Journal of Nonlinear Mathematical Physics

Volume 26, Issue 1, December 2018, Pages 69 - 90

On μ-symmetries, μ-reductions, and μ-conservation laws of Gardner equation

Özlem Orhan
Department of Mathematical Engineering, Istanbul Technical University, 34469 Maslak Istanbul-Turkey,
Özer Teoman
Department of Civil Engineering, Istanbul Technical University, 34469 Maslak Istanbul-Turkey,
Received 4 June 2018, Accepted 29 June 2018, Available Online 6 January 2021.
10.1080/14029251.2019.1544789How to use a DOI?
μ-symmetries; μ-conservation laws; μ-reductions; classification; Gardner equation

In this study, we represent an application of the geometrical characterization of μ-prolongations of vector fields to the nonlinear partial differential Gardner equation with variable coefficients. First, μ-symmetries and the corresponding μ-symmetry classification are investigated and then μ-reduction forms of the equations are obtained. Furthermore, μ-invariant solutions are determined and μ-conservation laws of Gardner equation are studied.

© 2019 The Authors. Published by Atlantis and Taylor & Francis
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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, 1720, 22, 2527 ]. 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 = ℝp, and then to the case of systems of PDEs.

Cicogna, Gaeta and Morando [46] 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,...,xp) and q dependent variables u = (u1,...,uq), μ function is a horizontal one-form μ = λidxi on the first-order jet space (J(1)M,π,M), where μ is a compatible, that is, DiλjDjλi = 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 λi 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

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 λi 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 ≃p and u ∈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,πk,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 = λidxi 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 λi : J(1)M → ℝ. μ-symmetries should satisfy the compatibility condition, that is DiλjDjλi = 0, where Di 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,ux)dx, where λ(x,u,ux) : J(1)M → ℝ is smooth real function.

Definition 2.1 ([16]).

Let us consider x-derivatives of function u in J(k)M. Then one can introduce a canonical contact conjecture in the jet space J(k)M, that is, the module generated by the set of canonical contact one-forms υaj:=dujauj,madxm.

Definition 2.2 ([7]).

Let Y be a vector field on JkM. The vector field Y preserves the contact structure if LY : εε.

Definition 2.3 ([7]).

A vector field X ∈ χ(M) can be written as


This can be extended to a vector field X(k) in J(k)M by requiring that it preserves the contact structure. Thus, the vector field in J(k)M can be written


Definition 2.4 ([7]).

Let X = ξ∂x +η∂t +ϕ∂u be a vector field on M and Y=X+n=1kΨn(/un) be a vector field on JkM. Let λ: J1M → ℝ be a smooth function. We say that Y is the λ-prolongation of X if its coefficients satisfy the λ-prolongation formula

for all n = 0,...,k − 1.

Definition 2.5 ([7,8]).

Let Δ be a k-th order ODE for u = u(x), u ∈ U = ℝ, and let (M = U ×B,π,B) be the corresponding variables bundle. Let the vector field Y in JkM be the λ-prolongation of the Lie-point vector field X in M. Then we say that X is a λ-symmetry of Δ if and only if Y is tangent to the solution manifold SΔ, that is, iff there is a smooth function Φ on JkM such that Y (Δ) = ΦΔ.

We take λ: J1M → ℝ, and then the λ-prolongation of a Lie-point vector field in M is a vector field in each JnM. We can consider λ: JrM → ℝ, obtaining obvious generalizations of the results. In this case the λ-prolongations of X would be generalized vector fields in each JnM with n > 0 even if X is a Lie-point vector field. The same procedure can be applied to the μ-prolongation.

Proposition 2.1.

The vector field Y ∈ χ(JkM), projecting to a vector field X ∈ χ(M) if and only if it preserves the contact structure in JkM.

Theorem 2.1 ([7,8]).

Let Δ be a scalar PDEs of order k for u = u(x1,...,xp). Let X = ξ∂x + η∂t + ϕ∂u be a vector field on M, with characteristic Q = ϕξuxτut and let Y be the μ-prolongation of order k of X. If X is a μ-symmetry for Δ, then Y : ϕXT ϕX , where ϕX ⊂ JkM is the solution manifold for the system ΔX made of Δ and EJDJQ = 0 for all J with |J| = 0,1,...,k − 1.

