Analysis of the Symmetries and Conservation Laws of the Nonlinear Jaulent-Miodek Equation

Abstract

Lie symmetry method is performed for the nonlinear Jaulent-Miodek equation. We will find the symmetry group and optimal systems of Lie subalgebras. The Lie invariants associated with the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. And conservation laws of the J-M equation are presented with two steps: firstly, finding multipliers for computation of conservation laws and, secondly, symbolic computation of conservation laws will be applied.

1. Introduction

The Jaulent-Miodek equation (J-M) is given by 𝑢_𝑡+𝑢_𝑥𝑥𝑥+32𝑣𝑣_𝑥𝑥𝑥+92𝑣_𝑥𝑣_𝑥𝑥−6𝑢𝑢_𝑥−6𝑢𝑣𝑣_𝑥−32𝑣²𝑢_𝑥𝑣=0,_𝑡+𝑣_𝑥𝑥𝑥−6𝑣𝑢_𝑥−6𝑢𝑣_𝑥−152𝑣²𝑣_𝑥=0. (1)

The coupled system of (1) is associated with the J-M spectral problem [1]. The relation between this system and Euler-Darboux equation was found by Matsuno [2]. In recent years, much work associated with the J-M equation has been done [3–5]. The symmetry group method plays a fundamental role in the analysis of differential equations. The theory of Lie symmetry groups of differential equations called classical Lie method was first developed by Lie [6] at the end of the nineteenth century. Nowadays, the application of Lie transformations group theory for constructing the solutions of nonlinear partial differential equations (PDEs) is regarded as one of the most active fields of research in the theory of nonlinear PDEs and applications.

Many PDEs in the applied sciences and engineering are continuity equations which express conservation of mass, momentum, energy, or electric charge. Such equations occur in, for example, fluid mechanics, particle and quantum physics, plasma physics, elasticity, gas dynamics, electromagnetism, magnetohydrodynamics, nonlinear optics, and so forth. In the study of PDEs, conservation laws are important for investigating integrability and linearization mappings and for establishing existence and uniqueness of solutions. They are also used in the analysis of stability and global behavior of solutions [7–10].

The present paper is organized as follows. In Section 1, we obtain the symmetry of (1) and Lie symmetry groups of J-M equation are found. In Section 2, we construct the optimal system of one-dimensional subalgebras of (1). Lie invariants and similarity reduced equations corresponding to the infinitesimal symmetries of (1) are obtained in Section 3. In Section 4, the conservation laws of (1) are obtained with finding multipliers, and finally some new conservation laws of (1) are obtained with symbolic computation of conservation laws.

2. Lie Symmetries of the J-M Equation

In this section, we draw your attention to the general procedure for determining symmetries for J-M equation; see [11–13]. We consider the one parameter Lie group of infinitesimal transformations on (𝑥¹=𝑥,𝑥²=𝑡,𝑢¹=𝑢,𝑢²=𝑣), 𝑥=𝑥+𝑠𝜉¹(𝑥,𝑡,𝑢,𝑣)+𝑂𝑠²,𝑡̃=𝑥+𝑠𝜉²(𝑥,𝑡,𝑢,𝑣)+𝑂𝑠²,𝑢=𝑥+𝑠𝜑¹(𝑥,𝑡,𝑢,𝑣)+𝑂𝑠²,𝑣̃=𝑥+𝑠𝜑²(𝑥,𝑡,𝑢,𝑣)+𝑂𝑠², (2) where 𝑠 is the group parameter and 𝜉¹, 𝜉², 𝜑¹, and 𝜑² are the infinitesimals of the transformations for the independent and dependent variables, respectively. The associated vector field is in the following form: 𝑉=𝜉¹(𝑡,𝑥,𝑢,𝑣)𝜕_𝑡+𝜉²(𝑡,𝑥,𝑢,𝑣)𝜕_𝑥+𝜑¹(𝑡,𝑥,𝑢,𝑣)𝜕_𝑢+𝜑²(𝑡,𝑥,𝑢,𝑣)𝜕_𝑣. (3) The Lie algebra 𝔤 of infinitesimal symmetry of (1) is spanned by three vector fields: 𝑋₁=𝜕_𝑡,𝑋₂=𝜕_𝑥,𝑋₃=𝑡𝜕_𝑡+13𝑥𝜕_𝑥−23𝑢𝜕_𝑢−13𝑣𝜕_𝑣. (4) The commutation relations of the 3-dimensional Lie algebra 𝔤 spanned by the vector fields 𝑋₁,𝑋₂,𝑋₃ are shown in Table 1.

Table 1 

The commutator table.

Theorem 1. If 𝑢=𝑓(𝑡,𝑥) and 𝑣=𝑔(𝑡,𝑥) are a solution of (1), then so are the functions 𝐺₁(𝑠)()(),𝐺⋅𝑓𝑡,𝑥=𝑓𝑡−𝑠,𝑥₁(𝑠)()(),𝐺⋅𝑔𝑡,𝑥=𝑔𝑡−𝑠,𝑥₂(𝑠)()(),𝐺⋅𝑓𝑡,𝑥=𝑓𝑡,𝑥−𝑠₂(𝑠)()(),𝐺⋅𝑔𝑡,𝑥=𝑔𝑡,𝑥−𝑠₃(𝑠)⋅𝑓(𝑡,𝑥)=𝑒^−(2/3)𝑠𝑓𝑒^−𝑠𝑡,𝑒^−(1/3)𝑠𝑥,𝐺₃(𝑠)⋅𝑔(𝑡,𝑥)=𝑒^−(1/3)𝑠𝑔𝑒^−𝑠𝑡,𝑒^−(1/3)𝑠𝑥. (5)

3. Optimal System of the Jaulent-Miodek Equation

In this section, we obtain the optimal system and reduced forms of (1) by using symmetry group properties obtained in previous section. Since the original partial differential equation has two independent variables, this partial differential equation transforms into the ordinary differential equation after reduction.

