Fractal structure in the scalar λ(φ²-1)² theory

Head-on collisions of kink and antikink solitons are investigated numerically in the classical one-dimensional λ(φ²-1)² model. It is shown that whether a kink-antikink interaction settles to a bound state or a two-soliton solution depends "fractally" on the impact velocity. We discuss the results using the framework of perturbation theory which helps to clarify the nature of the fractal structure in terms of resonances with the internal shape mode oscillations. We also review the technique of collective coordinates used to reduce the infinite-dimensional system to one with just two degrees of freedom. Although we do not expect exact agreement by using such a simplification, we show that the reduced system bears a striking qualitative resemblance to the full infinite-dimensional system, reproducing the fractal structure. The maximum Lyapunov exponents are computed for the bound-state oscillations and found to be ∼0.3 for both the full and reduced systems, demonstrating the chaotic nature of the bound state.