Research article

Lie symmetry reductions and exact solutions to a generalized two-component Hunter-Saxton system

  • Received: 03 August 2020 Accepted: 04 November 2020 Published: 09 November 2020
  • MSC : 37L20, 35C05, 35Q53

  • Based on the classical Lie group method, a generalized two-component Hunter-Saxton system is studied in this paper. All of the its geometric vector fields, infinitesimal generators and the commutation relations of Lie algebra are derived. Furthermore, the similarity variables and symmetry reductions of this new generalized two-component Hunter-Saxton system are derived. Under these Lie symmetry reductions, some exact solutions are obtained by using the symbolic computation. Moreover, a conservation law of this system is presented by using the multiplier approach.

    Citation: Huizhang Yang, Wei Liu, Yunmei Zhao. Lie symmetry reductions and exact solutions to a generalized two-component Hunter-Saxton system[J]. AIMS Mathematics, 2021, 6(2): 1087-1100. doi: 10.3934/math.2021065

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  • Based on the classical Lie group method, a generalized two-component Hunter-Saxton system is studied in this paper. All of the its geometric vector fields, infinitesimal generators and the commutation relations of Lie algebra are derived. Furthermore, the similarity variables and symmetry reductions of this new generalized two-component Hunter-Saxton system are derived. Under these Lie symmetry reductions, some exact solutions are obtained by using the symbolic computation. Moreover, a conservation law of this system is presented by using the multiplier approach.


    The Hunter-Saxton (HS) equation reads

    uxxt+uuxxx+2uxuxx2κux=0, (1.1)

    where u(x,t) depends on a time variable t and a space variable x, κ is a positive constant. This equation was derived as a model for propagation of orientation waves in a massive nematic liquid crystal director field [1]. In fact, it can be regarded as a short wave limit of the well known Camassa-Holm equation [2,3].

    The two-component Hunter-Saxton (2-HS) equation [1] is

    {uxxt+uuxxx+2uxuxx2κux=σρρx,ρt+(ρu)x=0, (1.2)

    where u(x,t) and ρ(x,t) depend on variables t and x, σ,κ are positive constants. The 2-HS equation has attracted much attention and it has been studied extensively and some results were obtained, we can see [4,5].

    Meanwhile, there is a generalized 2-HS system [6] as follow:

    {uxxt+uuxxx+(1α)uxuxxκρρx=0,ρt+uρx=αuxρ, (1.3)

    where α(α1),κ are constants. The model with (α,κ)=(1,1) in system (1.3) appeared initially in the work of Lenells [7]. The author showed that system (1.3) is the geodesic equation on a manifold K which admits a K¨ahler structure. The blow-up phenomena of system (1.3) was investigated in [4,8].

    Our goal is to study exact solutions of system (1.3) by applying classical Lie group method [9,10,11,12,13,14]. Firstly, the vector field for the system (1.3) will be given by Lie symmetry analysis. Secondly, similarity variables and its symmetry reductions equations are obtained. Thirdly, by solving the reduced equations, some exact solutions of the system (1.3) will be presented. Finally, we give a conservation law of system (1.3).

    First of all, let us consider a one-parameter Lie group of infinitesimal transformation:

    xx+ϵξ(x,t,u,ρ),
    tt+ϵτ(x,t,u,ρ),
    uu+ϵϕ(x,t,u,ρ),
    ρρ+ϵψ(x,t,u,ρ),

    with a small parameter ϵ1. The vector field associated with the above group of transformations can be written as

    V=ξ(x,t,u,ρ)x+τ(x,t,u,ρ)t+ϕ(x,t,u,ρ)u+ψ(x,t,u,ρ)ρ, (2.1)

    where the coefficient functions ξ(x,t,u,ρ),τ(x,t,u,ρ),ϕ(x,t,u,ρ) and ψ(x,t,u,ρ) of the vector field are to be determined later.

