Suppression of two-bounce windows in kink-antikink collisions

We consider a class of topological defects in $(1,1)$-dimensions with a deformed $\phi^4$ kink structure whose stability analysis leads to a Schr\"odinger-like equation with a zero-mode and at least one vibrational (shape) mode. We are interested in the dynamics of kink-antikink collisions, focusing on the structure of two-bounce windows. For small deformation and for one or two vibrational modes, the observed two-bounce windows are explained by the standard mechanism of a resonant effect between the first vibrational and the translational modes. With the increasing of the deformation, the effect of the appearance of more than one vibrational mode is the gradual disappearance of the initial two-bounce windows. The total suppression of two-bounce windows even with the presence of a vibrational mode offers a counterexample from what expected from the standard mechanism. For even larger deformation, some two-bounce windows reappear, but with a non-standard structure.


INTRODUCTION
Solitary waves are important objects of investigation in several areas of nonlinear physics in all scales, from low-energy [1] to high-energy physics [2]. The simplest solitary wave solution obtained with scalar fields is the (1, 1) dimensional kink. The embedding of a kinklike defect in three spatial dimensions gives rise to a domain wall, a topological defect separating a region of space in two volume domains. The initial physical conditions originating domain walls often favor the emergence of multiple domains separated by dynamical wall networks.
Deviations from standard domain wall models, allowing walls with different energy or topology or biased vacuum values modify considerably the network structure, originating different wall patterns [3]. A first-order phase transition in the early universe could generate bubbles of the broken-symmetry phase. The study of collapse of collapsing domain bubble [4] contributed for the discovery of breather solutions. In the regime of high bubble nucleation rate, one can consider the collision of two bubbles as in flat spacetime with SO(2, 1) symmetry [5]. For very large bubbles, the collision process can include in addition planar symmetry, an ingredient also used in the context of branes [6]. In a Minkowski background, this reduces the background dynamics of colliding domain walls to that of a KK pair in (1, 1) dimensions, as used for instance in the study of the effects of small initial quantum fluctuations in nucleated bubbles in collision [7].
Kink and antikink solutions can be obtained for instance in the renormalizable and nonintegrable φ 4 theory. Despite its simplicity, in this theory and in several other nonintegrable models the process of KK collisions can be surprisingly rich, when analyzed as a function of the initial velocity of approximation [8][9][10][11][12][13][14][15][16][17][18][19]. For large initial velocity the pair KK recedes from each other whereas for small initial velocity a KK bion state [20] is formed. For intermediate velocities, however, the richness of the collision is revealed with windows in velocity (called bounce windows) where an integer number of bounces do appear before the components of the KK pair recede from each other. If one zooms in the border of a region of a certain number of bounces, a new window shows up with a higher number of bounces in a kind of fractal structure [16]. The present work deals with the simplest effect of two-bounce windows. Two-bounce windows were also observed in collisions between kinks and defects [21][22][23][24] and in collisions of orthogonally polarized vector solitons in birefringent optical fibers [25][26][27]. For a good review for effects of nonlinearity in classical field theory for nonintegrable systems, see [28] and references therein.
According to Campbell, Schönfeld and Wingate (CSW) [13], a resonance effect is the mechanism behind the appearance of two-bounce windows. There the separation of the formed pair KK after the second bouncing is due to the change from the first vibrational mode and consequent restoration to the translational mode in a resonant mechanism. A counter-example of this mechanism was found for the φ 6 model and presented in Ref. [29], where it was shown that two-bounce windows could be obtained even in the absence of a vibrational mode, but as a result of collective mode produced by the pair KK. For more results with this model, see Ref. [30].
In this paper we present another counter-example of the CSW mechanism, in which twobounce windows disappear completely despite the presence of vibrational modes. For this, in section II we consider a class of deformed kinks [31] driven by a parameter b that recover the φ 4 kink (for b = 0) and in the limit b → ∞ leads to a double kink [32]. The model has always a zero mode and at least one vibrational mode. The number of vibrational modes grows with b. In section. III we consider the structure of KK scattering. The analysis shows the gradual disappearing of the two-bounce windows until their total suppression for a specific range of b values. In section IV we present our main conclusions, including the connection between our finding, the appearance of an additional vibrational mode and the structure of the potential of linear perturbations.

