In this work, we study the nonlinear self-adjointness and conservation laws for a class of wave equations with a dissipative source. We show that the equations are nonlinear self-adjoint. As a result, from the general theorem on conservation laws proved by Ibragimov and the symmetry generators, we find some conservation laws for this kind of equations.

1. Introduction

There have been abundant literatures that have contributed to the studies of Lie symmetry classification of various classes of (1 + 1)-dimensional nonlinear wave equations and their individual members. Probably, Barone et al. [1] were the first to study the nonlinear wave equation u_tt = u_xx + F(u) by means of symmetry method. Motivated by a number of physical problems, Ames et al. [2] investigated group properties of quasilinear u_tt = [f(u)u_x] _x. Later, Torrisi and Valenti [3, 4], generalizing above equations, have investigated the symmetries of the following equation:

()

Furthermore, classification results for the equation u_tt + K(u)u_t = [F(u)u_x] _x can be found in [5, 6]. An expanded form of the latter equation

()

was studied by Kingston and Sophocleous [7]. In the papers mentioned above, many interesting results including Lie point and nonlocal symmetries classification were systematically investigated.

In this paper, we consider a subclass of (2) with K(u) = α:

()

which is also viewed as a special case of (1) with an additional dissipation, where f(u) and g(u) are arbitrary differentiable functions, t is the time coordinate, x is the one-space coordinate, and α is a nonzero constant. Here, we will focus on the nonlinear self-adjointness and conservation laws for (3).

For (3), we cannot find easily the variational structure so it is inconvenient to apply the Noether theorem to construct conservation laws straightforward for this equation. However, it is fortunate that Ibragimov recently proved a result on conservation laws [8], which does not require the existence of a Lagrangian. The Ibragimov theorem on conservation laws provides an elegant way to establish local conservation laws for the equations under consideration.

Since the seminal work of Ibragimov [8], more and more works are dedicated to studying the self-adjointness and conservation laws for some equations in mathematical physics and there are many new developments, including strict self-adjointness [9–11], quasi-self-adjointness [12–15], weak self-adjointness [16, 17], and nonlinear self-adjointness [18–21] and some results which have been communicated in the recent literature; for more references, see [21] and references therein.

For the sake of completeness, we briefly present the notations, definition of nonlinear self-adjointness, and Ibragimov’s theorem on conservation laws. Let x = (x₁, x₂, …, x_n) be n independent variables, u = u(x) a dependent variable,

()

a symmetry of an equation

()

the adjoint equation to (5), where ℒ = vF is called formal Lagrangian, v = v(x) is a new dependent variable, and

()

denotes the Euler-Lagrange operator.

Now, let us state the definition of nonlinear self-adjointness for a equation.

Definition 1. Equation (5) is said to be nonlinearly self-adjoint if the equation obtained from the adjoint equation (6) by the substitution v = ϕ(x, u) with a certain function ϕ(x, u) ≠ 0 is identical with the original equation (5); in other words, the following equations hold:

()

for some differential function λ = λ(x, u, u₍₁₎, …).

Particularly, if (8) holds for a certain function ϕ such that ϕ_u ≠ 0 and for some x_i, (5) is called weak self-adjoint; if (8) holds for a certain function ϕ such that ϕ = ϕ(u) ≠ u and ϕ^′(u) ≠ 0, then (5) is called quasi-self-adjoint; if (8) holds for ϕ = u, then (5) is called (strictly) self-adjoint.

In the following, we recall the “new conservation theorem” given by Ibragimov in [8]. We will find conservation laws for (3) by this theorem.

Theorem 2. Any Lie point, Lie-Bäcklund, and nonlocal symmetry

()

of (5) provides a conservation law D_i(Cⁱ) = 0 for the system comprising (5) and its adjoint equation (6). The conserved vector is given by

()

where W = η − ξ^ju_j is the Lie characteristic function and ℒ = vF is the formal Lagrangian.

The paper is organized as the follows. In Section 2, we discuss the nonlinear self-adjointness of (3). In Section 3, we establish conservation laws for some particular cases of (3) using Theorem 2.