    If the vector field (2.1) generates a symmetry of the system (1.3), then V must satisfy the Lie symmetry condition

    {pr(3)V(Δ1)|Δ1=0=0,pr(1)V(Δ2)|Δ2=0=0, (2.2)

    where pr(3)V,pr(1)V denote the third and the first prolongation of V respectively, and Δ1=uxxt+uuxxx+(1α)uxuxxκρρx, Δ2=ρt+uρxαuxρ for system (1.3). Expanding (2.2), we find that the coefficient functions ξ,τ,ϕ and ψ must satisfy the symmetry condition

    {ϕxxt+ϕuxxx+uϕxxx+(1α)ϕxuxx+(1α)uxϕxxκψρxκρψx=0,ψt+ϕρx+uψxαϕxραuxψ=0, (2.3)

    where ϕ,ψ,ϕx,ψx,ψt,ϕxx,ϕxxx,ϕxxt are the coefficient functions given by

    ϕt=DtϕuxDtξutDtτ,  ψt=DtψρxDtξρtDtτ,ϕx=DxϕuxDxξutDxτ,  ψx=DxψρxDxξρtDxτ,ϕxx=D2xϕuxD2xξutD2xτ2uxxDxξ2uxtDxτ,ϕxxx=D3xϕuxD3xξutD3xτ3uxxD2xξ3uxtD2xτ3uxxxDxξ3uxxtDxτ,ϕxxt=DtD2xϕuxDtD2xξuxtD2xξ2uxxDtDxξ2uxxtDxξutDtD2xτuttD2xτ2uxtDtDxτ2uxttDxτuxxxDtξuxxtDtτ, (2.4)

    where Dx,Dt are the total derivatives with respect to x and t respectively.

    Substituting (2.4) into (2.3), combined with system (1.3) and setting the coefficients of the various monomials in u and v and their partial derivatives equal to zero one obtains the determining equations for the symmetry group of (1.3) as follows

    ξu=0, ξρ=0, τx=0, τu=0, τρ=0, ϕρ=0, ϕuu=0, ϕxxu=0, ψu=0,ρτt+ψ=0, ϕuξxψρ=0, (1α)(ϕuξx+τt)=0, ξxx2ϕxu=0,uϕxxx+ϕtxxκρψx=0, αρϕx+uψx+ψt=0, u(ξxτt)ξt+ϕ=0,uξxxx(1α)ϕxx=0, 2ξtx+(1α)ϕx+3u(ϕxuξxx)+ϕtu=0. (2.5)

    Solving these determining equations yields

    {ξ=(F1(t)+C1+C2)x+F2(t)+C3,τ=F1(t)α+C2t+C4,ϕ=F1(t)x+((1+α)F1(t)+C1)u+F2(t),ψ=(αF1(t)C2)ρ, (2.6)

    where F1(t),F2(t) are arbitrary functions of t, C1,C2,C3 and C4 are arbitrary constants.

    Thus, the Lie algebra of infinitesimal symmetries of system (1.3) is spanned by the following vector fields

    V1=F1(t)xxαF1(t)t+[F1(t)x+(1+α)uF1(t)]u+αρF1(t)ρ,V2=F2(t)x+F2(t)u,  V3=xx+uu,V4=xx+ttρρ,  V5=x,  V6=t,

    where V1 and V2 are the vector fields corresponding to the arbitrary functions F1(t) and F2(t) respectively.

    The commutation relations of Lie algebra determined by Vi(i=1,2,,6), which are shown as

    [Vi,Vi]=0, i=1,2,,6,[V1,V2]=[V2,V1]=V6(F1F2αF1F2),  [V1,V3]=[V3,V1]=[V2,V5]=[V5,V2]=0,[V3,V4]=[V4,V3]=[V3,V6]=[V6,V3]=[V5,V6]=[V6,V5]=0,[V1,V4]=[V4,V1]=V1(F1tF1),  [V1,V5]=[V5,V1]=V2(F1),[V1,V6]=[V6,V1]=V1(F1),  [V2,V3]=[V3,V2]=V2(F2),[V2,V4]=[V4,V2]=V2(F2tF2),  [V2,V6]=[V6,V2]=V2(F2),[V3,V5]=[V5,V3]=V5,  [V4,V5]=[V5,V4]=V5,  [V4,V6]=[V6,V4]=V6.