II. THE MODEL
We start with the action where the φ is a real scalar field and V (φ) is the potential. The equation of motion is The construction of static kinks φ S with the firstorder formalism requires the introduction of a superpotential W (φ). If the potential has the form V (φ) = 1/2W 2 φ , then the solutions of the first-order equation dφ/dx = ±W φ are also solutions of the second order equation of motion. The defects formed with this prescription minimize energy and are known as BP S defects [33,34]. The φ 4 model is the archetype of the construction of kink defects in non-integrable theories. It is characterized by a superpotential given by W φ = 1 − φ 2 , with a solution given by φ(x) = ± tanh(x − x 0 ), where x 0 is the center of the kink. Stability analysis is a standard procedure and considers small fluctuations . With the introduction of the superpotential, it can be shown that the Hamiltonian is positive definite and tachyonic modes are absent.
An interesting class of deformed kinks was considered in Ref. [31], where the properties of the defects where controlled by a parameter. The defect has the following scalar field where an antikink solution may be obtained by the space reflection to get φK = −φ K . The dimensionless parameter b regulates the appearance of a double kink character. Fig. 1a depicts some plots of φ(x) for several values of b. Such solution can be obtained by the superpotential [31] The Fig. 1b depicts the change in shape of the potential V (φ) corresponding to the several   vibrational modes with the increasing of b. We noted also that the energy of the second vibrational mode decreases for large b.

III. KK COLLISIONS
We considered the collisions to be symmetric, with a deformed kink coming from x → −∞ with velocity v and a deformed antikink coming from x → +∞ with velocity −v. The initial conditions are where the double sine-Gordon kink [14].
The CSW mechanism is described by the relation [13] ω 1 T = 2πm + δ.  Fig. 4a can be explained by the CSW mechanism. With the increasing of b, some of the two-bounce windows previewed are suppressed. We will show that one example of this corresponds to Fig. 4b for b = 1.3. With larger values of b a region of total suppression of two-bounce windows is achieved (see Fig. 4c for b = 1.6). For even larger values of b, some two-bounce windows are recovered (see Fig. 4d).
Tables I and II contains data of the first four observed two-bounce windows for b = 1.01 and b = 1.3, respectively (corresponding to Figs. 4a-b). Each window is characterized by a centerv and a thickness ∆v. From Table I  absent whereas Table II shows that for b = 1.3 the same occurs with the first three ones.
This shows that the suppression of the two-bounce starts in the lower value of m, labeled m min , and that m min grows with b. This is confirmed by Fig. 5.  Table II shows that for b = 1.3 an agreement between the obtained values of β is hardly possible, showing that there is no such scaling relation anymore.
In the region before suppression, we investigated some properties of the structure of the two-bounce windows, confirming the validity of Eq. (5) and the scaling relation [13] T ∝ (v 2 c − v 2 ) −1/2 . We also used CSW theory to understand the measured center of twobounce windows and to estimate values of lacked centers. Indeed, for each value of b, it is possible to predict the centers as given by [13] v Tables I to II show that the centers obtained numerically agree with those predicted by this method.

IV. CONCLUSIONS
Our analysis showed that the presence of at least one additional vibrational state concurs to spoil some of the resonant effect (according to the CSW mechanism) responsible for the formation of two-bounce windows. The effect of partial suppression of two-bounce windows was already previewed in the final considerations of Ref. [13] and observed for instance in the double sine-Gordon [14] and in φ 4 kink-impurity interactions [21]. In the latter case a qualitative explanation of the effect could be made using collective coordinates [9,[35][36][37]. Here, however, the quite intricate Schrödinger-like potential makes it impracticable windows. This makes sense with some known results, since total suppression of two-bounce windows was not observed in other models of two-kinks [38,39]. This signals that in the model analyzed here the total suppression of two-bounce windows in KK collisions requires beside the existence of more than one vibrational state a hybrid character between a kink and a double kink. Finally we stress that in a sense this work is a counterpart of what presented in Ref. [29], which showed that the formation of two-bounce windows in the φ 6 model even in the absence of an internal vibrational state. That is, this work also poses some limits on the applicability of the CSW mechanism to describe KK collisions.