A well-known standard procedure [11] allows us to classify all the one-dimensional subalgebras into subsets of conjugate subalgebras. This involves constructing the adjoint representation group, which introduces a conjugate relation in the set of all one-dimensional subalgebras. In fact, for one-dimensional subalgebras, the classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation. Since each one-dimensional subalgebra is determined by nonzero vector in 𝔤, this problem is attacked by the naive approach of taking a general element 𝑉 in 𝔤 and subjecting it to various adjoint transformations so as to “simplify” it as much as possible. Thus, we will deal with the construction of the optimal system of subalgebras of 𝔤. To compute the adjoint representation, we use the Lie series Adexp𝜀𝑋_𝑖𝑋_𝑗=𝑋_𝑗−𝜀𝑋_𝑖,𝑋_𝑗+𝜀²2𝑋_𝑖,𝑋_𝑖,𝑋_𝑗−⋯, (6) where [𝑋_𝑖,𝑋_𝑗] is the commutator for the Lie algebra, 𝜀 is a parameter, and 𝑖,𝑗=1,2,3. Then we have Table 2.

Table 2 

Adjoint representation of the infinitesimal generators.

Theorem 2. An optimal system of one-dimensional Lie algebras of the J-M equation is provided by (1) 𝑋₃, (2) 𝛼𝑋₁+𝑋₂, and (3) 𝑋₁, where 𝛼∈ℝ and 𝛼≠0.

Proof. Let 𝔤 be the symmetry group of (1), with adjoint representation determined in Table 2, and 𝑋=𝑎₁𝑋₁+𝑎₂𝑋₂+𝑎₃𝑋₃ (7) is a nonzero vector field of 𝔤. We will simplify as many of the coefficients of 𝑎_𝑖, 𝑖=1,2,3, as possible through judicious applications of adjoint maps to 𝑋.
Case 1. Suppose first that 𝑎₃≠0. Scaling 𝑋 if necessary, we can assume that 𝑎₃=1. Referring to Table 2, if we act on such a 𝑋 by Ad(exp(𝑎₁𝑋₁)) and Ad(exp(3𝑎₂𝑋₂)), respectively, we can make the coefficients of 𝑣₁ and 𝑋₂ vanish. Thus, every one-dimensional subalgebra generated by a 𝑋 with 𝑎₃≠0 is equivalent to the subalgebra spanned by 𝑋₃.
Case 2. The remaining one-dimensional subalgebras are spanned by vectors of the above form with 𝑎₃=0. If 𝑎₂≠0, we can scale to make 𝑎₂=1. Referring to Table 2, we cannot do anything in this case. Thus, every one-dimensional subalgebra generated by a 𝑣 with 𝑎₃=0 and 𝑎₂≠0 is equivalent to the subalgebra spanned by 𝛼𝑋₁+𝑋₂, where 𝛼 is arbitrary constant.
Case 3. Consider 𝑎₃=0,𝑎₂=0, and 𝑎₁≠0. Thus, every one-dimensional subalgebra generated by 𝑋 is equivalent to the subalgebra spanned by 𝑋₁.

4. Symmetry Reduction of the J-M Equation

We can now compute the invariants associated with the symmetry operators by integrating the characteristic equations. For example, for the operator characteristic equation 𝑋₃=𝑡𝜕_𝑡+(1/3)𝑥𝜕_𝑥−(2/3)𝑢𝜕_𝑢−(1/3)𝑣𝜕_𝑣, 𝑑𝑡𝑡=3𝑑𝑥𝑥=−3𝑑𝑢2𝑢=−3𝑣.𝑑𝑣 (8) The corresponding invariants are 𝜆=𝑥𝑡^−1/3, 𝑈=𝑢𝑡^2/3, and 𝑉=𝑣𝑡^1/3. Therefore, solution of our equation in this case is 𝑢=𝑈𝑡^−2/3,𝑣=𝑉𝑡^−1/3. Substituting derivatives of 𝑢 and 𝑣 in terms of 𝜆, 𝑈, and 𝑉 into (1), the coupled system of ordinary differential equation is obtained as follows: 4𝑈+2𝜆𝑈_𝜆−6𝑈_𝜆𝜆𝜆−9𝑉𝑉_𝜆𝜆𝜆−27𝑉_𝜆𝑉_𝜆𝜆+36𝑈𝑈_𝜆+36𝑈𝑉𝑉_𝜆+9𝑉²𝑈_𝜆=02𝑉+2𝜆𝑉_𝜆−6𝑉_𝜆𝜆𝜆+36𝑉𝑈_𝜆+36𝑈𝑉_𝜆+45𝑉²𝑉_𝜆=0. (9) And for the operator 𝛼𝑋₁+𝑋₂, we have 𝛼𝑈_𝜆+𝑈_𝜆𝜆𝜆+32𝑉𝑉_𝜆𝜆𝜆+92𝑉_𝜆𝑉_𝜆𝜆−6𝑈𝑈_𝜆−6𝑈𝑉𝑉_𝜆−32𝑉²𝑈_𝜆=0𝛼𝑉_𝜆+𝑉_𝜆𝜆𝜆−6𝑉𝑈_𝜆−6𝑈𝑉_𝜆−152𝑉²𝑉_𝜆=0, (10) and the corresponding invariants associated with the above operator are 𝜆=𝑥−𝛼𝑡, 𝑈=𝑢, and 𝑉=𝑣.

