1. Introduction

Symmetry group analysis is one of the most efficient methods in the analysis of differential equations and there are many studies in the literature dealing with the analysis of differential equations based on Lie symmetries, reductions and invariant solutions [1 , 4, 17–20, 22, 25–27 ]. In addition to application of Lie point symmetries to differential equations, λ-symmetry approach, which is first studied by Muriel and Romero [14, 15], plays important role for analyzing the analytical implicit solutions of ordinary differential equations (ODEs). In the literature, in order to obtain the extension of λ-symmetries for the case of partial differential equations (PDEs), it can be characterized by λ-prolongation in J⁽ⁿ⁾M, where (M,π,B) is the space B of independent variables seen as a bundle over the space B. Once this characterization is obtained, it is extended to the ordinary differential equations for the case B = ℝ and to the partial differential equations for the case B = ℝ^p, and then to the case of systems of PDEs.

Cicogna, Gaeta and Morando [4– 6] improved the λ-symmetry approach for PDEs. These symmetries are called μ-symmetries and the corresponding conservation laws for μ-symmetries are called μ-conservation laws. In this approach, for p independent variables x = (x¹,...,x^p) and q dependent variables u = (u¹,...,u^q), μ function is a horizontal one-form μ = λ_idxⁱ on the first-order jet space (J⁽¹⁾M,π,M), where μ is a compatible, that is, D_iλ_j − D_jλ_i = 0. If μ-prolongation is applied to differential equations then the determining equations are obtained for μ-symmetries and by solving these determining equations, the infinitesimals functions ξ, τ, ϕ and λ_i are determined. The application of μ-symmetry approach not only to ODEs but also to PDEs can be seen as the new application area of the symmetry group theory to differential equations.

In the literature, the Gardner equation was initially discussed by Miura in 1968 [13]. It is known that the Gardner equation is an extension of KdV equation and it introduces the identical features of the classical KdV equation, but it expands its order of availability to a larger interval of the parameters of the interior wave motion for a certain surrounding. The event of shallow-water wave is governed by Gardner equation, which is the KdV equation with dual power nonlinearity. Experimental investigations have shown that Gardner equation models deep ocean waves rather than shallow-water waves that is used by KdV equation. It is widely used in various areas of physics, such that plasma physics, fluid dynamics, quantum field theory, and it is a useful model for the description of a great variety of wave phenomena in plasma and solid state [2,3]. The μ-symmetries and μ-conservations laws of the extended KdV equation are investigated in the study [9].

In the studies [11] and [12], Johnpillai and Khalique investigated the optimal system of one-dimensional subalgebras of the Lie symmetry algebras of the classes

and where A₁(t) and A₄(t) represent the dispersion term and the linear damping term, respectively and u(x,t) denotes the amplitude of the relevant wave mode, which is a function of two independent variables x and t indicating the space variable in the direction of wave propagation and time parameter, respectively. Later, the same authors constructed conservation laws for Eq. (1.1) for some special forms of the functions A₁(t) and A₄(t). Additionally, Vaneeva [28] examined the variable-coefficient Gardner equation where A₂(t) and A₃(t) are smooth functions satisfying the condition A₁(t).A₃(t) ≠ 0. Bruzon and Rosa studied [23] the Gardner equation of the form where n is an arbitrary positive integer. In addition, the Lie point symmetries and the corresponding conservation laws of the Gardner equation with variable coefficients for n =1 are investigated in the study [24].

We propose herein a generalized variable-coefficient Gardner equation of the form

where m is an arbitrary positive integer. It is known that PDEs with two independent variables can be reduced to ODEs by using Lie point symmetry groups. If one obtains some solutions for the differential equations under the symmetries that leave invariant the equation itself then these solutions are called similarity solutions, which are invariant under the Lie group of transformations. In the literature, there are some studies in which exact solutions of PDEs are obtained from the similarity reductions, for example [20, 21]. For the cases of μ-symmetries, μ-reductions and μ-invariant solutions, the concept of the infinitesimal symmetry generator and the standard conservation laws should be redefined by taking account of μ-infinitesimal functions and λ_i functions [7, 8]. One of the aims of the present study is also to classify the μ-symmetries of the Gardner equation and to obtain μ-invariant solutions of the Gardner equation by using μ-symmetries and to determine μ-conservation laws.