    It is obvious that the vector fields Vi(i=1,2,,6) are closed under the Lie bracket.

    In this section, we will get similarity variables and its symmetry reductions. By solving the reduced equations, some exact solutions of the system (1.3) will be presented.

    Based on the infinitesimals (2.6), the similarity variables are found by solving the corresponding characteristic equations

    dxξ=dtτ=duϕ=dρψ.

    Case 1 Let C1=C2=F1(t)=0, C3(0) and C4 be arbitrary constants, F2(t) is an arbitrary functions of t, then by solving the characteristic equation one can get the similarity variables

    ω=xF2(t)+C4C3dt,  f(ω)=uF2(t)C3,  g(ω)=ρ,

    and the group-invariant solution is

    {u=F2(t)C3+f(ω),ρ=g(ω). (3.1)

    Substituting the group-invariant solution (3.1) into system (1.3), we reduce equation (1.3) to the following ODE:

    {C4fC3ff(1α)C3ff+C3κgg=0,C4g+αC3fgC3fg=0, (3.2)

    where f=df/dω,g=dg/dω.

    Case 2 Let C1,C3 be arbitrary non-zero constants, C2=C4=F1(t)=F2(t)=0, then by solving the characteristic equation one can get the similarity variables

    ω=xexp(C1tC3),  f(ω)=uexp(C1tC3),  g(ω)=ρ,

    and the group-invariant solution is

    {u=exp(C1tC3)f(ω),ρ=g(ω). (3.3)

    Substituting the group-invariant solution (3.3) into system (1.3), we reduce (1.3) to the following ODE:

    {C1ωfC3ff+C3(α1)ff+C1f+C3κgg=0,C1ωg+αC3fgC3fg=0, (3.4)

    where f=df/dω,g=dg/dω.

    Case 3 Let F1(t)=kt,F2(t)=0, C1,C2,C3,C4 and k be constants which satisfy C2αk0 and k+C1+C20, then by solving the characteristic equation one can get the similarity variables

    ω=[(k+C1+C2)x+C4](αkt+C2t+C3)k+C1+C2C2αkk+C1+C2,f(ω)=u[(C2αk)t+C3]αk+C1+kC2αk,g(ω)=ρ[(C2αk)t+C3],

    and the group-invariant solution is

    {u=[(C2αk)t+C3]αk+C1+kC2αkf(ω),ρ=g(ω)(C2αk)t+C3. (3.5)

    Substituting the group-invariant solution (3.5) into system (1.3), we reduce (1.3) to the following ODE:

    {(k+C1+C2)ωf+ff+(1α)ffκgg=0,(k+C1+C2)ωgαfg+fg=0, (3.6)

    where f=df/dω,g=dg/dω.

    In this section, we will derive the solutions of system (1.3) by using the symbolic computation [15,16,17]. Suppose that the solution of equation (3.2) is in the form

    f=a0+a1F+a2F2,g=b0+b1F+b2F2 (4.1)

    where F(ω) expresses the solution of the following generalized Riccati equation

    F=r+pF+qF2, (4.2)

    and r,p,q are real constants. Substituting (4.1) along with (4.2) into (3.2) and collecting all terms with the same power in Fi(i=0,1,,7) and setting the coefficients to zero yields a system of algebraic equations. Solving the algebraic equations and we can have the following results

    α=2,a0=±2κb2p4q2+C4C3,a1=±2κb22q,a2=0,b0=b2p24q2,b1=b2pq, (4.3)

    with b2,p,q,r,C3,C4 are constants and κ is a positive constant.

