Collective Coordinate Methods and Their Applicability to $$\phi ^4$$ Models

Abstract

Collective coordinate methods are frequently applied to study dynamical properties of solitons. These methods simplify the field equations -typically partial differential equations -to ordinary differential equations for selected excitations. More importantly though, collective coordinates provide a practical means to focus on particular modes of otherwise complicated dynamical processes. We review the application of collective coordinate methods in the analysis of the kink-antikink interaction within the ϕ 4 soliton model and illuminate discrepancies between these methods and the exact results from the field equations. Kink-antikink scatteringKink-antikink scattering can be simulated as a solution to the field equations with suitable initial conditions. Here we discuss this treatment and its results briefly.1 Here we restrain to the classical description, i.e. canonical commutation relations are not imposed. Then this factor has no dynamical relevance.

The ϕ4 model with the BPS preserving defect

“…11 c). Note, that in [37] it was conjectured that a similar disappearance of the shape modes of a colliding kink and antikink is responsible for the failure of the collective coordinate method applied to such a collision. On the other hand, this distance-dependence of the spectral structure is crucial for the existence of the resonance bounce structure in φ 6 model [29].…”

Resonance structures in kink-antikink collisions in a deformed sine-Gordon model

“…This mechanism can be reasonably well approximated in a truncated model, which takes into account only two dynamical degrees of freedom, the collective coordinates of the kink and the internal mode [13,15,19,20]. This approach is qualitatively effective but many attempts to derive it from first principles have been plagued by many ambiguities and discrepancies, see [20][21][22], although recent developments [23,24] show some promising results. Furthermore, some modifications of the φ 4 model, see e.g.…”

Quasinormal modes in kink excitations and kink–antikink interactions: a toy model

“…However, it is known that in many models the detailed pattern of the resonance windows cannot be captured by the collective coordinate approach. A recent approximate description of the resonance in the φ 4 and φ 6 models using the collective coordinates has been given in [46,47], where the authors discuss the applicability and the discrepancies between this approach and the exact results from the field equations. The studies in [46,47] give evidence to the fact that the energy exchange mechanism of Campbell et al is only an approximate description and it alone is not sufficient to create the resonance windows.…”

“…A recent approximate description of the resonance in the φ 4 and φ 6 models using the collective coordinates has been given in [46,47], where the authors discuss the applicability and the discrepancies between this approach and the exact results from the field equations. The studies in [46,47] give evidence to the fact that the energy exchange mechanism of Campbell et al is only an approximate description and it alone is not sufficient to create the resonance windows. In particular, it was pointed out in [48] that this failure might happen because the normal modes disappear into the continuum during a kink-antikink collision.…”