A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations

1 Introduction

The concept of a conservation law, which is a mathematical formulation of the familiar physical laws of conservation of energy, conservation of momentum, and so on, plays an important role to our understanding of the physical world [1]. For example, the conservation law of energy states that the total quantity of energy in an isolated system does not change, though it may change form. Conservation laws are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and ocean engineering.

Moreover, if the system under study has infinitely many independent conservation laws, then its integrability is quite possible. They are also used for existence, uniqueness, and Lyapunov stability analysis and construction of numerical schemes. Moreover, conservation laws are used in obtaining the new nonlocal symmetries, nonlocal conservation laws, and linearisation [2–4].

The first study in the literature for obtaining the conservation laws using the symmetries was given by Noether [5]. In this study, Noether states that for Euler–Lagrange differential equations, to each Noether symmetry associated with the Lagrangian, there corresponds a conservation law that can be determined explicitly by a formula. The application of Noether’s theorem depends on the knowledge of a suitable Lagrangian [5, 6].

In recent years, most authors have developed a lot of methods to find conservation laws of nonlinear evolution equations (NLEEs) such as direct method, partial Lagrangian method, Lax pairs approach, the characteristic method, the variational approach (multiplier approach), and nonlocal conservation theorem method [7–13].

According to the new conservation theorem method [12], if a differential equation (ordinary or partial) has a symmetry, then equation under study always admits a conservation law. In [12], Ibragimov introduced the concept of (strictly) self-adjoint differential equations, which enabled one to construct local conservation laws (e.g. in [12], he used this concept to construct conservation laws for the KdV equation). Later, in [14], he generalised that concept to nonlinearly self-adjoint differential equations.

Very recently, in [15], (see also, [16, 17]), Wang et al. presented a new way for obtaining the conservation laws of partial differential equations (PDEs) with the help of combination of multiplier and new conservation theorem method. In this new method, the most important point is that the solution of the adjoint equation is equivalent the multiplier function. Therefore, one can obtain local conservation laws of the system using the both method together.

In this work, we considered two famous evolution equations from the soliton theory. Solitons are of great importance in many physical areas, for example, in dislocation theory of crystals, plasma and fluid dynamics, magnetohydrodynamics, laser and fiber optics, and so on.

The propagation of longitudinal deformation waves in an elastic rod is modeled by the Pochammer–Chree (PC) equation

where a, b, and c are real constants. In the literature, some of the exact solitary wave solutions of (1) were obtained in [18–20].

The Kaup–Boussinesq type of coupled KdV equations

were introduced as a water-wave model in [21]. In [22], the inverse problem of the above-mentioned system was studied and soliton solutions which decay asymptotically were found. Very recently, Gürses and Pekcan [23] studied the traveling wave solutions of (2).

The plan of the paper is organised as follows: In Section 2, we give briefly the description of the nonlocal conservation and multiplier methods. In addition, the relationships of both methods were emphasised. Sections 3 and 4 are devoted to the conservation laws of (1) and (2) with the help of this combined method. In Section 5, some concluding remarks are given.

2 Necessary Preliminaries

In this section, we present the notations and some of the definitions below. For the details, see e.g. [4, 7, 12, 14, 15].

Consider the general s^th-order system of PDEs of n independent variables x=(x¹, x², …, xⁿ) and m dependent variables u=(u¹, u², …, u^m)

where u₍₁₎, u₍₂₎, …, u_(s) denote the collections of all first, second, …, sth-order partial derivatives, that is, uiα = Di(uα), uijα = DjDi(uα), … respectively, with the total differentiation operator with respect to xⁱ given by

where the summation convention is used whenever appropriate. As usual 𝒜 is the vector space of differential functions of finite orders. The basic operators defined in 𝒜 are stated below.

The n-tuple vector T=(T¹, T², …, Tⁿ), T^j∈A, j=1, …, n is a conserved vector of (3) if Tⁱ satisfies

Equation (5) defines a local conservation law of system (3). Every admitted consevation laws of (3) arises from multipliers Λ^α (x, u, u₍₁₎, …) such that

holds identically. In the multiplier approach for conservation laws, one takes the variational derivative of (6) that is,

holds for arbitrary functions of u(x¹, x², …, xⁿ) where

is the Euler–Lagrange operator.

In [15], the authors modified the Ibragimov’s new conservation theorem method in terms of the multiplier function. We now present the nonlocal conservation method using the multiplier function, which we desribed in the above. It is interesting to note that the higher order solutions of Ibragimov’s adjoint equation which is consequences of the notion of nonlinear self-adjointness can be obtained by the multiplier functions. According to [15], the solutions of the adjoint equation are constructed through the multipliers satisfying

For each known Q^μ , we define a formal Lagrangian function

and whose Euler–Lagrange equations inherit, all Lie symmetry generators (point, Lie–Backlund and nonlocal) of (3). Using the Noether theorem, we obtain the following conservation laws of (3)

where Nⁱ is the Noether operator,

and

is a symmetry generator of (3) and Wα = ηα − ξjujα is the characteristic function.

