Scattering of kinks in the $B{\phi }^4$ model

Highlights

• The $B{\phi }^4$ is a modified version of the ${\phi }^4$ theory in which a parameter $B$ is multiplied over the kink sector.

• The features of a single kink are independent of $B$ but the kink interaction depends on it.

• The critical velocity, fractal structure, and soliton behavior are studied as a function of B.

• In the regime 0.2 $\le$ B $\le$ 1, the existence of a quasi-fractal structure is confirmed.

• The lowest critical velocity and the peak of the chaotic behavior occur in B = 3.3.

Abstract

In this study, based on the ${\phi }^4$ model, a new model (called the $B{\phi }^4$ model) is introduced in which the potential form for the values of the field whose magnitudes are greater than 1 is multiplied by the positive number $B$. All features related to a single kink (antikink) solution remain unchanged and are independent of parameter $B$. However, when a kink interacts with an antikink in a collision, the results will significantly depend on parameter $B$. Hence, for kink–antikink collisions, many features such as the critical speed, output velocities for a fixed initial speed, two-bounce escape windows, extreme values, and fractal structure in terms of parameter $B$ are considered in detail numerically. The role of parameter $B$ in the emergence of a nearly soliton behavior in kink–antikink collisions at some initial speed intervals is clearly confirmed. The fractal structure in the diagrams of escape windows is seen for the regime $B\le 1$. However, for the regime $B>1$, this behavior gradually becomes fuzzing and chaotic as it approaches $B=3.3$. The case $B=3.3$ is obtained again as the minimum of the critical speed curve as a function of $B$. For the regime $3.3<B\le 10$, the chaotic behavior gradually decreases. However, a fractal structure is never observed. Nevertheless, it is shown that despite the fuzzing and shuffling of the escape windows, they follow the rules of the resonant energy exchange theory.

Introduction

Over the last few decades, nonlinear wave equations with solitary wave or especially soliton solutions have played a significant role in describing various phenomena in different branches of physics including optics [1], [2], [3], [4], condensed matter [5], [6], [7], high energy physics [8], [9], biophysics [10], and so on. Solitary wave solutions (also called defect structures) are the special solutions of nonlinear wave equations that can freely propagate without any distortion in their profiles. In addition, their corresponding energy densities are localized. Solitons are a special type of solitary wave solutions whose profiles and velocities are restored without any change after collisions. In general, solitary wave solutions can be classified into classical or relativistic and topological or non-topological. The topological property is important because by considering it the solitary wave solution will be inherently stable and minimally energized. As examples of the relativistic topological defects in $3+1$ dimensions, one can mention strings, vortices, magnetic monopoles, and skyrmions [8], [9], [11], [12], [13], [14], [15] which are applied in cosmology as well as hadron and nuclear physics. Q-balls can be mentioned in relation to relativistic non-topological solutions [16], [17], [18]. Moreover, there are a great number of non-relativistic defect structures (topological or non-topological) among which one can name the soliton solutions of the KdV, nonlinear Schrödinger, and Burgers equations [19], [20], [21].

The kink (antikink) solutions of the nonlinear Klein–Gordon equations in $1+1$ dimensions are the simplest relativistic topological defect structures [8], [9]. The importance of such solutions has been explicitly demonstrated in studying wave motion in DNA molecules and graphene sheets [10], [22], [23], [24], [25], [26], [27], [28], [29]. Furthermore, in cosmology, the structure and dynamics of domain walls in $3+1$ dimensions can be correctly described by $1+1$-dimensional kink-bearing theories [30], [31], [32], [33], [34], [35], [36], [37], [38]. Among the great variety of systems with kink (antikink) solutions, the two systems of ${\phi }^4$ and sine–Gordon (SG) have been particularly studied [8], [9]. The sine–Gordon system is important because it is the only integrable Klein–Gordon kink-bearing system which yields soliton solutions. The double SG (DSG) model which is a modified version of the SG model has also received much attention [39], [40], [41], [42]. Interestingly, the DSG system is used for the description of some physical systems such as gold dislocations [43], optical pulses [44], and Josephson structures [45].

