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

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.