This study is organized as follows. In section 2, we present the fundamental definitions about μ-prolongation and μ-symmetries. In section 3, we discuss some different forms of the Gardner equation and the corresponding determining equations. This section also includes different cases corresponding to different choices of coefficients. Furthermore, μ-symmetries for each different case are presented. Section 4 presents similarity reductions and some invariant solutions of a generalized variable coefficient Gardner equation. In the section 5, we examine μ-conservation laws of the Gardner equation. The last section summarizes some important results in the study.

2. Preliminaries

In this section, we introduce the concept of μ-symmetries for PDEs. Let us consider a space M = B×U with coordinates x ∈ B ≃ ℝ^p and u ∈ ℝ^q, x is an independent and u is a dependent variable. M is the total space of a linear bundle (M,π,B) over the base space B. The bundle M can be prolonged to the k-jet bundle (J^(k)M,π_k,B), with J⁽⁰⁾M ≡ M, the total space of the jet bundle is also called the jet space. Now we suppose any bundle P, the set of sections of this bundle is denoted by Γ(P) and the set of vector fields in P by χ(P). And μ = λ_idxⁱ be horizontal one-form on first-order jet space (J⁽¹⁾M,π,M) and compatible with contact structure ε on J^(k)M for k ≥ 2, i.e. dμ ∈ J(ε), where J(ε) is Cartan ideal generated by contact structure e and λ_i : J⁽¹⁾M → ℝ. μ-symmetries should satisfy the compatibility condition, that is D_iλ_j − D_jλ_i = 0, where D_i is total derivative of xⁱ. The horizontal one-form μ ∈ Λ¹(J¹M) on the vector bundle (M,π,ℝ) is the one-form μ = λ(x,u,u_x)dx, where λ(x,u,u_x) : J⁽¹⁾M → ℝ is smooth real function.

3. μ-symmetries of Gardner equation

In this section, we study μ-symmetry properties of the Gardner equations with variable coefficients. To deal with μ-symmetry analysis, the Gardner equation can be considered in different forms including a general form. For this purpose, we first, deal with the generalized variable-coefficient Gardner equation of the form

where n and m are non-zero integers. To determine μ-symmetries, first μ-prolongation of the vector field Y is applied to Gardner equation and then the determining equations are obtained. In order to determine μ-symmetries of PDEs, one can use the similar method as in the case for λ-symmetries of ODEs. Let us say X be a vector field on M and Y be μ-prolongation of μ = λ_idxⁱ. When the μ-prolongation of the vector field Y is applied to the differential Eq. (3.1) and then the determining equations can be obtained for μ-symmetries and the infinitesimal functions ξ, τ, ϕ, and λ_i. In addition, X = ξ∂_x +η∂_t + ϕ∂_u is a vector field and μ = λ₁dx+λ₂dt is a horizontal one-form. For λ₁ and λ₂ functions, the necessary condition to be satisfied is called the compatibility condition, which is represented by the formula D_tλ₁ = D_xλ₂. Thus, μ-prolongation of infinitesimal generator X (2.1) is written where

Now if we suppose μ = λ_idxⁱ is horizontal 1-form and V = exp(∫ μ)X is exponential vector field, where X is a vector field given by formula (2.1) then the characteristic function is defined as Q = ϕ − u_iξⁱ. Thus, one can write

If the μ-prolongation Y is acted on the Eq. (3.1) and by substituting (−A₂(t)u^mu_x −A₃(t)u²ⁿu_x − A₁(t)u_xxx − A₄(t)u) for u_t, we can use the property of functional independence of the variables u(x,t) and its derivatives and rearrange all polynomials, and separate independent functions, and equate the coefficients of various powers of the variables to zero. As a result, the following over-determined system of partial differential equations called determining equations are obtained in terms of η(x,t,u), ξ(x,t,u), ϕ(x,t,u), λ₁(x,t,u), and λ₂(x,t,u) functions