2. Nonlinear Self-Adjointness of (3)

In this section, we determine the nonlinearly self-adjoint subclasses of (3). Let

()

then we have the following formal Lagrangian for (3):

()

Computing the variational derivative of this formal Lagrangian

()

we obtain the adjoint equation of (3):

()

Assume that F^*|_v=h(t,x,u) = λF, for a certain function λ, where F is given by (11); then we have

()

The comparison of the coefficients of u_tt,

, u_t, and u_x in both sides of (15) yields

()

thus, (15) is reduced as

()

It follows from (17) and (18) that

()

Equation (22) splits into the following two cases.

Case 1 (λ = 0). In this case, from (16) we get that h_u = 0; thus, (16)–(20) are all satisfied. Noting that h does not depend on the function u, f and g are two functions of u, so from (21) we obtain that

()

Equation (23) can be satisfied by taking some appropriate function h = h(t, x). Therefore, we can take v = h(x, t) ≠ 0 such that F^*|_v=h(t,x,u) = λF holds. Thus, (3) is nonlinear self-adjoint in this case.

Case 2 (fu = 0). That is, f is a constant with respect to u; without loss of generality, we set f = C₀ ≠ 0. From (17), we assume that

()

Taking into account (16) and (19), we have a_t(t, x) = αa(t, x); thus, a(t, x) = k(x)e^αt. From (20), we deduce that

()

which implies that g(u) = C₂u + C₁ is a linear function of u and

, where C₂ and C₁ are constants. Hence, we have

()

Substituting (26) into (22) and using the original equation (3), we derive that b(t, x) satisfies

()

which is easy to be solved.

In this case, f(u) = C₀ is a constant, g(u) = C₂u + C₁, and ; therefore, (3) is a weak self-adjoint.

Here, we omit the tedious calculations to obtain the solutions of (23) and (27). In Table 1, we summarize the classification of nonlinear self-adjointness of (3) with the conditions that f(u) and g(u) should satisfy. In what follows, the symbol for all means that the corresponding function has no restrictions and c_i (1 ≤ i ≤ 5) are arbitrary constants.

Table 1. Nonlinear self-adjointness of (3).

f	g	v	Self-adjointness
∀(f_u ≠ 0)	∀	C₁ + C₂e^αt	Nonlinear
∀(f_u ≠ 0)	C₅	(C₁ + C₂e^αt)(C₃ + C₄x)	Nonlinear
∀(f_u ≠ 0)	C₅∫f(u)du		Nonlinear
C₀(≠0)	C₁ + C₂u		Weak

α is a nonzero constant; b(t, x) satisfies (27).

Thus, we have demonstrated the following statement.

Theorem 3. Let α be a nonzero constant, f(u) an arbitrary function of u, and the adjoint equation of (3) given by (14); then (3) is nonlinear self-adjointness with the substitution v and function g(u) given in Table 1.

3. Symmetries and Conservation Laws

In this section, we will apply Theorem 2 to construct some conservation laws for (3). First, we show the Lie classical symmetries for (3) for some special choice of f and g. Then applying formula in Theorem 2 to the formal Lagrangian (12), and to the symmetries X_i and eliminating v by the substitution v given in Table 1, we obtain the conservation law

()

3.1. Lie Symmetries

Now, let us show the Lie classical symmetries for (3). Based on Lie group theory [22], we assume that a Lie point symmetry of (3) is a vector field

()

on ℝ⁺ × ℝ × ℝ such that X⁽²⁾F = 0 when F = 0, and taking into account (3), the operator X⁽²⁾ is given as follows:

()

where

()

The condition X⁽²⁾F = 0, when F = 0, will yield determining equations. Solving these determining equations, we can obtain the symmetries of (3). For f(u) and g(u) arbitrary, the symmetries that are admitted by (3) are

()

For some special choices of the functions f(u) and g(u), here we omit the details of routine calculations and present the symmetries (besides X₁ and X₂) that are admitted by (3) in Table 2.