5. Conservation Laws for the J-M Equation

To deal with the conservation laws, many methods, such as the method based on the Noethers theorem and the multiplier method, are derived by the relationship between the conserved vector of the PDE and the Lie-Bäcklund symmetry generators of the PDE, the direct method, and so forth [7, 8, 11].

Definition 3. A local conservation law of the PDE system Δ_𝜈𝑥,𝑢^(𝑛)=0,𝜈=1,…,𝑙, (11) involving 𝑥=𝑥(𝑥¹,…,𝑥^𝑝),𝑢=(𝑢¹,…,𝑢^𝑞), and the derivatives of 𝑢 with respect to 𝑥 up to 𝑛, where 𝑢^(𝑛) represents all the derivatives of 𝑢 of all orders from 0 to 𝑛, is a divergence expression 𝐷_𝑖Φ^𝑖[𝑢]=𝐷₁Φ¹[𝑢]+⋯+𝐷_𝑛Φ^𝑛[𝑢]=0 (12) holding for all solutions of the system (11). Φ^𝑖[𝑢]=Φ^𝑖(𝑥,𝑢,𝜕𝑢,…,𝜕^𝑟_𝑢), 𝑖=1,…,𝑛, are called fluxes of the conservation law, and the highest-order derivative (𝑟) present in the fluxes Φ^𝑖[𝑢] is called the order of a conservation law [8].

Remark 4. If one of the independent variables of (11) is time 𝑡, the conservation law (12) takes the form 𝐷_𝑡Ψ[𝑢]+divΦ[𝑢]=0, (13) where divΦ[𝑢]=𝐷_𝑖Φ^𝑖[𝑢]=𝐷₁Φ¹[𝑢]+⋯+𝐷_𝑛Φ^𝑛[𝑢] is a spatial divergence and 𝑥=(𝑥¹,…,𝑥^𝑛−1) are 𝑛−1 spatial variables. Here Ψ[𝑢] is referred to as a density, and Φ^𝑖[𝑢] as spatial fluxes of the conservation law (13).

5.1. Computation of Conservation Laws with Finding Multiplier

In this study, we derive the conservation law from the multiplier method. In particular, a set of multipliers {Λ_𝜈}^𝑙_𝜈=1={Λ_𝜈(𝑥,𝑈,𝜕𝑈,…,𝜕^𝑟_𝑈)}^𝑙_𝜈=1 yields a divergence expression for the system (11) if the identity Λ_𝜈[𝑈]Δ_𝜈[𝑈]≡𝐷_𝑖Φ^𝑖[𝑈] (14) holds identically for arbitrary functions 𝑈(𝑥). Then, on the solutions 𝑈(𝑥)=𝑢(𝑥) of the system (11), if Δ_𝜈[𝑈] is nonsingular, one has local conservation law Λ_𝜈[𝑢]Δ_𝜈[𝑢]≡𝐷_𝑖Φ^𝑖[𝑢]=0.

Definition 5. The Euler operator with respect to 𝑈^𝑗 is the operator defined by 𝐸_𝑈^𝑗=𝜕𝜕𝑈^𝑗−𝐷_𝑖𝜕𝜕𝑈^𝑗_𝑖+⋯+(−1)^𝑠𝐷_𝑖1⋯𝐷_𝑖𝑠𝜕𝜕𝑈^𝑗_{𝑖1⋯𝑖𝑠}+⋯ (15) for 𝑗=1,…,𝑞 [8].

Theorem 6. A set of nonsingular local multipliers {Λ_𝜈(𝑥,𝑈,𝜕𝑈,…,𝜕^𝑟_𝑈)}^𝑙_𝜈=0 yields a local conservation law for the system Δ_𝜈(𝑥,𝑢^(𝑛)) if and only if the set of identities 𝐸_𝑈^𝑗Λ_𝜈𝑥,𝑈,𝜕𝑈,…,𝜕^𝑟𝑈Δ_𝜈𝑥,𝑢^(𝑛)=0,𝑗=1,…,𝑞, (16) holds for arbitrary functions 𝑈(𝑥) (Theorem 1.3.3, [8]).

The set of (16) yields the set of linear determining equations to find all sets of local conservation law multipliers of the system (11). Now, we consider all local conservation law multipliers of the forms Λ₁=𝛼(𝑡,𝑥,𝑢,𝑣,𝑢_𝑡,𝑣_𝑡,𝑢_𝑥,𝑣_𝑥,𝑢_𝑡𝑡,𝑣_𝑡𝑡,𝑢_𝑥𝑥,𝑣_𝑥𝑥) and Λ₂=𝛽(𝑡,𝑥,𝑢,𝑣,𝑢_𝑡,𝑣_𝑡,𝑢_𝑥,𝑣_𝑥,𝑢_𝑡𝑡,𝑣_𝑡𝑡,𝑢_𝑥𝑥,𝑣_𝑥𝑥) of (1). The determining equation (16) for J-M equation is 𝐸_𝑢Λ₁𝑢_𝑡+𝑢_𝑥𝑥𝑥+32𝑣𝑣_𝑥𝑥𝑥+92𝑣_𝑥𝑣_𝑥𝑥−6𝑢𝑢_𝑥−6𝑢𝑣𝑣_𝑥−32𝑣²𝑢_𝑥+Λ₂𝑣_𝑡+𝑣_𝑥𝑥𝑥−6𝑣𝑢_𝑥−6𝑢𝑣_𝑥−152𝑣²𝑣_𝑥𝐸≡0,_𝑣Λ₁𝑢_𝑡+𝑢_𝑥𝑥𝑥+32𝑣𝑣_𝑥𝑥𝑥+92𝑣_𝑥𝑣_𝑥𝑥−6𝑢𝑢_𝑥−6𝑢𝑣𝑣_𝑥−32𝑣²𝑢_𝑥+Λ₂𝑣_𝑡+𝑣_𝑥𝑥𝑥−6𝑣𝑢_𝑥−6𝑢𝑣_𝑥−152𝑣²𝑣_𝑥≡0, (17) where 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) are arbitrary functions. Equation (17) splits with respect to third order derivatives of 𝑢 to yield the determining PDE system whose solutions are the sets of local multipliers of all nontrivial local conservation laws of the J-M equation.