    The solutions of equation (4.2) are listed as follows:

    (a) When p24qr>0 and pq0 (qr0),

    F1=12q[p+p24qrtanh(p24qr2ω)],F2=12q[p+p24qrcoth(p24qr2ω)],F3=12q[p+p24qr[tanh(p24qrω)±isech(p24qrω)]],F4=12q[p+p24qr[A2+B2Acosh(p24qrω)]Asinh(p24qrω)+B],F5=12q[pp24qr[B2A2+Asinh(p24qrω)]Acosh(p24qrω)+B], B2A2>0,F6=2rcosh(p24qr2ω)p24qrsinh(p24qr2ω)pcosh(p24qr2ω),F7=2rsinh(p24qr2ω)p24qrcosh(p24qr2ω)psinh(p24qr2ω),

    where A,B are arbitrary constants.

    (b) When p24qr<0 and pq0 (qr0),

    F8=12q[p+4qrp2tan(4qrp22ω)],F9=12q[p+4qrp2cot(4qrp22ω)],F10=12q[p+4qrp2[tan(4qrp2ω)±sec(4qrp2ω)]],F11=12q[p+4qrp2[A2B2Acos(4qrp2ω)]Asin(4qrp2ω)+B], A2B2>0,F12=12q[p4qrp2[A2B2Asin(4qrp2ω)]Acos(4qrp2ω)+B], A2B2>0,F13=2rcos(4qrp22ω)4qrp2sin(4qrp22ω)+pcos(4qrp22ω),
    F14=2rsin(4qrp22ω)4qrp2cos(4qrp22ω)psin(4qrp22ω),

    where A,B are arbitrary constants.

    (c) When r=0 and pq0,

    F15=pCq[cosh(pω)sinh(pω)+C],F16=p[sinh(pω)+cosh(pω)]q[sinh(pω)+cosh(pω)+C],

    where C is an arbitrary constant.

    (d) When p=r=0 and q0,

    F17=1qω+C,

    where C is an arbitrary constant.

    Substituting (4.3) into (4.1) and (3.1), then we can obtain the following different exact solutions of system (1.3):

    (a1) If Δ=p24qr>0 and pq0 (qr0), then the solutions of system (1.3) with α=2 can be derived as

    {u1(x,t)=F2(t)+C4C3±2κb2Δ4q2tanh(Δ2ω),ρ1(x,t)=b2Δ4q2tanh2(Δ2ω), (4.4)

    where ω=xF2(t)C3dt.

    If we take F(t)=F2(t)+C4C3,Δ=2c1(c1>0),b=b24q2, then the above solution can be expressed as a simple form as

    {u1(x,t)=F(t)±22κbc1tanh(c1ω),ρ1(x,t)=4bc21tanh2(c1ω), (4.5)

    where ω=xF(t)dt, and c1(>0),b,κ are constants.

    Similarly, we can derive the other solutions of system (1.3) as

    {u2(x,t)=F(t)±22κbc1coth(c1ω),ρ2(x,t)=4bc21coth2(c1ω). (4.6)
    {u3(x,t)=F(t)±22κbc1[tanh(2c1ω)±isech(2c1ω)],ρ3(x,t)=4bc21[tanh(2c1ω)±isech(2c1ω)]2. (4.7)
    {u4(x,t)=F(t)±22κbc1A2+B2Acosh(2c1ω)Asinh(2c1ω)+B,ρ4(x,t)=4bc21[A2+B2Acosh(2c1ω)Asinh(2c1ω)+B]2, (4.8)

    where A,B are arbitrary constants.

    {u5(x,t)=F(t)±22κbc1B2A2+Asinh(2c1ω)Acosh(2c1ω)+B,ρ5(x,t)=4bc21[B2A2+Asinh(2c1ω)Acosh(2c1ω)+B]2, (4.9)

    where B2A2>0.