Table 2. Symmetries of (3) for some special choices of f and g.

f	g	Symmetries
u⁻²	lnu
uⁿ (n ≠ 0)	1
1	u	,
1	1	,

3.2. Conservation Laws

For the symmetries X, from the formula (10) in Theorem 2, we can obtain readily some conservation laws for (3). For example, we take f(u) = u⁻² and g(u) = ln u; let us construct the conserved vector corresponding to the time translation group with the generator

()

For this operator, we have ξ^t = 1, ξ^x = 0, η = 0, and W = −u_t. In this case, (3) is nonlinear self-adjoint and becomes

()

The formal Lagrangian is

()

and the adjoint equation of (34) is

()

Therefore, we obtain the following conserved vector:

()

where v satisfies (36). The reckoning shows that the vector (37) satisfies the conservation equation (28).

In this case, since (3) is nonlinear self-adjoint, from Table 1 we take v = C₁ + C₂e^αt. So, the conserved vector is simplified as follows:

()

Particularly, if setting v = 1, we have

()

Thus, the conserved vector can be reduced to the form

()

If setting v = e^αt, then the conserved vector (37) is simplified as

()

therefore, it can be reduced as

()

In what follows, we omit the tedious calculations and list only the conservation laws of (3) for some special choices of functions f(u) and g(u) in Table 3.

Table 3. Conservation laws of (3) for some special choices of f and g (n ≠ 0).

f	g	Conserved vector with substitution v
		C^t = (v_t − αv)(u + xu_x) − v(u_t + xu_xt)
	lnu	C^x = xvu_tt + αxvu_t + v
		v = C₁ + C₂e^αt


uⁿ	1
		v = (C₁ + C₂e^αt)(C₃ + C₄x)

1	u

		v = C₁ + C₂e^αt + (C₃ + C₄e^αt)e^x or b(t, x)

1	1



		v = (C₁ + C₂e^αt)(C₃ + C₄x) or C₁e^αtu + b(t, x)

In Table 3, the function b(t, x) satisfies (27), v and the symmetry are taken from Tables 1 and 2, respectively. The reckoning shows that the vector listed in Table 3 satisfies the conservation equation (28) with corresponding substitution v. In the same way above, we can simplify the conserved vector using corresponding substitution v.

4. Conclusions

Recently, the new outstanding concepts of nonlinear self-adjoint equations, containing quasi-self-adjoint and weak self-adjoint equations, which extend the self-adjointness to a more generalized meaning, have been introduced in order to find formal Lagrangians of differential equations without variational structure. Using these concepts and the general theorem on conservation laws that is, developed recently [8], nonlinear self-adjointness and conservation laws for (3) for different classes of f(u) and g(u) have been discussed. These conservation laws may be useful in mathematical analysis as they provide basic conserved quantity for obtaining various estimates for smooth solutions and defining suitable norms for weak solutions. Furthermore, it could make the construction of the bi-Hamiltonian form easier.

Acknowledgments

The authors would like to express their sincere gratitude to the referees for many valuable comments and suggestions which served to improve the paper. The first author is partly supported by NSFC under Grant no. 11101111. The second author is partly supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12A01003 and subjects research and development foundation of Hangzhou Dianzi University under Grant no. ZX100204004-6.

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Self-Adjointness, Symmetries, and Conservation Laws for a Class of Wave Equations Incorporating Dissipation

Abstract

1. Introduction

2. Nonlinear Self-Adjointness of (3)

3. Symmetries and Conservation Laws

3.1. Lie Symmetries

3.2. Conservation Laws

4. Conclusions

Acknowledgments

References

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References

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Self-Adjointness, Symmetries, and Conservation Laws for a Class of Wave Equations Incorporating Dissipation

Abstract

1. Introduction

2. Nonlinear Self-Adjointness of (3)

3. Symmetries and Conservation Laws

3.1. Lie Symmetries

3.2. Conservation Laws

4. Conclusions

Acknowledgments

References

Citing Literature

References

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