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 = λidxi. 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 λi. 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 Dtλ1 = Dxλ2. Thus, μ-prolongation of infinitesimal generator X (2.1) is written

Now if we suppose μ = λidxi 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 = ϕuiξi. Thus, one can write


If the μ-prolongation Y is acted on the Eq. (3.1) and by substituting (−A2(t)umuxA3(t)u2nuxA1(t)uxxxA4(t)u) for ut, 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 Dtλ1 = Dxλ2, are
λ1(x,t,u)=(2A1(t)c1(m2n)A2(t)13nm2nA3(t)1+3m2(m2n)Fx(x,t)4nA1(t)A3(t)A2(t)(F(x,t)xFx(x,t))+2mA1(t)A2(t)A3(t)(F(x,t)xFx(x,t))/(2F(x,t)(A1(t)c1(m2n)A2(t)13nm2nA3(t)1+3m2(m2n)+2nxA1(t)A3(t)A2(t)mxA1(t)A2(t)A3(t))) (3.7)
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,

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 mn, 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: mn. 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(4n5m)A2(t)2A3(t)2+4nA3(t)2((m5n)A2(t)2(m2n)A2(t)A2(t))+2mA2(t)A3(t)(6nA2(t)A3(t)+(m2n)A2(t)A3(t)))=0, (3.11)
    and Eq. (3.10) is written in the form

    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

    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
    where c1 is an arbitrary constant.

  • CASE II: m = n. To provide specific solutions for the last two equations, we now choose A1(t) = A1tk, where A1 and k are constants. Under this consideration, Eq. (3.9) gets


    Similarly, the equation (3.10) becomes


    The solution of (3) is written

    where c2 and c3 are arbitrary constants. Combining this solution (3.17) with the solution (3), the corresponding μ-infinitesimals are determined as below
    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) = A1tk. Hence, Eq. (3.9) gets

    and Eq. (3.10) reads

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

    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
    where c4 is an arbitrary constant. For this case, from solutions of equations (3.19), (3.20) and (3.6), μ-infinitesimal functions
    are determined, where c1 is a constant.

  • CASE IV: m = 5n. For the function A1(t) = A1tk and the Eqs. (3.9) and (3.10) are converted to the following differential equations


    Thus, the solution of Eq. (3.24) yields

    where c2 and c3 are arbitrary constants. From solution of Eq. (3.25), the function A4(t)
    is determined. Substituting Eqs. (3.26) and (3.27) into (3.6) leads to
    where c1 and c4 are constants.

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

    and the Eq. (3.10) is written

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

    where c2 and c3 are constants. By substituting A2(t) into (3.30), the function A4(t)
    is determined. Finally, the corresponding μ-infinitesimal functions are
    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


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

    are found, where c1 and c2 are constants. Similarly, for Eq. (1.2) of the form
    the μ-infinitesimals read
    where c1 is a constant. Finally, we deal with Eq. (1.3)
    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)
    are determined, where c3, c4, and c5 are constants. Thus, the μ-infinitesimal functions for the equation become
    where c2 is a constant.

4. μ-reduction forms and μ-invariant solutions of Gardner equation

In this section, we consider μ-reduced forms and corresponding μ-invariant solutions of Gardner equation by using the μ-symmetries, which are determined in the previous section. In the classical Lie symmetry group theory, the reduced forms of the original differential equations can be obtained by using their symmetry groups. In this manner, one can transform a PDE to a ODE after reduction if the original equation has two independent variables. In the case of μ-symmetries of PDEs the procedure is similar to the case for the Lie point symmetries of differential equations. To obtain the μ-reduced forms, we should write the characteristic equation based on the symmetry groups of the equation for different cases. As the first application to determine μ-reduced forms of the equation, we consider the case m = n. If the functions A1(t) and A3(t) are chosen as constants and then the functions A2(t) and A4(t) are determined from the related relations given in the previous section. As a result, for these infinitesimals for the case m = n, the characteristic equation

is written and then the similarity variable (invariant), which is the new independent variable for the reduced forms of the original equation is obtained by the integration of the first two terms of the characteristic equation
and the other invariant is obtained by the integration of equations of the other terms in the characteristic equation. Thus, the corresponding similarity variable becomes

Hereby, if we substitute similarity forms into the original equation, the μ-reduced equation, for example n = 1,

is found. Here, it is possible to determine other forms of similarity variables and invariants for other cases of the different choices of m and n.

