Analytical perspectives on localized solutions of the phi-4 theory - Dika, Alain (Department of Mathematics and Applied Mathematics, 2024)
We investigate the topological and non-topological solitons—kink and breather—in the φ4 model. Our first objective is to explain the chaotic motion of the resonantly driven kink. To this end, we consider the evolution of the kink's wobbling and rocking modes and their coupling. To improve the asymptotic expansion of the wobbling and rocking amplitudes, we introduce a slowly varying width of the kink. The analysis of the wobbling and rocking modes produces three-dimensional systems of equations, while their coupling results in five-dimensional systems for the amplitude of the wobbling, rocking, and the kink's velocity. Although these equations predict hysteretic transitions in the wobbling and rocking amplitudes (a conclusion verified by numerical simulations of the full partial differential equation), the asymptotic approach does not capture the chaotic behaviour of the kink. Numerical simulations indicate that the chaotic motion of the kink is associated with the spontaneous emission of a small-amplitude breather. This observation motivates our subsequent analysis of the kink-breather interaction. To study the kink-breather interaction, we use the variational method. We start with the development of the variational approach to a stand-alone breather. For the quiescent breather, we show that the resulting variational equations have continuous families of periodic orbits for each value of the breather width. For the moving breather, we compare two sets of Ansatze. The first set of trial functions is built by including a translation degree of freedom into the quiescent breather Ansatz. These trial functions turn out to be inconsistent with the conservation of momentum. To correct this inconsistency, we add one more collective coordinate to the previous Ansatze, which leads to two distinct and consistent trial functions for the moving breather. Having developed a variational approach to a stand-alone breather, we apply it to the kink- breather bound state. In our investigation, we implement two distinct Ansatze: the first trial function involves the amplitude of the breather and its distance from the kink, while the second trial function considers the breather's phase and the distance between the two objects. Local minima of the potential energy from both Ansatze are surrounded by families of closed level curves. The kink interacts with the breather when the initial conditions are chosen to be close enough to a local minimum. We observe that breathers with small amplitudes are trapped by the kink, while those with an amplitude greater than a critical value escape. The Ansatz based on the distance between the two objects and the breather's amplitude shows that when the trapping occurs, the kink and the breather move at the same speed. Finally, we test the robustness of the finite-dimensional model by allowing the kink's width to change. This results in a 6-dimensional phase space. The potential energy shows local minima surrounded by families of closed level surfaces. By selecting initial conditions that lie on the energy shell, the resulting trajectory will remain inside that level surface.