Moving nonradiating kinks in nonlocal φ4 and φ4-φ6 models.

We explore the existence of moving nonradiating kinks in nonlocal generalizations of φ(4) and φ(4)-φ(6) models. These models are described by nonlocal nonlinear Klein-Gordon equation, u(tt)-Lu+F(u)=0, where L is a Fourier multiplier operator of a specific form and F(u) includes either just a cubic term (φ(4) case) or cubic and quintic (φ(4)-φ(6) case) terms. The general mechanism responsible for the discretization of kink velocities in the nonlocal model is discussed. We report numerical results obtained for these models. It is shown that, contrary to the traditional φ(4) model, the nonlocal φ(4) model does not admit moving nonradiating kinks but admits solitary waves that do not exist in the local model. At the same time the nonlocal φ(4)-φ(6) model describes moving nonradiating kinks. The set of velocities allowed for these kinks is discrete with the highest possible velocity c(1). This set of velocities is unambiguously determined by the parameters of the model. Numerical simulations show that a kink launched at the velocity c higher than c(1) starts to decelerate, and its velocity settles down to the highest value of the discrete spectrum c(1).