As the second case, we take m = 5n, A1(t) and A3(t) are considered as constants and thus, A2(t) and A4(t) are determined and the infinitesimals are found. For this reason, the characteristic equation becomes

and the first invariant is
for n = 1, the other invariant is
and the corresponding μ-reduced equation reads

As the third case, we consider m = 4/5n, A3(t) is a constant and k = −1, that is A1(t) = A1/t and the functions A2(t) and A4(t) are determined according to these choices. Hence, one can write characteristic equation for the infinitesimal functions as below

which yields the corresponding invariants

For n = 1/2, the μ-reduced equation form is


In addition to the previous cases, it is possible to consider μ-reduced equations for other forms of the Gardner equation. For example, let’s consider Eq. (1.1) and then for the special choices of the function A1(t), the function A4(t) is determined and the characteristic equation becomes

giving the related invariants such that

The associated μ-reduced equation for Eq. (1.1) is


Solving μ-reduced equation and substituting into (4.13) lead to the invariant solution of the form


We now examine μ-reduced equation for the Gardner equation (1.2). For the special choice of A1(t), the function A4(t) is determined and the characteristic equation is written


The corresponding invariants

are defined and thus the μ-reduced equation for differential equation (1.2) becomes

The solution of μ-reduced equation yields the μ-invariant solution of the form


Finally, for the μ-reduced equation of the Gardner equation given by Eq. (1.3), we write the characteristic equation

thus, the similarity variables are
and the μ-reduced equation for Eq. (1.3) is
and solving μ-reduced equation, the μ-invariant solution
is determined.

Remark 4.1.

It is important to mention that μ-invariant solutions obtained in this section are both solutions of the μ-reduced equations and solutions of the original differential equations.

5. Lagrangians and μ-conservation laws

In this section, we deal with the Lagrangian functions of the Gardner equation and the corresponding μ-conservation laws. It is known that μ-symmetry is defined in the following form

where the functions λi are square matrices and it is called M-functions of the dependent variable u and independent variable x. SΔ is the solution manifold in JrM for given equation Δ of differential equation of order r. Thus, X is μ-symmetry of the equation Δ if Y=XMr:SΔTSΔ.

For a function M and one can have the relation M : JrM → ℝ. Under this relation, X is μ-symmetry of the equation Δ for the function H that H is μ-invariant under X if Y [H] = 0 and it is mentioned as the invariance of the level manifolds of H under Y . We now consider the vector fields X having ξi = 0 that is Qj = ϕj thus, it is represented by a standard formulation of the classical Noether’s theorem. It is clear that these assumptions simplify the procedure. For the applicability of Noether’s theorem, we need some form of variational structure in the systems. The construction of the Euler-Lagrange equations characterizes the minimizers of a variational problem.

Definition 5.1 ([10, 16]).

Let Ω ⊂ X be an open and variational problem that consists of finding maximum and minimum values of


The integrand L(x,u(n)) called the Lagrangian of the variational problem 𝔏, is depend on x, u and derivatives of u.

Definition 5.2 ([16]).

We suppose that f (x) is real-valued function and ▿ f (x) is the gradient of f (x). The change of gradient can be seen by

where 〈x,y〉 is the usual inner product. For functional 𝔏[u], the gradient is called variational derivative of 𝔏.

Proposition 5.1.

Let 𝔏[u] be a variational problem. The variational derivative of 𝔏 is


If u is an extremum of 𝔏[u], then


Theorem 5.1.