The solution of the determining system (17) for J-M equation is given by 𝛼=𝑐₁𝑥+6𝑐₁9𝑡𝑢+2𝑐₁𝑡𝑣²+𝑐₂𝑢𝑣+𝑐₃𝑢_1/6𝑐₂𝑣_𝑥𝑥+5𝑐12²𝑣³+34𝑐₃𝑣²+𝑐₄𝑣+𝑐₅,1𝛽=−6𝑐₂𝑢_𝑥𝑥+12𝑐₁5𝑥𝑣−𝑐24²𝑣²_𝑥−5𝑣12^𝑥𝑥−𝑐₂3𝑉−2𝑐₁𝑡𝑣_𝑥𝑥−14𝑐₃𝑣_𝑥𝑥+154𝑐₁𝑡𝑣³+9𝑐₁1𝑡𝑢𝑣+2𝑐₂𝑢²+54𝑐₂𝑢𝑣²+32𝑐₃𝑢𝑣+𝑐₄𝑢+35𝑐96²𝑣⁴+58𝑐₃𝑣³+34𝑐₄𝑣²+12𝑐₅𝑣+𝑐₆, (18) where 𝑐₁,𝑐₂,𝑐₃,𝑐₄,𝑐₅, and 𝑐₆ are arbitrary constants. So local multipliers are given by (1)(2)1𝛼=0,𝛽=1,𝛼=1,𝛽=2(3)3𝑣,𝛼=𝑣,𝛽=𝑢+4𝑣²,(4)3𝛼=𝑢+4𝑣²1,𝛽=−4𝑣_𝑥𝑥+325𝑢𝑣+8𝑣³,(5)9𝛼=𝑥+6𝑡𝑢+2𝑡𝑣²,1𝛽=23𝑥𝑣−2𝑡𝑣_𝑥𝑥+154𝑡𝑣³(6)1+9𝑡𝑢𝑣,𝛼=𝑢𝑣−6𝑣_𝑥𝑥+5𝑣12³,1𝛽=−6𝑢_𝑥𝑥−5𝑣24^2𝑥−512𝑣𝑣_𝑥𝑥+12𝑢²+54𝑢𝑣²+35𝑣96⁴. (19) Multipliers 𝛼 and 𝛽 determine a nontrivial local conservation law 𝐷_𝑡Ψ+𝐷_𝑥Φ=0 with the characteristic form 𝐷_𝑡Ψ+𝐷_𝑥Φ≡𝛼𝑢_𝑡+𝑢_𝑥𝑥𝑥+32𝑣𝑣_𝑥𝑥𝑥+92𝑣_𝑥𝑣_𝑥𝑥−6𝑢𝑢_𝑥−6𝑢𝑣𝑣_𝑥−32𝑣²𝑢_𝑥+𝛽𝑣_𝑡+𝑣_𝑥𝑥𝑥−6𝑣𝑢_𝑥−6𝑢𝑣_𝑥−152𝑣²𝑣_𝑥. (20) The total divergence operator must be inverted to calculate the conserved quantities Φ and Ψ. To do this, we need to integrate (by parts) one of the expressions in multidimensions involving arbitrary functions and its derivatives, which is a difficult task. The homotopy operator [14] is a powerful and useful algorithmic tool (explicit formula) that originates from homological algebra and variational bicomplexes.

Definition 7. The 2-dimensional homotopy operator is a vector operator with two components, (𝐻^(𝑥)_{𝐮(𝑥,𝑡)}𝑓,𝐻^(𝑡)_{𝐮(𝑥,𝑡)}𝑓), where 𝐻^(𝑥)_{𝐮(𝑥,𝑡)}𝑓=¹₀^𝑞_𝑗=0𝐼_𝑢^(𝑥)_𝑗𝑓[𝜆𝐮]𝑑𝜆𝜆,𝐻^(𝑡)_{𝐮(𝑥,𝑡)}𝑓=¹₀^𝑞_𝑗=0𝐼_𝑢^(𝑡)_𝑗𝑓[𝜆𝐮]𝑑𝜆𝜆. (21) The 𝑥-integrand, 𝐼^(𝑥)_{𝑢(𝑥,𝑡)𝑗}𝑓, is given by 𝐼_𝑢^(𝑥)_𝑗𝑓=^𝑀𝑗1_𝑘1^𝑀₌₁^𝑗2_𝑘2=0^𝑘1−1_𝑖1^𝑘₌₀²_𝑖2=0𝐵^(𝑥)𝑢^𝑗_{𝑥𝑖1𝑥𝑖2}−𝐷_𝑥^{𝑘1−𝑖1−1}×−𝐷_𝑡^{𝑘2−𝑖2}𝜕𝑓𝜕𝑢^𝑗_{𝑥𝑘1𝑡𝑘2}, (22) where 𝑀^𝑗₁,𝑀^𝑗₂ are the order of 𝑓 in 𝑢 to 𝑥 and 𝑡, respectively, with combinatorial coefficient 𝐵^(𝑥)=𝐵(𝑖₁,𝑖₂,𝑘₁,𝑘₂), where 𝐵𝑖₁,𝑖₂,𝑘₁,𝑘₂=^{𝑖1+𝑖2𝑖1𝑘1+𝑘2−𝑖1−𝑖2𝑘−11−𝑖1−1}_{𝑘1+𝑘𝑖2𝑘1}. (23) Similarly, 𝑡-integrand, 𝐼^(𝑡)_{𝑢(𝑥,𝑡)𝑗}𝑓, defined as 𝐼_𝑢^(𝑡)_𝑗𝑓=^𝑀𝑗1_𝑘1^𝑀₌₀^𝑗2_𝑘2=1^𝑘1_𝑖1^𝑘₌₀²⁻¹_𝑖2=0𝐵^(𝑡)𝑢^𝑗_{𝑥𝑖1𝑥𝑖2}−𝐷_𝑥^{𝑘1−𝑖1}×−𝐷_𝑡^{𝑘2−𝑖2−1}𝜕𝑓𝜕𝑢^𝑗_{𝑥𝑘1𝑡𝑘2}, (24) where 𝐵^(𝑡)=𝐵(𝑖₂,𝑖₁,𝑘₂,𝑘₁).

