Some exact solutions to the sine-Gordon equations

We present some new exact (solitary and traveling) solutions to the sine-Gordon equations c2uxx - utt = α sin u and cuxt + utt = α sin u, and the dissipative sine-Gordon equation c2Uxx = utt - γu, = α1 sin u + α2 sin(2u). Here, α, α1, and α2 are the coupling constants, real numbers; c is the intrinsic speed of the propagating wave determined by the system; and γ is the damping factor. The physical implication is briefly discussed