If 𝔏[u]=ΩL(x,u(n))dx is variational problem, then it must be a solution of the Euler-Lagrange equations


Definition 5.3 ([16]).


is a differential operator, its adjoint is the differential operator D* is

The differential operator D : 𝒜k𝒜 has adjoint D*: 𝒜𝒜k, the adjoint of the transposed entries of D. An operator D is self-adjoint if D* = D and it is skew-adjoint if D* = −D.

Note that, if P ∈ 𝒜, its Frechet derivative has adjoint Dp*:𝒜l𝒜q.

Theorem 5.2.

Let Δ is the Euler-Lagrange expression for some variational problem 𝔏 = ∫ Ldx, i.e. Δ = E(L), if and only if the Frechet derivative DΔ is self-adjoint: DΔ*=DΔ. The function Lagrangian for Δ can be found using the formula L[u]=01u.Δ[λu]dλ.

Definition 5.4.

A standard conservation law is

where Pi is a p-dimensional vector and we know that μ = λidxi is a horizontal one-form and it satisfies the compatibility condition i.e. Diλj = Djλi. Now, using these property we can define μ-conservation law for this μ-symmetry [4] as
where Pi is a M-vector and it is called μ-conserved vector.

As a result, one can obtain μ-conservation law using the following Theorem:

Theorem 5.3 ([8]).

Let X be a vector field and ℒ = L(x,u(n)) be the n-th order Lagrangian. X is a μ-symmetry for ℒ if and only if the equation has the M-vector Pi and it satisfies the μ-conservation law (Di + λi)Pi = 0.

Using this Theorem, the M-vector Pi can be found using Lagrangian. Firstly, we suppose ℒ (x,t,u,ux,ut) is first order Lagrangian, the vector field X = ϕ(∂/∂ u) is a μ-symmetry for ℒ . Then, we can define the M-vector Pi = ϕ(∂ ℒ/∂ui). For second-order Lagrangian ℒ and the vector field X = ϕ(∂/∂u) is a μ-symmetry for ℒ and then, the M-vector

is a μ-conserved vector.

5.1. Lagrangian of Gardner equation

Now, we consider the Gardner equation given by (1.3)

where A1(t), A2(t) and A3(t) are smooth functions satisfying the condition A1.A3 ≠ 0. It is a fact that the Gardner equation does not have standard Lagrangian function itself and the equation has Lagrangian if and only if its Frechet derivative is self-adjoint. However, the Frechet derivative of Gardner equation is not self-adjoint since it is of odd order. To prove this fact, let us take the following Frechet derivative of Δ

The simplification gives

and then, the adjoint operator becomes

From the equations (5.12) and (5.13), one can say that it is not a variational problem since it is not self-adjoint, that is DΔ*DΔ. Therefore, the potential form of Gardner equation having a standart Lagrangian should be considered. For this purpose, one can consider the differential substitution u = vx to Eq. (1.3) to write


This equation is called “the Gardner equation in potential form” having the Frechet derivative of Δv of the form


Thus, one can see easily that the Gardner equation in potential form is self-adjoint, that is DΔ*=DΔ. By the Theorem 5.2. it has a Lagrangian of the form


By reducing the order of Lagrangian, one can write


Hence, the Lagrangian of equation Δv becomes


5.2. μ-conservation laws of Gardner equation

In order to analyze μ-conservation laws, let us consider the vector field in the form

Y=ϕv+ψxvx+ψtvt+ψxxvxx+ψxtvxt+ψttvtt, (5.19)

In order to determine the μ-conservation forms, we deal with the application of μ-prolongation of the vector field Y (5.19) to the Lagrangian of the Gardner equation in potential form (5.18) by considering following two different cases.

  • Case I: In the first case, the function λi is considered not only functions of x, t, v and but also functions of the derivatives of v, i.e the function λi can be written as λi = λi(x,t,v,vx,vt), which is the most general case. In this case, the coefficients of the μ-prolongation of the vector field Y should be reformulated by considering the dependence of λi functions with respect to the derivatives of the dependent function. Hence, if the μ-prolongation of the vector field Y acts on Eq. (5.18) by substituting vt=(A1(t)vxx2A2(t)vx3/3A3(t)vx4/6)/vx and after straight forward but rigorous calculations one can get the following determining equation


    In this equation, one can consider λi functions as linear in terms of vx and vt since the functions λi depend on derivatives of v function, which satisfies the compatibility condition Dtλ1 = Dxλ2. Then, the solution of the determining equation is

    where K(x,t), C(x), h(t) and G(t) are arbitrary functions.