We apply homotopy operator to find conserved quantities Ψ and Φ which yield multipliers 𝛼=0 and 𝛽=1. We have 𝑓=𝛼𝑢_𝑡+𝑢_𝑥𝑥𝑥+32𝑣𝑣_𝑥𝑥𝑥+92𝑣_𝑥𝑣_𝑥𝑥−6𝑢𝑢_𝑥−6𝑢𝑣𝑣_𝑥−32𝑣²𝑢_𝑥+𝛽𝑣_𝑡+𝑣_𝑥𝑥𝑥−6𝑣𝑢_𝑥−6𝑢𝑣_𝑥−152𝑣²𝑣_𝑥=𝑣_𝑡+𝑣_𝑥𝑥𝑥−6𝑣𝑢_𝑥−6𝑢𝑣_𝑥−152𝑣²𝑣_𝑥. (25) The integrands (22) and (24) are 𝐼_𝑢^(𝑡)𝑓=0,𝐼_𝑢^(𝑥)𝑓=−6𝑢𝑣,𝐼_𝑣^(𝑡)𝐼𝑓=𝑣,_𝑣(𝑥)𝑓=−6𝑢𝑣−152𝑣³+𝑣_𝑥𝑥. (26) Apply (21) to the integrands (26); therefore Φ=𝐻^(𝑥)_{𝐮(𝑥,𝑡)}𝑓=6𝑢𝑣−𝑣_𝑥𝑥+52𝑣³,Ψ=𝐻^(𝑡)_{𝐮(𝑥,𝑡)}𝑓=−𝑣. (27) So, we have the conservation law of the J-M equation with respect to multipliers 𝛼=0 and 𝛽=1: 𝐷_𝑡(−𝑣)+𝐷_𝑥6𝑢𝑣−𝑣_𝑥𝑥+52𝑣³=0. (28) And similarly, conservation laws with respect to other multipliers are given as follows: (1)𝛼=1 and 𝛽=(1/2)𝑣: 1Φ=−4𝑣²−𝑢,Ψ=15𝑣16⁴+92𝑢𝑣²−2𝑣𝑣_𝑥𝑥−54𝑣²_𝑥+3𝑢²−𝑢_𝑥𝑥; (29) (2)𝛼=𝑣 and 𝛽=𝑢+(3/4)𝑣²: 1Φ=−4𝑣³9−𝑢𝑣,Ψ=8𝑣⁵+6𝑢𝑣³−94𝑣²𝑣_𝑥𝑥+6𝑢²𝑣−𝑢𝑣_𝑥𝑥−𝑣𝑢_𝑥𝑥+𝑢_𝑥𝑣_𝑥; (30)(3)𝛼=𝑢+(3/4)𝑣² and 𝛽=−(1/4)𝑣_𝑥𝑥+(3/2)𝑢𝑣+(5/8)𝑣³: 5Φ=−𝑣32⁴−34𝑢𝑣²+18𝑣𝑣_𝑥𝑥−12𝑢²,Ψ=25𝑣32⁶+398𝑢𝑣⁴−78𝑣³𝑣_𝑥𝑥+152𝑣²𝑢²−32𝑢𝑣²_𝑥−3𝑢𝑣𝑣_𝑥𝑥−34𝑣²𝑢_𝑥𝑥+32𝑣𝑢_𝑥𝑣_𝑥+2𝑢³+12𝑢²_𝑥+18𝑣²_𝑥𝑥−𝑢𝑢_𝑥𝑥−18𝑣𝑣_𝑡𝑥+18𝑣_𝑡𝑣_𝑥; (31) (4)𝛼=𝑢𝑣−(1/6)𝑣_𝑥𝑥+(5/12)𝑣³ and 𝛽=−(1/6)𝑢_𝑥𝑥−(5/24)𝑣²_𝑥−(5/12)𝑣𝑣_𝑥𝑥+(1/2)𝑢²+(5/4)𝑢𝑣²+(35/96)𝑣⁴: 7Φ=−𝑣96⁵−512𝑢𝑣³+572𝑣𝑣²_𝑥−12𝑣𝑢²+5𝑣36²𝑣_𝑥𝑥=112𝑣𝑢_𝑥𝑥+112𝑢𝑣_𝑥𝑥,Ψ=25𝑣64⁷+4516𝑢𝑣⁵−25𝑣48³𝑣²_𝑥+234𝑣³𝑢²−95𝑣96⁴𝑣_𝑥𝑥−114𝑢𝑣²𝑣_𝑥𝑥−5𝑣12³𝑢_𝑥𝑥+3𝑣𝑢³−54𝑢𝑣𝑣²_𝑥−12𝑢²𝑣_𝑥𝑥−5𝑣36²𝑣_𝑡𝑥+536𝑣𝑣_𝑡𝑣_𝑥+13𝑣𝑣²_𝑥𝑥+5𝑣24^2𝑥𝑣_𝑥𝑥−𝑢𝑣𝑢_𝑥𝑥+1𝑢12^𝑡𝑣_𝑥−112𝑣𝑢_𝑡𝑥+1𝑣12^𝑡𝑢_𝑥+16𝑢_𝑥𝑥𝑣_𝑥𝑥−112𝑢𝑣_𝑡𝑥; (32) (5)𝛼=𝑥+6𝑡𝑢+(9/2)𝑡𝑣² and 𝛽=(1/2)𝑥𝑣−(3/2)𝑡𝑣_𝑥𝑥+(15/4)𝑡𝑣³+9𝑡𝑢𝑣: Φ=−1516𝑡𝑣⁴−92𝑡𝑢𝑣²+34𝑡𝑣𝑣_𝑥𝑥−14𝑥𝑣²−3𝑡𝑢²−𝑥𝑢,Ψ=7516𝑡𝑣⁶+1174𝑡𝑢𝑣⁴+1516𝑥𝑣⁴+45𝑡𝑢²𝑣²−212𝑡𝑣³𝑣_𝑥𝑥+9𝑡𝑣𝑢_𝑥𝑣_𝑥−9𝑡𝑢𝑣²_𝑥+92𝑥𝑢𝑣²−18𝑡𝑢𝑣𝑣_𝑥𝑥−92𝑡𝑣²𝑢_𝑥𝑥+12𝑡𝑢³+54𝑣𝑣_𝑥+34𝑡𝑣²_𝑥𝑥−2𝑥𝑣𝑣_𝑥𝑥+34𝑡𝑣_𝑡𝑣_𝑥−6𝑡𝑢𝑢_𝑥𝑥+3𝑥𝑢²−54𝑥𝑣²_𝑥+3𝑡𝑢²_𝑥−34𝑡𝑣𝑣_𝑡𝑥−𝑥𝑢_𝑥𝑥+𝑢_𝑥. (33)