    {u6(x,t)=F(t)±22κbc1[psinh(c1ω)2c1cosh(c1ω)2c1sinh(c1ω)pcosh(c1ω)],ρ6(x,t)=4bc21[psinh(c1ω)2c1cosh(c1ω)2c1sinh(c1ω)pcosh(c1ω)]2. (4.10)
    {u7(x,t)=F(t)±22κbc1[pcosh(c1ω)2c1sinh(c1ω)2c1cosh(c1ω)psinh(c1ω)],ρ7(x,t)=4bc21[pcosh(c1ω)2c1sinh(c1ω)2c1cosh(c1ω)psinh(c1ω)]2. (4.11)

    (a2) When Δ=p24qr<0 and pq0 (qr0), if we denote F(t)=F2(t)+C4C3,Δ=2c1(c1>0),b=b24q2, then the solutions of system (1.3) with α=2 can be derived as

    {u8(x,t)=F(t)±22κbc1tan(c1ω),ρ8(x,t)=4bc21tan2(c1ω). (4.12)
    {u9(x,t)=F(t)±22κbc1cot(c1ω),ρ9(x,t)=4bc21cot2(c1ω). (4.13)
    {u10(x,t)=F(t)±22κbc1[tan(2c1ω)±isech(2c1ω)],ρ10(x,t)=4bc21[tan(2c1ω)±isech(2c1ω)]2. (4.14)
    {u11(x,t)=F(t)±22κbc1A2B2Acos(2c1ω)Asin(2c1ω)+B,ρ11(x,t)=4bc21[A2B2Acos(2c1ω)Asin(2c1ω)+B]2, (4.15)

    where A,B are arbitrary constants and A2B2>0.

    {u12(x,t)=F(t)±22κbc1A2B2Asin(2c1ω)Acos(2c1ω)+B,ρ12(x,t)=4bc21[A2B2Asin(2c1ω)Acos(2c1ω)+B]2, (4.16)

    where A2B2>0.

    {u13(x,t)=F(t)±22κbc1[psin(c1ω)2c1cos(c1ω)2c1sin(c1ω)+pcos(c1ω)],ρ13(x,t)=4bc21[psin(c1ω)2c1cos(c1ω)2c1sin(c1ω)+pcos(c1ω)]2. (4.17)
    {u14(x,t)=F(t)±22κbc1[pcos(c1ω)+2c1sin(c1ω)2c1cos(c1ω)psin(c1ω)],ρ14(x,t)=4bc21[pcos(c1ω)+2c1sin(c1ω)2c1cos(c1ω)psin(c1ω)]2. (4.18)

    (a3) When r=0 and pq0, if we denote F(t)=F2(t)+C4C3,b=b24q2, then the solutions of system (1.3) with α=2 can be derived as

    {u15(x,t)=F(t)±2κbp(12Ccosh(pω)sinh(pω)+C),ρ15(x,t)=bp2(12Ccosh(pω)sinh(pω)+C)2, (4.19)

    where ω=xF(t)dt, F(t) is an arbitrary function and b,C are constants.

    {u16(x,t)=F(t)±2κbp(12Csinh(pω)+cosh(pω)+C),ρ16(x,t)=bp2(12Csinh(pω)+cosh(pω)+C)2. (4.20)

    (a4) When p=r=0 and q0, if we denote F(t)=F2(t)+C4C3,b=b24q2, then the solution of system (1.3) with α=2 can be derived as

    {u17(x,t)=F(t)±22κb1ω+C,ρ17(x,t)=4b(1ω+C)2, (4.21)

    where ω=xF(t)dt, F(t) is an arbitrary function and b,C are constants.

    In order to show the properties of the above solutions visually, we plot the 2D-graphs of some typical solutions. Some wave figures are given as follows (Figures 15):

    Figure 1.  (a) 2D figure of solution u1 with F=0.05,κ=2,b=0.1,c1=1, (b) 2D figure of solution ρ1 with F=0.05,κ=2,b=0.1,c1=1.
    Figure 2.  (a) 2D figure of solution u2 with F=0.05,κ=2,b=0.1,c1=1, (b) 2D figure of solution ρ2 with F=0.05,κ=2,b=0.1,c1=1.
    Figure 3.  (a) 2D figure of solution u4 with F=0.05,κ=2,b=0.1,c1=1,A=2,B=2, (b) 2D figure of solution ρ4 with F=0.05,κ=2,b=0.1,c1=1,A=2,B=2.
    Figure 4.  (a) 2D figure of solution u6 with F=1,κ=2,b=0.1,c1=1,p=1, (b) 2D figure of solution ρ6 with F=1,κ=2,b=0.1,c1=1,p=1.
    Figure 5.  (a) 2D figure of solution u13 with F=0.02,κ=2,b=0.1,c1=1,p=1, (b) 2D figure of solution ρ13 with F=0.02,κ=2,b=0.1,c1=1,p=1.