Equation (3) is said to be strictly self-adjoint if the adjoint equation

becomes equivalent to the original equation (3) by the substitution w=u. Moreover, (3) is said to be nonlinearly self-adjoint if its adjoint equation becomes equivalent to the original equation after the substitution

w = ϕ

where L is the formal Lagrangian for (3) defined by

and ϕ is a nonzero function depending on the independent variables, the dependent variable as well as the partial derivatives of the dependent variable. In other words, the following identities holding for undetermined coefficients λαβ,

which will be applicable in the computations.

In the following, for (1), Λ^μ =Λ, and for system (2), we will write (Λ¹, Λ²) as (Λ₁, Λ₂). For two independent variables, (x¹, x²) will be written as (x, t) so that (T¹, T²) will be written as (T^x, T^t).

3 Conservation Laws of (1)

We first consider PC equation. The determining equation for the zeroth order multiplier Λ(x, t, u) is (with the help of [4])

Expanding and then separating (17) with respect to different combinations of derivatives of u yields the following overdetermined system for the multipliers:

Λtt = 0,   Λxx = 0,   Λu = 0.

After solving this system, we get the multiplier

where c₁, c₂, c_3, and c₄ are constants. Corresponding to the above multiplier, we have the following conserved vectors of (3):

According to the Ibragimov’s method, (1) has the formal Lagrangian

L = w(utt − uttxx − auxx − 2bux2 − 2buuxx − 6cux2u − 3cu2uxx)

where w(x, t) is the new dependent variable. The adjoint equation for (1) is

We can conclude that (1) is not strictly self-adjoint.

We note that it can be shown that each multiplier Λ_i satisfies the adjoint equation (23). Therefore, if the Lie symmetries of (1) are known, local conservation laws can be obtained. Symmetries of (1) can be easily found using Maple program [4] as the following

X1 = ∂∂x,   X2 = ∂∂t.

On the basis of the stated multipliers and symmetries, some conserved vectors are given below.

RemarkIt is easily seen that using the divergence condition (5), the above conserved vectors are nontrivial. Obtained from other cases (1, X₁), (t, X₁), (1, X₂), and (x, X₂) are trivial conserved vectors, for this reason, it is not given.

4 Conservation Laws of (2)

The determining equation for multiplier Λ₁(t, x, u, v) and Λ₂(t, x, u, v) from (7) is (with the help of [4])

The standard Euler operator δ/δu and δ/δv can be defined as

δδu = ∂∂u − Dx∂∂ux − Dt∂∂ut + Dx2∂∂uxx + DxDt∂∂uxt + …δδv = ∂∂v − Dx∂∂vx − Dt∂∂vt + Dx2∂∂vxx + DxDt∂∂vxt + …

and total derivative operators D_x and D_t are

Dx = ∂∂x + ux∂∂u + uxt∂∂ut + uxx∂∂ux + …Dt = ∂∂t + ut∂∂u + utt∂∂ut + uxt∂∂ux + …

After expansion of (25) and solving determining equations, the multipliers Λ₁(x, t, u, v) and Λ₂(x, t, u, v) are obtained as

Λ1(x, t, u, v) = (3tu24 + xu2 + vt)c1 + u2c2 + (3u24 + v)c3 + c4,Λ2(x, t, u, v) = (tu + x)c1 + c2 + uc3.

Thus, for the determination of conservation laws of (2), we found four multipliers. Corresponding to the each multiplier, we constructed the following local conserved vectors of (2).

RemarkThe multiplier approach gave rise to four local conservation laws for (2).

The formal Lagrangian for (2) is

where p(x, t) and r(x, t) are adjoint variables. The adjoint equations for (2) are

If one substitutes u instead of p and v instead of r in (28), (2) is not obtained. Consequently, (2) is not strictly self-adjoint. However, it is interesting that each multiplier found by multiplier method satisfies the adjoint equations (28).

Equation (2) admits the following Lie-point symmetry generators:

The conservation laws associated with the generators (29) are in the following equation.

It is readily seen that using the divergence condition (5), we obtain the trivial conserved vectors in (30) for the case of (X₃, T^t, T^x).

It is readily seen that using the divergence condition (5), we obtain the trivial conserved vectors in (31) for the cases of (X₁; T^t, T^x) and (X₂; T^t, T^x).

It is readily seen that using the divergence condition (5), we obtain the trivial conserved vectors in (32) for the case of (X₁; T^t, T^x) and (X₂; T^t, T^x).

It is readily seen that using the divergence condition (5), we obtain the trivial conserved vectors in (33) for the all cases.

5 Concluding Remarks

In this work, we have constructed conservation laws of the PC equation and the Kaup–Boussinesq type of coupled KdV system which is not derivable from a variational principle. Namely, no recourse to a Lagrangian formulation is made.

Using the combined approach which is a union of multiplier and Ibragimov’s new conservation theorem method, abundant local conservation laws (some of them are trivial) were obtained. We conclude that the solutions of adjoint equation can be obtained by the multiplier functions. In this article, we restricted ourselves to zeroth order multiplier functions. However, one can also consider the higher order multiplier functions. The conserved vectors obtained here can be used in double reductions and solutions of the underlying equations [24]. In future work, with the aid of conservation laws of the equations, nonlocal symmetries such as potential and nonclassical potential symmetries will be obtained.