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

  • Dumitru Baleanu EMAIL logo , Mustafa Inc , Abdullahi Yusuf and Aliyu Isa Aliyu
Published/Copyright: June 26, 2018
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Open Physics
From the journal Open Physics Volume 16 Issue 1

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+butuxx−uxt−uxx=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=c2−12auc3+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=−at∂t−(au−2x)∂u+ax∂x.(12)

The corresponding commutator table of the infinitesimal generators is given by

Table 1

Commutator table of the Lie algebra for GSWWE

[Xi, Xj]X1X2X3X4
X1000aX1 + 2X2
X2000−aX2
X3000−aX3
X4−aX1 − 2X2aX2aX30

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=cμvλlv∂∂lλ,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:

  1. Case 1: For μ = 1, v = 4, and λ = 1, 2 in Table 1. [X1, X4] = c141X1c142X2 and the non vanishing structure constants are ( cμvλ ) are c141 = a, c142 = 2.

  2. Case 2: For μ = 2, v = 4, and λ = 2 in Table 1. [X2, X4] = c242X2 and the non vanishing structure constants ( cμvλ ) are c242 = −a.

  3. Case 3: For μ = 3, v = 4, and λ = 3 in Table 1. [X3, X4] = c343X3 and the non vanishing structure constants ( cμvλ ) are c343 = ∗a.

  4. Case 4: For μ = 4, v = 1, and λ = 1, 2 in Table 1. [X4, X1] = c411X1c412X2 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] = c522X2 and c422 = a, Setting v = 3, λ = 3 row four column three, we get [X4, X3] = c433X3 and c433 = a.

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

E1=al4∂∂l1+2l4∂∂l2,E2=−al4∂∂l2,E3=−al4∂∂l3,E4=−al1∂∂l1−2l1∂∂l2+al2∂∂l2+al3∂∂l3.(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.

  1. 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 = aϵ1l4 + l1, l2 = 2ϵ1l4 + l2, l3 = l3, l4 = l4.

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

  2. For E2, we obtain l1 = l1, l2 = −aϵ2l4 + l2, l3 = l3, l4 = l4.

  3. For E3, we have l1 = l1, l2 = l2, l3 = −aϵ3l4 + l3 and l4 = l4.

  4. For E4, we have l1¯=l11+aϵ4,l2¯=−2ϵ1+l21−aϵ4,l3¯=l31−aϵ4 and l4 = l4.

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

aϵ1l4+l12ϵ1l4+l2l3l4l1−aϵ2l4+l2l3l4l1l2−aϵ3l4+l3l4l11+aϵ4−2ϵ1+l21−aϵ4l31−aϵ4l4

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+2bFζ−aζFζζ+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,−a−b−c1ζ−c2ζ)c4+Q0Q1−Q2+Q32ζL1−L2,(22)

    where

    L1=−(a+b)BesselJ(9,−a−b−c1ζ−c2ζ),L2=(a+b)BesselJ(9,−a−b−c1ζ−c2ζ)c4),Q0=−a−b−c1ζ−c2,Q1=−2Bessel(10,−a−b−c1ζ−c2x),Q2=BesselJ(8,−a−b−c1ζ−c2ζ)c1,Q3=BesselJ(10,−a−b−c1ζ−c2ζ)c4.

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

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

    where

    E0=29BesselJ(9,−a−b−c1ζ−c2ζ)c4+Q0Q1−Q2+Q32ζx(L1−L2),

    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

    2−2a2bζFζ2−aζFζζ+a2ζ2Fζζ+a2F(ζ)(2+2bFζ−aζFζζ)+Fx(2b−2a2ζ+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=−(aα2−bα2)92ζ72BesselK[9,−aα2−bα2ζa23]4608a183a18,G0=−(aα2−bα2)5ζ4(BesselK[8,−aα2−bα2ζa23]−G1)248832a20,G1=BesselK[10,−aα2−bα2ζa23],G2=−c43(−aα2−bα2)92BesselK[9,−aα2−bα2ζa23]8a18(a+b)92α9,G3=c435(−aα2−bα2)5ζ4G448a18(a+b)92α9a20,G4=(BesselK[8,−aα2−bα2ζa23]+G5,G5=BesselK[10,−aα2−bα2ζa23]),G6=(−aα2−bα2)92BesselK[9,−aα2−bα2ζa23]20736a183−G7,G7=35(−aα2−bα2)92BesselK[9,−aα2−bα2ζa23]c443a18(a+b)92α9.

    and the exact solution is

    u(x,t)=x21+ax−4aζ(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=(at−α)xa and F(ζ) satisfies the following

    F(ζ)(2+2bFζ−aζFζζ+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,−a−b−c1ζ−c2ζ)c4+Q0Q1−Q2+Q32ζx(L1−L2)

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

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

    α(1+α−aβ+(a+b)Fζ)Fζζ−Fζζζζ)=0,(30)

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

    Fζ−14(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=αt−xα.

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.

Figure 1 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

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

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

Figure 3 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

Figure 4 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+aυtuxx−bυtuxx+−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,(a−b)ϕu+3ϕxuu=0,2bϕu+3ϕxuu=0,(a+b)ϕuu+3ϕxuuu=0,aϕt−bϕt+3ϕxtu=0,−ϕuu+aϕtu+3ϕxtuu=0,2bϕx+3ϕxxu=0,−ϕuu+aϕxu+2bϕxu+3ϕxxuu=0,−ϕtu−2ϕxu+aϕxt+3ϕxxtu=0,−ϕxu+bϕxx+ϕxxxu=0,−ϕxt−ϕxx+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(a−b)b≠0,(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=14−2+4buxuxt+uxxxt,C2=−14−2+4buxuxx+uxxxx.(41)
C1=12(a−2b)uxt,C2=−12(a−2b)uxx.(42)

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

C1=14utt2−2aux+4−2autuxt−3uxxtt,C2=142−1+auxuxt+2−2+autuxx+3uxxxt.(43)
C1=14−8+8aux+2atutt−1+aux
=−4atuxt+2axuxt−8bxuxt(44)
=−2a2u(x,t)uxt+4abu(x,t)uxt(45)
=+4abxuxuxt+ut8b−8abux+2a2tuxt(46)
=−auxxt+3atuxxtt+axuxxxt,(47)
C2=−144+4a2ux2−2atuxt=−4atuxx+2axuxx−8bxuxx−2a2u(x,t)uxx=+4abu(x,t)uxx+2a2tutuxx+2aux−4+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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Received: 2017-08-19
Accepted: 2018-01-09
Published Online: 2018-06-26

© 2018 Dumitru Baleanu et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2018-0049
Keywords for this article
GSWW; optimal system; Cls; infinitesimal generators; NSA
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Articles
  2. A modified Fermi-Walker derivative for inextensible flows of binormal spherical image
  3. Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics
  4. Three-dimensional atom localization via probe absorption in a cascade four-level atomic system
  5. Determination of the energy transitions and half-lives of Rubidium nuclei
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  7. Three phase heat and mass transfer model for unsaturated soil freezing process: Part 2 - model validation
  8. Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity
  9. Constructing analytic solutions on the Tricomi equation
  10. Feynman diagrams and rooted maps
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  63. Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
  64. Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
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  66. Rank correlation between centrality metrics in complex networks: an empirical study
  67. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
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  69. Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
  70. Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
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  75. Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
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  79. Pulsed excitation terahertz tomography – multiparametric approach
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  81. Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
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  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
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  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
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  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
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  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
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