5.2. Symbolic Computation for Finding Conservation Laws Equation

This subsection covers the application of the homotopy operator to the computation of conservation laws of J-M equation. Finding a conservation law needs computing the density Ψ first, followed by computing of the flux Φ. Computing flux Φ will require using homotopy operator. Following the approach by Hereman et al. [9, 14, 15], a candidate density is built as a linear combination (with undetermined coefficients of differential terms) which is invariant under the scaling symmetry of the given PDE. By determining Ψ we can compute 𝐷_𝑡Ψ and remove all time derivatives; 𝐷_𝑡Ψ must be a divergence. Thus, using Theorem 4.4 of [10], one requires that 𝐸_𝑈^𝑗𝐷_𝑡Ψ=0,𝑗=1,…,𝑁. (34) This leads to a linear system for the undetermined coefficients. Substituting its solution into the candidate for Ψ gives the actual density. Finally, the Φ=div⁻¹(𝐷_𝑡Ψ) is computed with the homotopy operator.

Jaulent-Miodek equation is invariant under the scaling (dilation) symmetry (4): (𝑡,𝑥,𝑢,𝑣)⟶𝜆³𝑡,𝜆𝑥,𝜆⁻²𝑢,𝜆⁻¹𝑣. (35) Conservation law (13) must hold on solutions of (1). Therefore, we search for polynomial conservation laws that obey the scaling symmetry of the PDE. Indeed, we have to find a polynomial conservation law that does not adhere to the scaling symmetry. We choose a scaling factor for one of the components of (13). The selected scaling factor will be called the rank (𝑅) of that component. Then, we construct a candidate for that component as a linear combination of monomial terms (all of rank 𝑅) with undetermined coefficients. If we remove divergence and divergence-equivalent terms dynamically that candidate will be shorted and of low order.