The standard but somewhat troublesome calculations for above determining equations yield unknown functions as infinitesimal functions, namely, η, ξ, ϕ of the infinitesimal generator (2.1) of the form

and the corresponding λ₁ and λ₂ functions, which satisfy the compatibility condition D_tλ₁ = D_xλ₂, are and where m ≠ 2n, F(x,t) is an arbitrary function and c₁, c₂, c₃, c₄ and c₅ are arbitrary constants. In addition, we have following differential relations among the coefficients A₁(t), A₂(t), A₃(t) and A₄(t) for the μ-invariance condition of the Eq. (3.1), which are obtained from solutions of determining equations, and

Therefore, it is possible to carry out a one-type of μ-symmetry classification from Eq. (3.9) and Eq. (3.10) based on the relations between coefficients A₁(t), A₂(t), A₃(t) and A₄(t) of the Gardner equation. In addition, the other type of classification is possible with respect to the relations between parameters m and n in (3.9) and (3.10). From these two equations, it is clear that five different cases can be considered to examine μ-symmetries for different choice of integer parameters m and n, which are m ≠ n, m = n, m = −2n, m = 5n, and m = (4/5)n since Eq. (3.9) and Eq. (3.10) have different forms and different solutions for A₁(t), A₂(t), A₃(t), and A₄(t) for each case. In this study, we only examine the μ-symmetry classification with respect to relations between the parameters m and n. We consider each case in order below.

• CASE I: m ≠ n. From the mathematical point of view, it is not possible to determine the general solutions of A₁(t), A₂(t), A₃(t) and A₄(t) from Eqs. (3.9) and (3.10). For this reason, we consider some special chooses of the functions A₁(t), A₂(t), A₃(t) and A₄(t). In order to obtain a specific solution for (3.9) and (3.10), for example, lets choose A₁(t) = A₁, where A₁ is a constant. Under this consideration, Eq. (3.9) becomes

  A12m(4n−5m)A2(t)2A3′(t)2+4nA3(t)2((m−5n)A2′(t)2−(m−2n)A2(t)A2″(t))+2mA2(t)A3(t)(6nA2′(t)A3′(t)+(m−2n)A2(t)A3″(t)))=0, (3.11)
  and Eq. (3.10) is written in the form
  A1(4(m−5n)A3(t)2A2′(t)2+2A2(t)A3(t)(3(m+2n)A2′(t)A3′(t)+2(m−2n)A3(t)(3nA4(t)A2′(t)−A2″(t)))+2A2(t)2(−5m+4n)A3′(t)2+2(m−2n)2A3(t)2A4′(t)−(m−2n)A3(t)(3mA4(t)A3′(t)−2A3″(t))=0.(3.12)

  It is clear that from Eq. (3.11), first, the solution of function A₃(t) can be determined and then the function A₂(t) is found from the equation (3.12) as below

