Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons

This paper applies the direct construction method, symmetry/adjoint symmetry pair method (SA method), symmetry action on a known conservation law method, Ibragimov’s conservation theorem (which always yields the same results as the SA method) and a recursion formula to calculate several conservation laws for nonlinear telegraph systems. In addition, a comparison is made between these methods for conservation laws admitted by nonlinear telegraph systems.

In the SA method, every conservation law results from a bilinear skew-symmetric identity and includes the use of any pair consisting of a local symmetry and an adjoint symmetry of a given PDE system.The symmetry action on a known conservation law method [5,10] directly seeks new conservation laws for any PDE system from an admitted symmetry action on a known conservation law.In Ibragimov's conservation theorem [11], a general formula on conservation laws for arbitrary PDEs is stated by combining the Lie symmetry generators and adjoint equations with formal Lagrangians.
A system of m-th-order PDEs in N-dependent variables u = (u 1 , u 2 , . . ., u N ) and n-independent variables x = (x 1 , x 2 , . . ., x n ) is given as: where ∂ x u, ∂ 2 x u, etc., are all given order derivatives of u σ with respect to x up to m.We suppose U = (U 1 , U 2 , . . ., U N ) are arbitrary functions of x, and Corresponding total derivatives are written by: We let F[U] represent a function depending on x, U and derivatives of U with respect to x.

Definition 2. A local conservation law of PDE System (1) is a divergence expression:
yielding for all solutions u(x) of (1), and Φ i [u] are called the conserved densities (fluxes).
The layout of the rest of this paper is as follows: In Section 2, the direct construction method and the recursion formula are discussed.In Sections 3 and 4, the SA method and symmetry action on a known conservation law method for obtaining conservation laws for PDEs are reviewed, respectively.In Section 5, these approaches are applied to seek conservation laws for a nonlinear telegraph system, and a comparison is made between the SA method, Ibragimov's conservation theorem and the symmetry action on a known conservation law method for conservation laws admitted by such a nonlinear telegraph equation.Finally, conclusions are summarized in Section 6.

Direct Construction Method
In general, for a given PDE system (1) expressed in a standard Cauchy-Kovalevskaya form, all non-trivial conservation laws arise from linear combinations of the PDE system (1) with multipliers that hold non-trivial divergence expressions [4,5,9,18].In this paper, we suppose that (1) is expressed in a standard Cauchy-Kovalevskaya form.Definition 3. A set of multipliers (factors, characteristics) {Λ α [U]} yields a divergence expression for the PDE system (1) if the identity: holds for arbitrary functions U(x).
For all solutions U σ = u σ (x) of given System (1), if {Λ α [U]} is non-singular, one has a local conservation law: of System (1).There are many algorithms for determining the conservation laws of a given PDE system.The direct construction method especially, introduced in [3-5,7-9], yields the formulas and multipliers for the corresponding conservation laws when no variational principle exists.Specifically, {Λ α [U]} holds a set of multipliers for a conservation law of System (1) if and only if each Euler operator (variational derivative): annihilates the left-hand side of (9), i.e., for arbitrary U σ , U σ i , U σ ij , . . ., etc. Condition (12) can be separated in regards to G α [U] and its differential consequences to hold a set of over-determined linear homogeneous PDEs named the determining system (adjoint invariance conditions) for multipliers Λ α [U].If (1) is self-adjoint system, i.e., PDE System (1) has a Lagrangian function, then its multipliers are generators of its admitted continuous (point, contact, higher order) symmetries in characteristic form subject to additional conditions.For any multiplier {Λ α [U]} of adjoint invariance conditions (12), one can directly calculate the corresponding fluxes Φ j [u] under an integral formula (see [5,[7][8][9]).Zhang [19] has studied successfully the existence of multipliers for conservation law of PDEs by applying the property of nonlinear self-adjointness with differential substitution.
Unlike the other methods reviewed in this paper, the direct calculation method yields all local conservation laws and any order multipliers for PDE systems written in Cauchy-Kovalevskaya form.
Recently, a recursion formula [13] for the construction of local conservation laws of PDEs was presented by Cheviakov and Naz, which produces a divergence expression including an arbitrary function of all independent variables, summarized by the following lemma.
Lemma 1. Suppose that PDE System (1) admits a non-trivial local conservation law (8).Then, an arbitrary differentiable function f = f (x), following the formal divergence expression, vanishes on any given solution u(x) of (1):