    Therefore, the Theorem 5.3 yields μ-conservation vectors such that

    where v(x,t) = ∫u(x,t)dx. By considering Noether Theorem for μ-symmetries, one can write μ-conservation law for the Gardner equation in the following potential form

    In order to write μ-conservation law for the Gardner equation in its original form (1.3) (not potential), first, the equation is written in following form

    and then substituting u in vx gives
    where R(t) is an arbitrary function. Hence, one can compute μ-conserved vectors in the forms

    As a consequence, the corresponding μ-conservation law of the Gardner equation in the original form is written as


  • Case II: As a second case, we consider the case λi = λi(x,t,v) and then the solution of determining equation (5.20) yields

    where S(x,t) is an arbitrary function and it can be shown that it satisfies the condition ℒ [v] = 0. Firstly, the corresponding λ1 and λ2 functions
    are determined. It is clear that λ1 and λ2 satisfy the compatibility condition. Then, μ-conservation vectors Pi
    are obtained and the corresponding μ-conservation law for the potential form
    is found. Similar to the first case, μ-conservation vectors for the original Gardner equation is determined in the following form
    where H(t) is an arbitrary function. Finally, μ-conservation law of the original Gardner equation is of the form

Remark 5.1.

It is also important to express that the μ-conservation laws of the Gardner equations with variable coefficients obtained in this section are new. In addition, the corresponding conservation laws of the Gardner equations with variable coefficients are obtained in two different forms by using μ-symmetry approach for the first time in the literature.

6. Concluding Remarks

In this study the concept of μ-symmetry is represented to analyze μ-symmetry classification, μ-reductions, μ-invariant solutions, and μ-conservations laws of nonlinear partial differential Gardner equation with variable coefficients. The important object in this method is a horizontal one-form μ = λi(x,u,ux)dxi, which must satisfy compatibility conditions. Firstly, some important properties of μ-symmetries are presented. Then we show that the method of μ-reduction can also be interpreted in terms of the formulation of the Noether theorem when μ-symmetries are considered to find the invariant solutions of partial differential equations, which are called μ-invariant solutions. In this view, we investigate μ-symmetries for different cases of equation parameters, namely m and n, μ-reduction forms and some invariant solutions of the Gardner equations.

Furthermore, we analyze μ-conservation laws of Gardner equation. It is known that Gardner equation does not have standard Lagrangian function itself. We know that the equation has Lagrangian if and only if its Frechet derivative is self-adjoint. In addition, the Frechet derivative of the equation is not self-adjoint since it is of odd order. However, the equation in potential form is self-adjoint that is DΔ*=DΔ that this means it has a Lagrangian.

The μ-conservations laws of the Gardner equation in potential form are considered in two different cases. Firstly the new conservation law is determined by considering the dependence of λi functions with respect to the derivatives of the dependent function that is λi = λi(x,t,v,vx,vt), which is the most general case. For this case, μ-conservations laws are determined not in the potential form but also in the form of original variables. As a second case, the function λi is considered only functions of x, t, and v variables. In this case, we also obtain new conservation laws of the equation both in the potential form and in the form of the original variables. We show that the both conservation laws corresponding to these two cases have different forms.


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Journal of Nonlinear Mathematical Physics
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Cite this article

AU  - Özlem Orhan
AU  - Özer Teoman
PY  - 2021
DA  - 2021/01/06
TI  - On μ-symmetries, μ-reductions, and μ-conservation laws of Gardner equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 69
EP  - 90
VL  - 26
IS  - 1
SN  - 1776-0852
UR  -
DO  - 10.1080/14029251.2019.1544789
ID  - Orhan2021
ER  -