For J-M equation we will compute the density Ψ of a fixed rank; for example, 𝑅=−3. We construct a list of differential terms which contains all powers of dependent variables and their derivatives and products of them of rank −3: 𝒬=𝑢³_𝑥,𝑣³,𝑡𝑢³,𝑥𝑢²,𝑢𝑢_𝑥,𝑢𝑣,𝑢²𝑢_𝑥𝑥𝑥,𝑢³_𝑥𝑢_𝑥𝑥,𝑣³𝑢_𝑥𝑥,𝑣𝑢²_𝑥,𝑣_𝑥𝑢³_𝑥,𝑣³𝑣_𝑥,𝑡𝑢³𝑢_𝑥𝑥,𝑥𝑢²𝑢_𝑥𝑥,𝑢𝑢_𝑥𝑢_𝑥𝑥,𝑢𝑣𝑢_𝑥𝑥,𝑢²𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥,𝑣𝑢²𝑢_𝑥𝑥,𝑣²𝑢_𝑥𝑢_𝑥𝑥,𝑡𝑢³𝑣_𝑥,𝑥𝑢²𝑣_𝑥,𝑢𝑣_𝑥𝑢_𝑥,𝑢𝑣𝑣_𝑥,𝑢²𝑣_𝑥𝑢_𝑥𝑥𝑥,𝑣𝑣_𝑥𝑢²_𝑥,𝑣²𝑢_𝑥𝑣_𝑥. (36) By removing all terms that are divergences or divergence-equivalent to other terms in 𝒬, we have 𝒬=𝑢³_𝑥,𝑣³,𝑡𝑢³,𝑥𝑢²,𝑣²𝑢_𝑥,𝑣³𝑢_𝑥𝑥,𝑣𝑢²_𝑥,𝑣_𝑥𝑢³_𝑥,𝑡𝑢³𝑢_𝑥𝑥,𝑥𝑢²𝑢_𝑥𝑥,𝑢𝑣𝑢_𝑥𝑥,𝑢²𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥,𝑣𝑢²_𝑥𝑢_𝑥𝑥,𝑣²𝑢_𝑥𝑢_𝑥𝑥,𝑡𝑢³𝑣_𝑥,𝑥𝑢²𝑣_𝑥,𝑢𝑢_𝑥𝑣_𝑥,𝑢²𝑣_𝑥𝑢_𝑥𝑥𝑥,𝑣𝑣_𝑥𝑢²_𝑥,𝑣²𝑣_𝑥𝑢_𝑥. (37) Now, by forming a candidate density combining the terms in 𝒬 linearly with undetermined coefficients 𝑐_𝑖, Ψ=𝑐₁𝑢³_𝑥+𝑐₂𝑣³+𝑐₃𝑡𝑢³+𝑐₄𝑥𝑢²+𝑐₅𝑣²𝑢_𝑥+𝑐₆𝑣³𝑢_𝑥𝑥+𝑐₇𝑣𝑢²_𝑥+𝑐₈𝑣_𝑥𝑢³_𝑥+𝑐₉𝑡𝑢³𝑢_𝑥𝑥+𝑐₁₀𝑥𝑢²𝑢_𝑥𝑥+𝑐₁₁𝑢𝑣𝑢_𝑥𝑥+𝑐₁₂𝑢²𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥+𝑐₁₃𝑣𝑢²_𝑥𝑢_𝑥𝑥+𝑐₁₄𝑣²𝑢_𝑥𝑢_𝑥𝑥+𝑐₁₅𝑡𝑢³𝑣_𝑥+𝑐₁₆𝑥𝑢²𝑣_𝑥+𝑐₁₇𝑢𝑢_𝑥𝑣_𝑥+𝑐₁₈𝑢²𝑣_𝑥𝑢_𝑥𝑥𝑥+𝑐₁₉𝑣𝑣_𝑥𝑢²_𝑥+𝑐₂₀𝑣²𝑣_𝑥𝑢_𝑥. (38) Compute the total derivative with respect to 𝑡 of (38), and set 𝐹=−𝐷_𝑡Ψ. (39) After replacing 𝑢_𝑡 with −𝑢_𝑥𝑥𝑥−(3/2)𝑣𝑣_𝑥𝑥𝑥−(9/2)𝑣_𝑥𝑣_𝑥𝑥+6𝑢𝑢_𝑥+6𝑢𝑣𝑣_𝑥+(3/2)𝑣²𝑢_𝑥 and 𝑣_𝑡 by −𝑣_𝑥𝑥𝑥+6𝑣𝑢_𝑥+6𝑢𝑣_𝑥+(15/2)𝑣²𝑣_𝑥, (39) must be a divergence, use (34), and require 𝐸_{𝑢(𝑡,𝑥)}𝐹=0,𝐸_{𝑣(𝑥,𝑡)}𝐹=0. (40)

The solution of system (40) is 𝑐₁=0,𝑐₂=0,𝑐₃=0,𝑐₄𝑐=0,₅=0,𝑐₆=𝑐₂₀,𝑐₇=𝑐₁₁,𝑐₈=13𝑐₁₃,𝑐₉=0,𝑐₁₀=0,𝑐₁₂=0,𝑐₁₄=𝑐₁₉,𝑐₁₅=0,𝑐₁₆=0,𝑐₁₇=𝑐₁₁,𝑐₁₈=0, (41) where 𝑐₁₁, 𝑐₁₃, 𝑐₁₉, and 𝑐₂₀ is arbitrary.

Case 1. Substitute (41) and 𝑐₁₁=1, 𝑐₁₃=0, 𝑐₁₉=0, and 𝑐₂₀=0 into (38) and (39) given Ψ=𝑣𝑢²_𝑥+𝑢𝑣𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑣_𝑥 and 𝐹=𝑢_𝑥𝑣_𝑥𝑢_𝑥𝑥𝑥+2𝑣𝑢_𝑥𝑢_{𝑥𝑥𝑥𝑥}+92𝑢_𝑥𝑣²_𝑥𝑣_𝑥𝑥+9𝑣𝑢_𝑥𝑣²_𝑥𝑥+𝑢𝑢_𝑥𝑣_{𝑥𝑥𝑥𝑥}+3𝑣²𝑢_𝑥𝑣_{𝑥𝑥𝑥𝑥}+92𝑢𝑣_𝑥𝑣_𝑥𝑥+𝑢𝑣_𝑥𝑢_{𝑥𝑥𝑥𝑥}+6𝑢𝑣²_𝑥𝑣_𝑥𝑥𝑥+32𝑢𝑣²𝑣_{𝑥𝑥𝑥𝑥𝑥}+32𝑣²𝑢_𝑥𝑥𝑣_𝑥𝑥𝑥+𝑣𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥+𝑣𝑢𝑢_{𝑥𝑥𝑥𝑥𝑥}+𝑢𝑢_𝑥𝑥𝑣_𝑥𝑥𝑥−12𝑢²𝑣_𝑥𝑢_𝑥𝑥−92𝑣³𝑢_𝑥𝑢_𝑥𝑥−32𝑢𝑣³𝑢_𝑥𝑥𝑥−6𝑢²𝑣²𝑣_𝑥𝑥𝑥−6𝑣𝑢²𝑢_𝑥𝑥𝑥−6𝑢²𝑢_𝑥𝑣_𝑥𝑥−30𝑢𝑢²_𝑥𝑣_𝑥−27𝑣²𝑣_𝑥𝑢²_𝑥+𝑢²_𝑥𝑣_𝑥𝑥𝑥−6𝑢²𝑣³_𝑥−18𝑣𝑢³_𝑥+273𝑣𝑢_𝑥𝑣_𝑥𝑣_𝑥𝑥𝑥+92𝑣𝑣_𝑥𝑣_𝑥𝑥𝑢_𝑥𝑥+9𝑢𝑣𝑣_𝑥𝑣_{𝑥𝑥𝑥𝑥}+15𝑢𝑣𝑣_𝑥𝑥𝑣_𝑥𝑥𝑥−692𝑢𝑣²𝑢_𝑥𝑣_𝑥𝑥−27𝑢𝑣²𝑣_𝑥𝑢_𝑥𝑥−48𝑢𝑣𝑢_𝑥𝑢_𝑥𝑥−57𝑢𝑣𝑢_𝑥𝑣²_𝑥−24𝑣𝑢²𝑣_𝑥𝑣_𝑥𝑥. (42) Since 𝐹=divΦ, the flux Φ can be computed with the 1D homotopy operator which inverts divergences. Applying 1D homotopy operator formulas in (21) and removing curl term of flux Φ yield 3Φ=−2𝑣³𝑢²_𝑥−32𝑢𝑣³𝑢_𝑥𝑥−6𝑣²𝑢²𝑣_𝑥𝑥−452𝑢𝑣²𝑢_𝑥𝑣_𝑥−18𝑢𝑣𝑢²_𝑥−32𝑣²𝑢_𝑥𝑣_𝑥𝑥𝑥+92𝑣𝑣_𝑥𝑢_𝑥𝑢_𝑥𝑥+32𝑢𝑣²𝑣_{𝑥𝑥𝑥𝑥}−6𝑣𝑢²𝑢_𝑥𝑥+6𝑢𝑣𝑣_𝑥𝑣_𝑥𝑥𝑥+92𝑢𝑣𝑣²_𝑥𝑥−6𝑢²𝑢_𝑥𝑣_𝑥+𝑣𝑢_𝑥𝑢_𝑥𝑥𝑥+𝑢𝑣𝑢_{𝑥𝑥𝑥𝑥}+𝑢𝑢_𝑥𝑣_𝑥𝑥𝑥. (43)

