given as

Eμ=cμvλlvlλ,μ=1,...,4.(15)

and the structure constants of the Lie algebra L4 defined by cμvλ is given as

[Xμ,Xv]=cμvλXλ.(16)

Consider the following cases:

  • Case 1: For μ = 1, v = 4, and λ = 1, 2 in Table 1. [X1, X4] = c141 X1 c142 X2 and the non vanishing structure constants are ( cμvλ ) are c141 = a, c142 = 2.

  • Case 2: For μ = 2, v = 4, and λ = 2 in Table 1. [X2, X4] = c242 X2 and the non vanishing structure constants ( cμvλ ) are c242 = −a.

  • Case 3: For μ = 3, v = 4, and λ = 3 in Table 1. [X3, X4] = c343 X3 and the non vanishing structure constants ( cμvλ ) are c343 = ∗a.

  • Case 4: For μ = 4, v = 1, and λ = 1, 2 in Table 1. [X4, X1] = c411 X1 c412 X2 and the non vanishing structure constants ( cμvλ ) are c411 = −a, c412 = −2. Setting v = 2, λ = 2 row four column two, we get [X4, X2] = c522 X2 and c422 = a, Setting v = 3, λ = 3 row four column three, we get [X4, X3] = c433 X3 and c433 = a.

Now, Substituting the values of the non-vanishing structure constants in Eq. (15) for μ = 1, 2, …, 4, we obtain

E1=al4l1+2l4l2,E2=al4l2,E3=al4l3,E4=al1l12l1l2+al2l2+al3l3.(17)

2.3 Lie equation

To obtain the Lie equation, we integrate the generators E1, E2, E3, E4 in Eq. (17) using the initial condition l|ϵ=0 = l.

  • For the generator E1, the Lie equation with the parameter ϵ are given by l1¯ϵ=al4¯,l2¯ϵ=2l4¯l3¯ϵ=0,l4¯ϵ=0. Integrating and using the initial condition we obtain l1 = 1 l4 + l1, l2 = 2ϵ1 l4 + l2, l3 = l3, l4 = l4.

    similarly for the other generators by following the same approach we get

  • For E2, we obtain l1 = l1, l2 = −2 l4 + l2, l3 = l3, l4 = l4.

  • For E3, we have l1 = l1, l2 = l2, l3 = −3 l4 + l3 and l4 = l4.

  • For E4, we have l1¯=l11+aϵ4,l2¯=2ϵ1+l21aϵ4,l3¯=l31aϵ4 and l4 = l4.

using SYM package [31], we can get the optimal system raw data in matrix form as:

aϵ1l4+l12ϵ1l4+l2l3l4l1aϵ2l4+l2l3l4l1l2aϵ3l4+l3l4l11+aϵ42ϵ1+l21aϵ4l31aϵ4l4

and the number of the functionally invariants, which is found to be l4. The corresponding one-dimensional optimal system of subalgebras are found to be the following:

  1. X3,

  2. αX2 + X3,

  3. X4,

  4. X1 + X4,

  5. αX3 + X4,

  6. αX1 + βX2 + X3,

where α, β ∈ ℝ. In the following, we list the corresponding similarity variables, similarity solutions as well as the reduced PDEs obtained from the generators of optimal system and their exact solutions.

  1. Similarity variable related to X3 is u(x, t) = F(x) and F(x) satisfies Fxx = 0 two times integration implies that F(x) = c1 + xc2 and we have the exact solution

    u(x,t)=c1+xc2.(18)

  2. Similarity variable related to αX2 + X3 is u(x, t) = αt + F(x) and F(x) satisfies Fxx = 0 which after integrating twice gives F(x) = c1 + xc2 and we have the exact solution

    u(x,t)=αt+c1+xc2.(19)

  3. Similarity variable related to X4 is u(x, t) = x2+aF(ζ)ax, ζ = tx and F(ζ) satisfies

    F(ζ)(2+2bFζaζFζζ+ζ[2bFζ2+Fζ(2+(a+b)ζFζζ)+ζ(Fζζ+ζFζζζζ)]=0,(20)

    thrice integration of Eq. (20) and letting c1 = 0 yields

    4ζFζ+14ζ2((a+b)F2(ζ)36F(ζ))+(c3ζ+c2)ζ=0,(21)

    solving for F(ζ) in Eq. (21) we obtain

    F(ζ)=29BesselJ(9,abc1ζc2ζ)c4+Q0Q1Q2+Q32ζL1L2,(22)

    where

    L1=(a+b)BesselJ(9,abc1ζc2ζ),L2=(a+b)BesselJ(9,abc1ζc2ζ)c4),Q0=abc1ζc2,Q1=2Bessel(10,abc1ζc2x),Q2=BesselJ(8,abc1ζc2ζ)c1,Q3=BesselJ(10,abc1ζc2ζ)c4.

    Hence by back substituting the similarity variables we get the exact solution as

    u(x,t)=xa+E0,(23)

    where

    E0=29BesselJ(9,abc1ζc2ζ)c4+Q0Q1Q2+Q32ζx(L1L2),

    L1, L2 are as stated above and ζ = tx.

