Abstract
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 -dimensional nonlinear wave equations and their individual members. Probably, Barone et al. [1] were the first to study the nonlinear wave equation by means of symmetry method. Motivated by a number of physical problems, Ames et al. [2] investigated group properties of quasilinear . Later, Torrisi and Valenti [3, 4], generalizing above equations, have investigated the symmetries of the following equation: Furthermore, classification results for the equation 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 : which is also viewed as a special case of (1) with an additional dissipation, where and are arbitrary differentiable functions, is the time coordinate, is the one-space coordinate, and 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 be independent variables, a dependent variable, a symmetry of an equation the adjoint equation to (5), where is called formal Lagrangian, 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 with a certain function is identical with the original equation (5); in other words, the following equations hold:
for some differential function .
Particularly, if (8) holds for a certain function such that and for some , (5) is called weak self-adjoint; if (8) holds for a certain function such that and , then (5) is called quasi-self-adjoint; if (8) holds for , 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 for the system comprising (5) and its adjoint equation (6). The conserved vector is given by where is the Lie characteristic function and 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 , for a certain function , where is given by (11); then we have The comparison of the coefficients of , , , , and 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 . In this case, from (16) we get that ; thus, (16)–(20) are all satisfied. Noting that does not depend on the function , and are two functions of , so from (21) we obtain that Equation (23) can be satisfied by taking some appropriate function . Therefore, we can take such that holds. Thus, (3) is nonlinear self-adjoint in this case.
Case 2 . That is, is a constant with respect to ; without loss of generality, we set . From (17), we assume that
Taking into account (16) and (19), we have ; thus, . From (20), we deduce that
which implies that is a linear function of and , where and are constants. Hence, we have
Substituting (26) into (22) and using the original equation (3), we derive that satisfies
which is easy to be solved.
In this case, is a constant, , 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 and should satisfy. In what follows, the symbol for all means that the corresponding function has no restrictions and () are arbitrary constants.
Thus, we have demonstrated the following statement.
Theorem 3. Let be a nonzero constant, an arbitrary function of , and the adjoint equation of (3) given by (14); then (3) is nonlinear self-adjointness with the substitution and function 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 and . Then applying formula in Theorem 2 to the formal Lagrangian (12), and to the symmetries and eliminating by the substitution 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 such that when , and taking into account (3), the operator is given as follows: where The condition , when , will yield determining equations. Solving these determining equations, we can obtain the symmetries of (3). For and arbitrary, the symmetries that are admitted by (3) are For some special choices of the functions and , here we omit the details of routine calculations and present the symmetries (besides and ) that are admitted by (3) in Table 2.
3.2. Conservation Laws
For the symmetries , from the formula (10) in Theorem 2, we can obtain readily some conservation laws for (3). For example, we take and ; let us construct the conserved vector corresponding to the time translation group with the generator For this operator, we have , , , and . 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 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 . So, the conserved vector is simplified as follows: Particularly, if setting , we have Thus, the conserved vector can be reduced to the form If setting , 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 and in Table 3.
In Table 3, the function satisfies (27), 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 . In the same way above, we can simplify the conserved vector using corresponding substitution .
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 and 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.