Case 2. Substitute𝑐₁₁=0, 𝑐₁₃=1, 𝑐₁₉=0, and 𝑐₂₀=0 into (38), given 1Ψ=3𝑣_𝑥𝑢³_𝑥+𝑣𝑢²_𝑥𝑢_𝑥𝑥,1Φ=−6𝑢²_𝑥36𝑢𝑣²𝑣_𝑥𝑥+9𝑣³𝑢_𝑥𝑥+69𝑣²𝑢_𝑥𝑣_𝑥+36𝑢𝑣𝑣²_𝑥−36𝑣𝑣_𝑥𝑣_𝑥𝑥𝑥+12𝑢𝑢_𝑥𝑣_𝑥−27𝑣𝑣²_𝑥𝑥+36𝑢𝑣𝑢_𝑥𝑥+48𝑣𝑢²_𝑥−9𝑣²𝑣_{𝑥𝑥𝑥𝑥}−6𝑣𝑢_{𝑥𝑥𝑥𝑥}−2𝑢_𝑥𝑣_𝑥𝑥𝑥. (44)

Case 3. Substitute 𝑐₁₁=0, 𝑐₁₃=0, 𝑐₁₉=1, 𝑐₂₀=0, Ψ=𝑣²𝑢_𝑥𝑢_𝑥𝑥+𝑣𝑣_𝑥𝑢²_𝑥,1Φ=−2𝑣𝑢_𝑥12𝑢𝑣²𝑣_𝑥𝑥+3𝑣³𝑢_𝑥𝑥+33𝑣²𝑢_𝑥𝑣_𝑥+12𝑢𝑣𝑣²_𝑥−12𝑣𝑣_𝑥𝑣_𝑥𝑥𝑥+12𝑢𝑣𝑢_𝑥𝑥+12𝑢𝑢_𝑥𝑣_𝑥−3𝑣²𝑣_{𝑥𝑥𝑥𝑥}+24𝑣𝑢²_𝑥−9𝑣𝑣²_𝑥𝑥−2𝑣𝑢_{𝑥𝑥𝑥𝑥}−2𝑢_𝑥𝑣_𝑥𝑥𝑥. (45) And finally, 𝑐₁₁=0, 𝑐₁₃=0, 𝑐₁₉=0, 𝑐₂₀=1, 1Ψ=3𝑣³𝑢_𝑥𝑥+𝑣²𝑢_𝑥𝑣_𝑥,1Φ=−6𝑣²3𝑣³𝑢_𝑥𝑥+12𝑢𝑣²𝑣_𝑥𝑥+63𝑣²𝑣_𝑥𝑢_𝑥+12𝑢𝑣𝑣²_𝑥+48𝑣𝑢²_𝑥−9𝑣𝑣²_𝑥𝑥−12𝑣𝑣_𝑥𝑣_𝑥𝑥𝑥+36𝑢𝑢_𝑥𝑣_𝑥−3𝑣²𝑣_{𝑥𝑥𝑥𝑥}+12𝑢𝑣𝑢_𝑥𝑥−2𝑣𝑢_{𝑥𝑥𝑥𝑥}−6𝑢_𝑥𝑣_𝑥𝑥𝑥. (46)

6. Conclusion

In this paper, we studied Jaulent-Miodek equation using the Lie symmetry group of infinitesimal transformations of the equation. We found that the underlying equation admits a three-dimensional Lie algebra. We obtained the optimal system of one-dimensional subalgebras of the Lie algebra of the equation. These subalgebras were then used to reduce the underlying equation to nonlinear third order ordinary system of differential equations. Further conservation laws are constructed for this equation in two methods. First, conservation laws of the equation are obtained by finding multipliers; then some other conservation laws of J-M equation are obtained with symbolic computation of conservation laws.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publishing of this paper.