  4. Similarity variable related to X1 + X4 is u(x, t) = x2+F(ζ)1+ax, ζ = t(1 + ax) and F(ζ) satisfies the following

    22a2bζFζ2aζFζζ+a2ζ2Fζζ+a2F(ζ)(2+2bFζaζFζζ)+Fx(2b2a2ζ+a2(a+b)ζ2Fζζ)+a3ζ3Fζζζζ=0,(24)

    three times integration and letting c1 = c2 = c3 = 0 in the above equation, gives

    (4aζFζ+(a+b)F2(ζ)36aF(ζ))14a2ζ2+ζ33=0,(25)

    solving for F(ζ) we have the following equation in Bessel function form

    F(ζ)=4aζ(P0+G0+G2+G3)G6,(26)

    where

    P0=(aα2bα2)92ζ72BesselK[9,aα2bα2ζa23]4608a183a18,G0=(aα2bα2)5ζ4(BesselK[8,aα2bα2ζa23]G1)248832a20,G1=BesselK[10,aα2bα2ζa23],G2=c43(aα2bα2)92BesselK[9,aα2bα2ζa23]8a18(a+b)92α9,G3=c435(aα2bα2)5ζ4G448a18(a+b)92α9a20,G4=(BesselK[8,aα2bα2ζa23]+G5,G5=BesselK[10,aα2bα2ζa23]),G6=(aα2bα2)92BesselK[9,aα2bα2ζa23]20736a183G7,G7=35(aα2bα2)92BesselK[9,aα2bα2ζa23]c443a18(a+b)92α9.

    and the exact solution is

    u(x,t)=x21+ax4aζ(P0+G0+G2+G3)G6,(27)

    where ζ = t(1 + ax).

  5. Similarity variable related to βX3 + X4 is u(x, t) = x2+aF(ζ)ax,ζ=(atα)xa and F(ζ) satisfies the following

    F(ζ)(2+2bFζaζFζζ+ζ[2bFζ2+Fζ(2+(a+b)ζFζζ)+ζ(Fζζ+ζFζζζζ)=0,(28)

    here the reduced PDE is the same as that in reduction 3, the only different is the variable ζ. Therefore, we get

    u(x,t)=xa+E(29)

    where

    E=29BesselJ(9,abc1ζc2ζ)c4+Q0Q1Q2+Q32ζx(L1L2)

    and L1, L2, Q0, Q1, Q2, and Q3 are as stated in Eq. (23) and ζ = (atα)xa.

  6. Similarity variable related to αX1 + βX2 + X3 is u(x, t) = βx+αF(ζ)α,ζ=αtxα and F(ζ) satisfies the following

    α(1+αaβ+(a+b)Fζ)FζζFζζζζ)=0,(30)

    three times integration of Eq. (30) with c1 = 0 leads

    Fζ14(a+b)αF2(ζ)+(c3+ζc2)ζ=0(31)

    solving for Fζ, we obtain

    F(ζ)=4(aα+bα)223(c2(aα+bα))23[A1+A2(aα+bα)(A3+A4)],(32)

    where

    A1=AiryBiPrime[243(14c3(aα+bα)+14c2(aα+bα)ζ)(c2(aα+bα)23)],A2=AiryAiPrime[243(14c3(aα+bα)+14c2(aα+bα)ζ)c4(c2(aα+bα)23)],A3=AiryBiPrime[243(14c3(aα+bα)+14c2(aα+bα)ζ)(c2(aα+bα)23)],A4=AiryAiPrime[243(14c3(aα+bα)+14c2(aα+bα)ζ)c4(c2(aα+bα)23)],

    Thus, by back substituting the similarity variables we get

    u(x,t)=βxα+4(aα+bα)223(c2(aα+bα))23[A1+A2(aα+bα)(A3+A4)](33)

    where ζ=αtxα.

3 Physical interpretation of the solutions (23) and (33)

In order to have clear and proper understanding of the physical properties of the power series solution, the 3-D, 2-D and contour plots for the solution Eqs. (23) and (33) are plotted in Figures 1-4 by using suitable parameter values.

3D plot of (33) α = 4, a = 1, b = 2.5, c4 = 13
Figure 1

3D plot of (33) α = 4, a = 1, b = 2.5, c4 = 13

contour plot of (33) α = 80, c1 = 5, c2 = 0.5
Figure 2

contour plot of (33) α = 80, c1 = 5, c2 = 0.5

3D plot of (23) α = 80, β = 10, a = 3, b = 2.5, c3 = 0.5, c2 = c4 = 10
Figure 3

3D plot of (23) α = 80, β = 10, a = 3, b = 2.5, c3 = 0.5, c2 = c4 = 10

contour plot of (23) α = 80, β = 10, a = 3, b = 2.5, c3 = 0.5, c2 = c4 = 10
Figure 4

contour plot of (23) α = 80, β = 10, a = 3, b = 2.5, c3 = 0.5, c2 = c4 = 10

4 Nonlinear self-adjointness

The system of m [20, 21] differential equations

Fα¯(x,u,u(1),...,u(s))=0,(34)