  A3(t)=c3A2(t)2nm(c2−3mt2m−4n)4n−2m3m,A2(t)=c5exp(∫c4+mA4(t)(2mc2−4nc2−3mt)2mc2−4nc2−3mtdt),(3.13)
  where c₂, c₃, c₄ and c₅ are constants. From equations (3.6) and (3.13), the corresponding μ-infinitesimals are written in the following form
  ξ(x,t,u)=F(x,t),η(x,t,u)=−(A1(c5exp(∫c4+mA4(t)(2mc2−4nc2−3mt)2mc2−4nc2−3mtdt))3nm−2n(2mc2−4nc2−3mt)F(x,t))/(−2mm−2nc1(2c2(m−2n)−3mt)(2−4n3mc3(c5exp(∫c4+mA4(t)(2mc2−4nc2−3mt)2mc2−4nc2−3mtdt))2nm(2mc2−4nc2−3mtm−2n)4n−2m3m)3m2(m−2n)+A1mx(c5exp(∫c4+mA4(t)(2mc2−4nc2−3mt)2mc2−4nc2−3mtdt))3nm−2n)),ϕ(x,t,u)=(A1(c5exp(∫c4+mA4(t)(2mc2−4nc2−3mt)2mc2−4nc2−3mtdt))3nm−2nu(c4−2m+mA5(t)(2mc2−4nc2−3mt)F(x,t))/(m(−2mm−2nc1(2c2(m−2n)−3mt)(2−4n3mc3(c5exp(∫c4+mA4(t)(2mc2−4nc2−3mt)2mc2−4nc2−3mtdt))2nm(2mc2−4nc2−3mtm−2n)4n−2m3m)3m2(m−2n)+A1mx(c5exp(∫c4+mA4(t)(2mc2−4nc2−3mt)2mc2−4nc2−3mtdt))3nm−2n)),(3.14)
  where c₁ is an arbitrary constant.

• CASE II: m = n. To provide specific solutions for the last two equations, we now choose A₁(t) = A₁t^k, where A₁ and k are constants. Under this consideration, Eq. (3.9) gets

  −n2t−2+2kA12(16t2A3(t)2A2′(t)2−4tA2(t)A3(t)(3tA2′(t)A3′(t)+A3(t)(2kA2′(t)+tA2″(t)))A2(t)2((−2+k)kA3(t)2+t2A3′(t)2+2tA3(t)(2kA3′(t)+tA3″(t)))=0.(3.15)

  Similarly, the equation (3.10) becomes

  2ntk−1A1(A2(t)A3(t)(9tA2′(t)A3′(t)+A3(t))((k−6ntA4(t))A2′(t)+2tA2″(t)))−2nt−1+kA18tA3(t)2A2′(t)2−A2(t)2(−tA3′(t)2+nA3(t)2(kA4(t)+2tA4′(t)))−A2(t)2A3(t)((k−3ntA4(t))A3′(t)+2tA3″(t))=0.(3.16)

  The solution of (3) is written

  A3(t)=c3A2(t)2nm(c2−3mt2m−4n)4n−2m3m,(3.17)
  where c₂ and c₃ are arbitrary constants. Combining this solution (3.17) with the solution (3), the corresponding μ-infinitesimals are determined as below
  η(x,t,u)=3A1(c2+tk+1)F(x,t)3c1c33tk/2A1tk+A1(k+1)tkx,        ξ(x,t,u)=F(x,t),ϕ(x,t,u)=A1F(x,t)u((3c2k+(k−2)tk+1)A3(t)−3t(c2+tk+1)A3′(t))2ntA1tkA3(t)(3c1c33tk/2+(k+1)xA1tk),(3.18)
  where c₁ is a constant.

• CASE III: m = −2n. To consider Eqs. (3.9) and (3.10) together, the function A₁(t) can be considered in the same form with the previous case as A₁(t) = A₁t^k. Hence, Eq. (3.9) gets

  −4n2t−2+2kA12(7t2A3(t)2A2′(t)2−2tA2(t)A3(t)(−3tA2′(t)A3′(t)+2A3(t)(2kA2′(t)+tA2″(t)))A2(t)2(4(−2+k)kA3(t)2+7t2A3′(t)2−4tA3(t)(2kA3′(t)+tA3″(t)))=0,(3.19)
  and Eq. (3.10) reads
  4nt−1+kA1(−7tA3(t)2A2′(t)2+2A2(t)A3(t)2((k−6ntA4(t)A2′(t)2tA2′(t))A2(t)2(7tA3′(t)2+8nA3(t)2(kA4(t)+2tA4′(t))−2A3(t)((k+6ntA4(t)A3′(t)2tA3′(t)))=0.(3.20)

  From the solution of Eq. (3.19), the function A₂(t)

