The Sine-Gordon equation in (1+3) dimensions has N-traveling front ("kink", "domain wall")- solutions for all N ≥ 1. A nonlinear functional of the solution, which vanishes on a single-front, maps multi-front solutions onto sets of infinitely long, but laterally bounded, rods, which move in space. Each rod is localized in the vicinity of the intersection of two Sine-Gordon fronts. The rod system...

This paper presents the perturbation theory for the double–sine–Gordon equation. We received the system of differential equations that shows the soliton parameters modification under perturbation’s influence. In particular case λ = 0 the results of the research transform into well-known perturbation theory for the sine–Gordon equation.

We describe and characterize rigorously the homoclinic structure of the perturbed sine{ Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine{Gordon equation is established. Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.

We describe and characterize rigorously the chaotic behavior of the sine– Gordon equation. The existence of invariant manifolds and the persistence of homoclinic orbits for a perturbed sine–Gordon equation are established. We apply a geometric method based on Mel’nikov’s analysis to derive conditions for the transversal intersection of invariant manifolds of a hyperbolic point of the perturbed ...