α = 1, …, m, with m dependent variables u = (u1,…,um) is said to be NSA if the adjoint equations

Fα(x,u,u(1),v(1)...,u(s),v(s))δ(vβ¯Fβ¯)δuα=0,(35)

α = 1, …,m, are satisfied for all solutions u of the original system Eq. (34) upon a substitution

vα¯=φα¯(x,u),(36)

α = 1, …, m, such that

φ(x,u)0.(37)

On the other hand, the equation below holds:

Fα(x,u,φ(x,u),...,u(s),φ(s))=λαβ¯Fβ¯(x,u,...,u(s)),(38)

α = 1, …, m, where λαβ¯ are undetermined coefficients, and φ(σ) are derivatives of Eq. (36),

φ(σ)={Di1...Diσ(φα¯(x,u))},

,σ = 1, …, s. Here v and φ are the m-dimensional vectors v = (v1, …, v(m)), φ = (φ1, …, φm), and also, it is worth noting that not all components φα(x, u) of φ vanish simultaneously from Eq. (37).

4.1 Test for self-adjointness for GSWW

Here, we want to test the self-adjointness of Eq. (1). The adjoint equation for Eq. (1) is given by

F=2bυxuxt+1+auxυxt+aυtuxxbυtuxx+1+butυxx+υxxxt=0.(39)

Let v = ϕ(x, t, u), after some calculations and equating the coefficients of the derivatives ut, ux, uxt, uxx, uxxt and uxxx to zero, we have

ϕuu=03ϕ,uu=03ϕ,uuu=0,ϕuuuu=0,ϕtu=0,3ϕtuu=0,ϕtuuu=0,3ϕxu=0,(ab)ϕu+3ϕxuu=0,2bϕu+3ϕxuu=0,(a+b)ϕuu+3ϕxuuu=0,aϕtbϕt+3ϕxtu=0,ϕuu+aϕtu+3ϕxtuu=0,2bϕx+3ϕxxu=0,ϕuu+aϕxu+2bϕxu+3ϕxxuu=0,ϕtu2ϕxu+aϕxt+3ϕxxtu=0,ϕxu+bϕxx+ϕxxxu=0,ϕxtϕxx+ϕxxxt=0.

The solution for ϕ(t, x, u) from the above equation is simply found to be the following

ϕ(t,x,u)=C1for(ab)b0,(40)

where C1 is an arbitrary constant. Therefore, Eq. (1) is NSA with the substitution in Eq. (40).

5 Conservation laws for GSWW

In this section, we establish Cls for Eq. (1) [19,20,21].

The reality that Eq. (1) is NSA with the obtained differential substitution in Eq. (40), we can use the Noether operator 𝓝 to obtain its conserved vectors (C1, C2) [17,18,19,20]. The obtained conserved vectors will satisfy the conservation equation DxC1 + DtC2 = 0. Moreover, the non local variables appearing in that formula must be substituted according to equation (40).

The following are the conserved vectors obtained from the four infinitesimals say X1, X2, X3, and X4 when C1 = 1 respectively.

C1=142+4buxuxt+uxxxt,C2=142+4buxuxx+uxxxx.(41)

C1=12(a2b)uxt,C2=12(a2b)uxx.(42)

Similarly, one can verify and see that DxC1 + DtC2 = 0 which is a trivial conservation laws.

C1=14utt22aux+42autuxt3uxxtt,C2=1421+auxuxt+22+autuxx+3uxxxt.(43)

C1=148+8aux+2atutt1+aux

=4atuxt+2axuxt8bxuxt(44)

=2a2u(x,t)uxt+4abu(x,t)uxt(45)

=+4abxuxuxt+ut8b8abux+2a2tuxt(46)

=auxxt+3atuxxtt+axuxxxt,(47)

C2=144+4a2ux22atuxt=4atuxx+2axuxx8bxuxx2a2u(x,t)uxx=+4abu(x,t)uxx+2a2tutuxx+2aux4+atuxt=+2bxuxx+4auxxx+3atuxxxt+axuxxxx}.

6 Conclusion

In this study, Lie symmetry analysis, one dimensional optimal system and Cls for GSWW equation were studied. Some reductions and their solutions were reported from the obtained one dimensional optimal system. We presented some figures for some of the obtained exact solutions. The exact solutions include an expression with a Bessel function and an Airy function. We verified the authenticity of the solution by substitution into the original equation. The GSWW equation is a NSA with the obtained differential substitution, we obtained Cls using the new conservation theorem presented by Ibragmov.

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About the article

Received: 2017-08-19

Accepted: 2018-01-09

Published Online: 2018-06-26


Citation Information: Open Physics, Volume 16, Issue 1, Pages 364–370, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0049.

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© 2018 Dumitru Baleanu et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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