SOME EXACT-SOLUTIONS TO THE SINE-GORDON EQUATIONS

We present some new exact (solitary and traveling) solutions to the si ne-Gordon equations c(2)u(xx) - u(tt) = alpha sin u and cu(xt) + u(tt) = alpha sin u, and the dissipative sine-Gordon equation c(2)u(xx) - u (tt) - gamma u(t) = alpha(1) sin u + alpha(2) sin(2u). Here, alpha, al pha(1), and alpha(2) are the coupling constants, real numbers; c is th e intrinsic speed of the propagating wave determined by the system; an d gamma is the damping factor. The physical implication is briefly dis cussed.