The collective coordinates approximation for the kink/anti-kink scattering in the $1+1$ dimensional $ϕ^4$ model is considered and we discuss how the results found in the current literature on the topic can be improved by giving the analytical expression for the lagrangian in the space of parameters. A comprehensive discussion of the role of the collective coordinates approximation in this particular situation is given for completeness.
The fractal velocity pattern in symmetric kink-antikink collisions in $ϕ^4$ theory is shown to emerge from a dynamical model with two effective moduli, the kink-antikink separation and the internal shape mode amplitude. The shape mode usefully approximates Lorentz contractions of the kink and antikink, and the previously problematic null-vector in the shape mode amplitude at zero separation is regularized.
For kink-antikink scattering within the ϕ^4 non--linear field theory in one space and one time dimension resonance type configurations emerge when the relative velocity between kink and antikink falls below a critical value. It has been conjectured that the vibrational excitation of the kink would be the source for these resonances because (simplified) collective coordinate calculations, that emphasized on this excitation, qualitatively reproduced those resonances. Surprisingly a numerical study in the ϕ^6 field theory also exhibited such resonances even though it does not contain the vibratio…
We compare numerical solutions to the full field equations to simplified approaches based on implementing three collective coordinates for kink-antikink interactions within the $φ^4$ and $ϕ^6$ models in one time and one space dimensions. We particularly pursue the question whether the collective coordinate approximation substantiates the conjecture that vibrational modes are important for resonance structures to occur in kink-antikink scattering.
Moduli spaces - finite-dimensional, collective coordinate manifolds - for kinks and antikinks in $ϕ^4$ theory and sine-Gordon theory are reconsidered. The field theory Lagrangian restricted to moduli space defines a reduced Lagrangian, combining a potential with a kinetic term that can be interpreted as a Riemannian metric on moduli space. Moduli spaces should be metrically complete, or have an infinite potential on their boundary. Examples are constructed for both kink-antikink and kink-antikink-kink configurations. The naive position coordinates of the kinks and antikinks sometimes need to b…
We study kink scattering processes in the (1+1)-dimensional $φ^6$ model in the framework of the collective coordinate approximation. We find critical values of the initial velocities of the colliding kinks. These critical velocities distinguish different regimes of collisions. The exact equation of motion for the $φ^6$ model is also solved numerically with the same initial conditions. We discuss advantages and disadvantages of the collective coordinate approximation, and also outline its applicability limits. Resonance phenomena and the so-called escape windows are also observed in the kink co…
In this paper, we investigate the scattering of two $ϕ^6$ kinks and derive the real dynamics by solving the appropriate field equation numerically employing a Runga-Kutta method. We also use a collective coordinate approximation to find approximate dynamics, with the objective being to compare the approximate motion to the real dynamics in order to test the validity of this kind of approximation, which is used extensively in the study of solitons.
The symmetric dynamics of two kinks and one antikink in classical (1+1)-dimensional $ϕ^4$ theory is investigated. Gradient flow is used to construct a collective coordinate model of the system. The relationship between the discrete vibrational mode of a single kink, and the process of kink-antikink pair production is explored.
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 re…
We investigate the role that quasinormal modes can play in kink-antikink collisions, via an example based on a perturbation of the $ϕ^4$ model. We find that narrow quasinormal modes can store energy during collision processes and return it back to the translational degrees of freedom. Quasinormal modes also decay, which leads to energy leakage, causing a closing of resonance windows and an increase of the critical velocity. We observe similar phenomena in an effective model, a small modification of the collective-coordinate approach to the $ϕ^4$ model.
Recently a linearized perturbation theory has been formulated for soliton sectors of quantum field theories. While it is more economical than alternative formalisms, such as collective coordinates, it is currently limited to solitons which stay close to a base point, about which the theory is linearized. As a result, so far this formalism has only been applied to stationary solitons. In spite of this limitation, we construct kink states with fixed nonzero momenta and also moving, normalizable kink wave packets. The former are nonnormalizable, coherent superpositions of kinks at all spatial pos…
Antikink-kink ( $\bar{\rm K} $$ {\rm K}$) collisions in the $ϕ^6$ model exhibit resonant scattering although the $ϕ^6$ kinks do not support any bound states to which energy could be transferred. In Phys. Rev. Lett. 107 (2011) 091602 it was conjectured that, instead, energy is transferred to a collective bound mode of the full $\bar{\rm K} $$ {\rm K}$ configuration. Here we present further strong evidence for this conjecture. Further, we construct a collective coordinate model (CCM) for $\bar{\rm K} $$ {\rm K}$ scattering based on this collective bound mode trapped between the $\bar{\rm K} $$ {…
Collective coordinates provide a powerful tool for separating collective and elementary excitations, allowing both to be treated in the full quantum theory. The price is a canonical transformation which leads to a complicated starting point for subsequent calculations. Sometimes the collective behavior of a soliton is simple but nontrivial, and one is interested in the elementary excitations. We show that in this case an alternative prescription suffices, in which the canonical transformation is not necessary. The use of a nonperturbative operator which creates a soliton state allows the theor…
We review in detail the construction of {\em all} stable static fermion bags in the 1+1 dimensional Gross-Neveu model with $N$ flavors of Dirac fermions, in the large $N$ limit. In addition to the well known kink and topologically trivial solitons (which correspond, respectively, to the spinor and antisymmetric tensor representations of O(2N)), there are also threshold bound states of a kink and a topologically trivial soliton: the heavier topological solitons (HTS). The mass of any of these newly discovered HTS's is the sum of masses of its solitonic constituents, and it corresponds to the te…