  A2(t)=c3t2kA3(t)(tk+1+c2)43,(3.21)
  is found, where c₂ and c₃ are arbitrary constants. If one substitutes the function A₂(t) into Eq. (3.20) and the function A₄(t) can be obtained from the solution of Eq. (3.20) in the form
  A4(t)=tkc2+t1+k(c4+∫1/(2nA32(t))t−2−k(c2k(1+k)A32(t)−t2(c2+t1+k)(A3′)2(t)+tA3(t)((t1+k−c2k)A3′(t)+t(c2+t1+k)A3″(t))dt),(3.22)
  where c₄ is an arbitrary constant. For this case, from solutions of equations (3.19), (3.20) and (3.6), μ-infinitesimal functions
  ξ(x,t,u)=F(x,t),η(x,t,u)=3A3(t)A1tk(c2+t1+k)(c1+t1+k)4/3F(x,t)(1+k)tkx(c2+t1+k)4/3A3(t)A1tk+3c1(c2+t1+k)(A3(t)c2t2k)3/4,ϕ(x,t,u)=−A1tk(c1+t1+k)F(x,t)(−(3c2k+(k−2)t1−k)A3(t)+3t(c2+t1+k)A3′(t)2ntA3(t)((1+k)tkxA1tk+3c1(c2+t1+k)(c2t2k)3/4),(3.23)
  are determined, where c₁ is a constant.

• CASE IV: m = 5n. For the function A₁(t) = A₁t^k and the Eqs. (3.9) and (3.10) are converted to the following differential equations

  3n2t−2+2kA12A2(t)−4tA3(t)(−5tA2′(t)A3′(t)+A3(t)(2kA2′(t)+tA2″(t)))+A2(t)(−3(−2+k)kA3(t)2−35t2A3′(t)2+10tA3(t)(2kA3′(t)+tA3″(t)))=0,(3.24)
  and
  6ntk−1A1A2(t)(A3(t)(7tA2′(t)A3′(t)−A3(t))((k−6ntA4(t))A2′(t)+2tA2″(t)))A2(t)(−7tA3′(t)2+3nA3(t)2(kA4(t)+2tA4′(t))+A3(t)((k−15ntA4(t))A3′(t)+2tA3″(t)))=0.(3.25)

  Thus, the solution of Eq. (3.24) yields

  A2(t)=t−3k/2A3(t)5/2((1+k)c2+t1+kc3)1+k,(3.26)
  where c₂ and c₃ are arbitrary constants. From solution of Eq. (3.25), the function A₄(t)
  A4(t)=2nc4tk+tk∫(t−2−k(c2k(1+k)2A3(t)2−t2(c2+c2k+c3t1+k)A3′(t)2/A3(t)2)dt2n(c2+c2k+c3t1+k)++tk∫(tA3(t)(−c2k(1+k)+c3t1+k)A3′(t)+(t(c2+c2k+c3t1+k)A3″(t)))/A3(t)2)dt2n(c2+c2k+c3t1+k),(3.27)
  is determined. Substituting Eqs. (3.26) and (3.27) into (3.6) leads to
  η(x,t,u)=3A1(c2+c2k+c3t1+k)F(x,t)A1tk(1+k)(3c1tk/2+c3xA1tk),        ξ(x,t,u)=F(x,t),ϕ(x,t,u)=A1tk(c2+c2k+c3t1+k)u(x,t)F(x,t)((3c1k(1+k)+c2(k−2)tk+1)A3(t))2ntA3(t)(3c1(1+k)t3k/2(c1+c1k+c2tk+1)+A1tk(c2+c2k+c3t1+k)c2(1+k)xtk)−A1tk(c2+c2k+c3t1+k)u(x,t)F(x,t)(3t(c1+c1k+c2t1+k)A3′(t))2ntA3(t)(3c1(1+k)t3k/2(c1+c1k+c2tk+1)+A1tk(c2+c2k+c3t1+k)c2(1+k)xtk),(3.28)
  where c₁ and c₄ are constants.