The ${\phi }^4$ system and its deformed versions have been widely studied due to their simplicity and applications [26], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66]. In relation to the study of kink–antikink scattering and the quasi-fractal structure, the ${\phi }^4$ system has been considered as a representative of such non-integrable systems [59], [60], [61], [62], [63], [64], [65]. It has been studied from different aspects including the interaction of kinks with impurities [46], [47], [48], [49], maximum values of quantities in a kink–antikink collision [50], scattering between wobbling kink and antikink [51], [52], [53], the standard deformed versions [54], [55], [56], [57], [58], and collective coordinate models for kink–antikink interactions [63], [64], [65]. There is also an informative book on the subject that specifically examines the ${\phi }^4$ system from various angles [66]. Recently, models with polynomial potentials of higher degrees have also received a lot of attention including the ${\phi }^6$ model [67], [68], [69], the ${\phi }^8$ model [70], [71], [72], [73], and the models that yield kinks with power-law tails [74], [75], [76], [77], [78], [79]. Two scalar field models have also been of interest to researchers [31], [32], [33], [80], [81], [82], [83], [84], [85], [86].

Kink–antikink collisions exhibit a richer behavior in non-integrable systems (such as the ${\phi }^4$ system) than in integrable ones [59], [60], [61], [63]. In this respect, except for the SG and radiative systems, the study of output velocity in terms of the incoming (initial) velocity for non-integrable kink-bearing systems has usually led to the similar results. First, kink–antikink collisions face the two different fates of scattering or bion formation. Bion is a metastable non-topological object which decays slowly and radiates energy in the form of small-amplitude waves. Second, there is always a critical velocity ($v_{cr}$) after which a bionic state can no longer be formed. Third, for incoming velocities less than the critical velocity, the typical fate is the formation of a bionic state. However, there are many wide and narrow intervals of initial velocities that lead to escape, are labeled with integer numbers $\mathcal{N}>1$, and are called $\mathcal{N}$-bounce escape windows. More precisely, an $\mathcal{N}$-bounce escape window is an interval of incoming velocities in which the kink and antikink collide $\mathcal{N}$ times before escaping. In such a regime of initial velocities (i.e. $v<v_{cr}$), a quasi-fractal structure is created from the arrangement of different escape windows.

In this paper, the potential of the ${\phi }^4$ system is deformed in a new and different way using parameter $B$ ($0.2\le B\le 10$) so that no change is made to the kink solution itself and the effect of this parameter can only be seen in the interactions. In fact, the potential of the new system (called the $B{\phi }^4$ system) can be divided into two parts. Part I is located between the two vacua $\phi =-1$ and $\phi =1$ and is the same as the ${\phi }^4$ potential so that only this part of the potential is important for determining the kink (antikink) solution. Part II (the rest) is the ${\phi }^4$ potential multiplied by parameter $B$ at $\phi <-1$ and $\phi >1$ (see Fig. 1). Indeed, studying different quantities related to kink–antikink collisions in terms of parameter $B$ is interesting and can enhance our knowledge of systems with kink solutions. Also, from the physical point of view, the existence of systems with different potentials only over the kink sector is equivalent to the existence of the same structure (such as a DNA molecule, a graphene sheet or a domain wall) but with different results in interactions. For example, if kink and antikink are considered as two hypothetical classical particles in $1+1$ dimensions, different values of parameter $B$ lead to different interactions for the same particles, a situation which has no equivalent in particle physics and can be interesting from this point of view.

Our study is structured as follows: In the next section, the $B{\phi }^4$ model is introduced. In addition, some necessary details and general properties of kink-bearing systems are briefly reintroduced. In Section 3, the necessary numerical considerations for the obtained results are presented. Section 4 deals with all numerical results obtained for many systems in the range of $0.2\le B\le 10$. This section contains six subsections in which several features related to kink–antikink collisions are studied in detail. In particular, the critical velocity, the escape windows, the fractal structure, the two-bounce escape windows, the output velocity for a fixed initial velocity, and the maximum values at the center of mass point are all studied in detail in terms of parameter $B$ in this section. The last section is devoted to summary and conclusions. It should be noted that a lot of information is presented graphically in the form of high-resolution figures and can only be accessed by zooming in on them. Therefore, using the printed version of this article is not recommended at all.