Symmetry/Adjoint Symmetry Pair Method
The symmetry/adjoint symmetry pair method [5][6][7][8]20] (SA method) contains the following steps: (a) linearize the given PDE system using Fréchet derivatives; (b) find the adjoint system of the linearized system by applying adjoint Fréchet derivatives; (c) find solutions of the linearized system; (d) find adjoint symmetries of the adjoint system; (e) for any pair, consisting of a adjoint symmetry and a local symmetry, seek a conservation law directly through the given conservation identity.
The linearizing operator (Fréchet derivative) associated with the PDE system (1) is given by: with respect to arbitrary functions Suppose also that the PDE system (1) has a point symmetry (3) with an infinitesimal generator (4), then the symmetry components V ρ = ηρ [U] arise from the solutions ηρ [u] of the symmetry determining equations, i.e., the linearized system: for any solution u(x) of ( 1).The adjoint operator (adjoint Fréchet derivative) L * [U] connected with the PDE system (1) is obtained formally through integration by parts and is given by: with respect to an arbitrary function W(x) = (W 1 (x), ..., W N (x)).
For any solution U(x) = u(x) of (1), a set of functions {ω σ [u]} N σ=1 that satisfies the adjoint linearized system: is named an adjoint symmetry [21,22] of System (1).As a rule, the adjoint system is a subset of the linear determining equations for local multipliers in the direct construction method.
Theorem 1.For a PDE system (1), any pair consisting of a symmetry components ηρ [u] and an adjoint symmetry {ω σ [u]} N σ=1 , with solution u(x) replaced by an arbitrary function U(x), satisfies the conservation laws' identity [5,7,20]: with: 18) and ( 19), we obtain a local conservation law D i ψ i [u] = 0 of the PDE system (1), resulting from a pair of symmetry and adjoint symmetry for (1).
Recently, Ma [23] has utilized the pairs of symmetries and adjoint symmetries for presenting conservation laws of discrete evaluation equations without a Lagrangian.

Symmetry Action on a Known Conservation Law Method
More than a decade ago, symmetry action on a known conservation law method [5,10] was introduced firstly by Bluman, Temuerchaolu and Anco.Since this method establishes a significant relationship between the symmetries and conservation laws of given DE systems without a Lagrangian, we review it in this section.In order to demonstrate the application of this method, we will introduce the process of constructing conservation laws of a nonlinear telegraph system in Section 5. 3.
In symmetry action on a known conservation law method, the authors have presented two important formulas related to constructing new conservation laws for any system of PDEs from the known conservation law through the action of an invertible transformation.The first formula transforms any conservation law of a PDE system (1) to the corresponding conservation law of the system obtained under a contact transformation.When the contact transformation is a symmetry, the second formula checks a priori whether the action of a (continuous or discrete) symmetry on a conservation law of (1) yields one or more new conservation laws of (1).

Contact Transformation Action on a Conservation Law
Consider an invertible contact transformation [5,10] acting on (z, W, ∂ z W)-space: with U σ i given by the formula: with respect to the inverse of the Jacobian matrix of (20) given by: and the total derivative operators: in terms of the independent variables z i .Here, W = (W 1 , . . ., W N ) expresses arbitrary functions of z = (z 1 , . . ., z n ); ; and ∂ k z W denotes all k-th-order partial derivatives of W σ with respect to z i .
The contact transformation (20) naturally extends to the (z, W, ∂ z W, ..., ∂ k z W)-space: Through a contact transformation (20), a function G α [U] is transformed to some function after the coordinates of G α [U] are represented in regard to (24).If U σ = u σ (x) solves PDE System (1), then correspondingly, W σ = w σ (z) solves PDE system: involving dependent variables w σ (z) and independent variables z i .The following theorem and its proof appear in [5,10].
From Theorem 2 above, one easily obtains the next important result.
Corollary 1. Through a contact transformation (20), a conservation law (10) for PDE System (1) is transformed to the conservation law: for PDE system (26) with fluxes Ψ i [w] given by (29) for any W = w(z) solving PDE System (26).

Symmetry Action on a Conservation Law
In this subsection, we show that the action of a symmetry on a conservation law of PDE System (1) could yield new conservation laws of (1) when the contact transformation ( 20) is a symmetry.Since a symmetry of (1) leaves invariant the solution manifold of (1), there exist specific functions A β α [W] so that ( 25) is of the form: Therefore, through Formulas ( 27) and ( 29), one obtains: 20) is a symmetry of PDE System (1), then the local conservation law (10) of System ( 1) is transformed to a conservation law: of ( 1) with fluxes Ψ i [u] where Ψ i [W] is given by (29).
} is a set of multipliers of ( 1) admitting (20), then { Λβ [U]} yields a set of multipliers of (1) where Λβ [U] is given by (34) after replacing z i by x i , W σ by U σ , W σ i by U σ i , etc.
Suppose contact transformation is a one-parameter (ε) Lie group of point transformation [2,24], then one has: The extended infinitesimal generator X associated with (35) can be written as: with: where In terms of (36), one has is a point symmetry of (1).Then, one has an important result: for some functions a . Consequently, it is easy to see that: where and: The following theorem and its proof appear in [5,10].
The NLT system (48) admits three continuous translation symmetries (49) and the point symmetry with infinitesimal generator: resulting from (51).

Ibragimov's Conservation Theorem for NLT Systems (48)
In order to further reveal the relationship between the SA method and Ibragimov's conservation theorem in the next subsection, the construction of conservation laws of NLT System (48) is firstly viewed by using Ibragimov's conservation theorem in this subsection.