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

and Applied Analysis 3 Table 1: Nonlinear self-adjointness of (3). f g V Selfadjointness ∀(f u ̸ = 0) ∀ C 1 + C 2 eαt Nonlinear ∀(f u ̸ = 0) C 5 (C 1 + C 2 eαt)(C 3 + C 4 x) Nonlinear ∀(f u ̸ = 0) C 5 ∫f(u)du C 1 + C 2 eαt +(C 3 + C 4 e αt )e x/C5 Nonlinear C 0 ( ̸ = 0) C 1 + C 2 u C 1 e (C2/C0)x+αtu + b(t, x) Weak α is a nonzero constant; b(t, x) satisfies (27). 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 selfadjoint in this case. Case 2 (f u = 0). That is, f is a constant with respect to u; without loss of generality, we set f = C 0 ̸ = 0. From (17), we assume that h (t, x, u) = a (t, x) u + b (t, x) . (24) Taking into account (16) and (19), we have a t (t, x) = αa(t, x); thus, a(t, x) = k(x)e. From (20), we deduce that g u (u) = C 0 k󸀠 (x) k (x) , (25) which implies that g(u) = C 2 u + C 1 is a linear function of u and k(x) = C 1 e20, where C 2 and C 1 are constants. Hence, we have h (t, x, u) = C 1 e (C 2 /C 0 )x+αt u + b (t, x) . (26) Substituting (26) into (22) and using the original equation (3), we derive that b(t, x) satisfies b tt − αb t − C 0 b xx + C 2 b x = 0, (27) which is easy to be solved. In this case,f(u) = C 0 is a constant, g(u) = C 2 u+C 1 , and λ = h u = C 1 e20; 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. 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 Table 2: Symmetries of (3) for some special choices of f and g.


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   =   + () 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.
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   (  ) = 0 for the system comprising (5) and its adjoint equation (6).The conserved vector is given by where  =  −     is the Lie characteristic function and L = V 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.

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  * | V=ℎ(,,) = , for a certain function , where  is given by ( 11); then we have The comparison of the coefficients of   ,  2  ,  2  ,   , 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.
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   (1 ≤  ≤ 5) 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 V and function () given in Table 1.

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 V by the substitution V given in Table 1, we obtain the conservation law   (  ) +   (  ) = 0.
Table 3: Conservation laws of (3) for some special choices of  and  ( ̸ = 0).  Symmetries Conserved vector with substitution V 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 () =  −2 and () = ln ; let us construct the conserved vector corresponding to the time translation group with the generator For this operator, we have   = 1,   = 0,  = 0, 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 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 =  1 +  2   .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 =   , 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), 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.

Conclusions
Recently, the new outstanding concepts of nonlinear selfadjoint 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.

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