Solitary wave solutions of nonlocal sine-Gordon equations

In this paper a nonlocal generalization of the sine-Gordon equation, u(tt)+sin u=( partial differential / partial differential x) integral (- infinity ) (+ infinity )G(x-x('))u(x(') )(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 2pi-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)= summation operator (j=1) (N)kappa(j)e(-eta(j)mid R:ximid R:), eta(j)>0, j=1,2, em leader,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 in (c(*)- varepsilon,c(*)+ varepsilon ) for a certain value of varepsilon. 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) approximately K(0)(mid R:ximid R:/lambda) (lambda is a parameter) that have applications in the Josephson junction theory. (c) 1998 American Institute of Physics.