• CASE V: m = (4/5)n. For the same function A₁(t) considered above, Eq. (3.9) becomes

  −1225n2t2k−2A12A3(t)(35t2A3(t)A2′(t)2−10tA2(t)(2tA2′(t)A3′(t)+A3(t)(2kA2′(t)+tA2″(t)))+A2(t)2(3(−2+k)kA3(t)+4t(2kA3′(t)+tA3″(t))))=0,(3.29)
  and the Eq. (3.10) is written
  1225nt−1+kA1A3(t)(−35tA3(t)A2′(t)2+5A2(t)(7tA2′(t)A3′(t)+A3(t)((k−6ntA4(t))A2′(t)+2tA2″(t)))+A2(t)2((−5k+12ntA4(t))A3′(t)+6nA3(t)(kA4(t)+2tA4′(t))−10tA3″(t)))=0.(3.30)

  Additionally, the solution of Eq. (3.29) leads to

  A2(t)=c3t3k/5A3(t)2/5(t1+k+c2)2/5,(3.31)
  where c₂ and c₃ are constants. By substituting A₂(t) into (3.30), the function A₄(t)
  A4(t)=c4tk+tk∫(t−2−k(c2k(1+k)2A3(t)2−t2(c2+c2k+c3t1+k)A3′(t)2/2nA3(t)2)dtc2+t1+k+tk∫(tA3(t)(−c2k(1+k)+c3t1+k)A3′(t)+(t(c2+c2k+c3t1+k)A3″(t)))/2nA3(t)2)dtc2+t1+k,(3.32)
  is determined. Finally, the corresponding μ-infinitesimal functions are
  η(x,t,u)=3t(c2+t1+k)A3(t)F(x,t)A3(t)t3k/2((1+k)x+3c1c35/2),        ξ(x,t,u)=F(x,t),ϕ(x,t,u)=tk/2u(x,t)A3(t)F(x,t)((3c2k+(k−2)tk+1)A3(t)−3t(c2+t1+k)A3′(t))2ntx(1+k)t3k/2A3(t)2+3c1c35/2t3k/2A3(t),(3.33)
  where c₁ and c₄ are constants.

  In addition to these cases analyzed above, it is possible to investigate the other different forms of Gardner equations, which are represented in the introduction section. Firstly, we examine Eq. (1.1) ([11]–[12]) of the form

  ut+uux+A1(t)uxxx+A4(t)u=0.

  For Eq. (1.1), by using the same form for A₁(t) function, the infinitesimal functions

  η(x,t,u)=(c1−3tA1)F(x,t)c2−x,      ξ(x,t,u)=F(x,t),       ϕ(x,t,u)=2uF(x,t)(c2−x),(3.34)
  are found, where c₁ and c₂ are constants. Similarly, for Eq. (1.2) of the form
  ut+u2ux+A1(t)uxxx+A4(t)u=0,
  the μ-infinitesimals read
  η(x,t,u)=(3t−3tlog(t))F(x,t)c1−x,      ξ(x,t,u)=F(x,t),       ϕ(x,t,u)=(3log(t)−1)uF(x,t)2c1−x,
  where c₁ is a constant. Finally, we deal with Eq. (1.3)
  ut+A2(t)uux+A3(t)u2ux+A1(t)uxxx=0,
  and by considering the function of form A₁(t) = 1/(a + bt), where a and b are arbitrary constants, the functions A₂(t) and A₃(t)
  A2(t)=c5(3log(a+bt)−bc1)−2b+b2c1+c36ba+bt,         A3(t)=c4(3log(a+bt)−bc1)−b2c1+c33ba+bt,(3.35)
  are determined, where c₃, c₄, and c₅ are constants. Thus, the μ-infinitesimal functions for the equation become
  η(x,t,u)=(bc1−3log(a+bt))(a+bt)F(x,t)b(c2−x),         ξ(x,t,u)=F(x,t),ϕ(x,t,u)=(c3+c1b2−2b)uF(x,t)2b(c2−x),(3.36)
  where c₂ is a constant.