Solitary wave solutions of nonlocal sine-Gordon equations

In this paper a nonlocal generalization of the sine-Gordon equation, u(tt) + sin u =(delta/delta x)integral(-infinity)(+infinity)G(x - x')u(xt)(x', t)dx' is considered. We present a brief review of the applications of such equations and show that involving such a nonlocality can change features of the model. In particular, some solutions of the sine-Gordon model (for example, traveling 2 pi-kink solutions) may disappear in the nonlocal model, furthermore, some new classes of solutions such as traveling topological solitons with topological charge greater than 1 may arise. We show that the lack of Lorenz invariancy of the equation under consideration can lead to a phenomenon of discretization of kink velocities. We discussed this phenomenon in detail for the special class of kernels G(xi) = Sigma(j=1)(N) kappa(,)e(-eta j/xi/), eta(j) > 0, j = 1,2,...,N. We show that, generally speaking, in this case the velocities of kinks (i) are determined unambiguously by a type of kink and value(s) of kernel parameter(s); (ii) are isolated i.e., if c* is the velocity of a kink then there are no other kink solutions of the same type with velocity c is an element of (c* - epsilon, c* + epsilon) for a certain value of epsilon. We also used this special class of kernels to construct approximations for analytical and numerical study of the problem in a more general case. Finally, we set forth results of the numerical investigation of the problem with the kernel that is the McDonald function G(xi) similar to K-0(\xi\/lambda) (lambda is a parameter) that have applications in the Josephson junction theory. (C) 1998 American Institute of Physics.