    For the solution (4.5), if we take the integration constant as 0 in ω=xF(t)dt, then we plot the solution for the plus sign in u1 as

    For the solution (4.6), if we take the integration constant as 0 in ω=xF(t)dt, then we plot the solution for the plus sign in u2 as

    For the solution (4.8), if we take the integration constant as 0 in ω=xF(t)dt, then we plot the solution for the plus sign in u4 as

    For the solution (4.10), if we take the integration constant as 0 in ω=xF(t)dt, then we plot the solution for the plus sign in u6 as

    For the solution (4.17), if we take the integration constant as 0 in ω=xF(t)dt, then we plot the solution for the plus sign in u13 as

    Remark 1 If we take F(t) as a constant, then all of the above solutions of system (1.3) are traveling wave solutions.

    Remark 2 For the reduced equations (3.4) and (3.6), there exist a power series solutions [18,19]. We omit the details here for brevity.

    In this section, we use the direct multiplier method [20] to derive a conservation law for system (1.3). The zero-order multipliers Λ1(t,x,u,ρ), Λ2(t,x,u,ρ) for the system (1.3) are determined by

    {δδu[Λ1(uxxt+uuxxx+(1α)uxuxxκρρx)+Λ2(ρt+uρxαuxρ)]=0δδρ[Λ1(uxxt+uuxxx+(1α)uxuxxκρρx)+Λ2(ρt+uρxαuxρ)]=0, (5.1)

    where δδu,δδρ are Euler-Lagrange operators defined by

    δδu=uDxux+D2xuxxDtD2xuxxtD3xuxxx,δδρ=ρDtρtDxρx. (5.2)

    Expanding (5.1) and splitting with respect to derivative of u,ρ, we obtain the following determining equations

    Λ1u=0, Λ1ρ=0, Λ1x=0, Λ2u=0, Λ2t=0, Λ2x=0,αρΛ2ρ+uΛ2u+(1+α)Λ2=0. (5.3)

    Then we obtain the solution

    Λ1(t,x,u,ρ)=H(t), Λ2(t,x,u,ρ)=Aρ1+αα, (5.4)

    where A is an arbitrary constant, H(t) is an arbitrary functions with respect to t. From the solution (5.4), we can see that system (1.3) has one zero-order multiplier in the form of Λ1=H(t), Λ2=ρ1+αα. So a conservation law of system (1.3) is

    Dt(αρ1α)+Dx(H(t)uxt+H(t)uuxxα2H(t)u2xκ2H(t)ρ2αuρ1α)=0. (5.5)

    In this paper, a generalized 2-HS system is investigated by using the classical Lie group method. First, Lie symmetry analysis was performed for the generalized 2-HS system, and its infinitesimal generator, geometric vector fields and commutation table of Lie algebra were obtained. Then, all of the similarity variables and its symmetry reductions of this equation are obtained. And by solving the reduced equations, some new exact solutions including traveling wave solutions of this generalized 2-HS system are constructed successfully. These are new solutions for the generalized 2-HS system. Finally, a conservation law of the generalized 2-HS system are shown by using the multiplier method.

    This work is supported by the National Natural Science Foundation of China (No.11461022) and Applied Basic Research Foundation of Yunnan Province (Nos. 2018FH001-013 and 2018FH001-014), the Science Research Foundation of Yunnan Education Bureau (No. 2018JS479) and the Second Batch of Middle and Young Aged Academic Backbone of Honghe University (No. 2015GG0207).

    The authors declare that there are no conflict interests regarding the publication of this paper.



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