A new look at the double sine-Gordon kink-antikink scattering
Ekaterina Belendryasova, Vakhid A. Gani, Aliakbar Moradi Marjaneh, Danial Saadatmand, Alidad Askari
AA new look at the double sine-Gordon kink-antikinkscattering
E Belendryasova , V A Gani , , A Moradi Marjaneh ,D Saadatmand and A Askari National Research Nuclear University MEPhI (Moscow Engineering Physics Institute),115409 Moscow, Russia National Research Center Kurchatov Institute, Institute for Theoretical and ExperimentalPhysics, 117218 Moscow, Russia Young Researchers and Elite Club, Quchan Branch, Islamic Azad university, Quchan, Iran Department of Physics, University of Sistan and Baluchestan, Zahedan, Iran Department of Physics, Faculty of Science, University of Hormozgan, P.O.Box 3995, BandarAbbas, IranE-mail: [email protected]
Abstract.
We study the kink-antikink scattering within the double sine-Gordon model. In thenumerical simulations we found a critical value v cr of the initial velocity v in , which separates twodifferent scenarios: at v in < v cr the kinks capture each other and form a bound state, while at v in > v cr the kinks pass through each other and escape to infinities. We obtain non-monotonousdependence of v cr on the model parameter R . Besides that, at some initial velocities below v cr we observe formation and interaction of the so-called oscillons (new phenomenon), as well asescape windows (well-known phenomenon).
1. Introduction
Field-theoretic models with polynomial and non-polynomial potentials are of great importancefor various physical systems, e.g., in high energy physics, cosmology, condensed matter, etc.,[1–4]. Significant progress has been made in studying topological solitons in models with onereal scalar field [5–19], as well as in more complex models with two or more fields [20–34]. Bothanalytical and numerical methods are successfully applied to studying kink-antikink interactions.In particular, the collective coordinate method with one or more degrees of freedom enables tomodel kink-(anti)kink interactions [11, 35, 36]. Many interesting results has been obtained bynumerical modeling of collision of kink and antikink, of kink with a defect [37], and of severalkinks in one point [38–40].In this work we study the (1 + 1)-dimensional double sine-Gordon (DSG) model [6, 41–43].Although this model has been well-investigated, we have obtained new results. Firstly, wehave found a non-monotonous dependence of the critical velocity v cr on the model parameter R . Secondly, at some initial velocities from the range v in < v cr in the final state we observedcomplex oscillating structures — oscillons. The latter phenomenon is rather new [16, 17] andwas not reported for the DSG model previously. a r X i v : . [ h e p - t h ] O c t . Scattering of kinks Consider the (1 + 1)-dimensional double sine-Gordon model. Within this model the dynamicsof a real scalar field φ ( x, t ) is described by the Lagrangian L = 12 (cid:18) ∂φ∂t (cid:19) − (cid:18) ∂φ∂x (cid:19) − V ( φ ) (1)with the potential V R ( φ ) = tanh R (1 − cos φ ) + 4cosh R (cid:18) φ (cid:19) , R > , (2)see figure 1a. From the Lagrangian (1) it is easy to obtain the equation of motion — partialdifferential equation of the second order: ∂ φ∂t − ∂ φ∂x + dVdφ = 0 . (3)In the static case φ = φ ( x ), and we have the ordinary differential equation d φdx = dVdφ ⇔ dφdx = √ V , (4)which yields static kink and antikink (see figure 1b): φ k(¯k) ( x ) = 4 πn ± x cosh R . (5)Note that the kink can be viewed as a superposition of two subkinks separated by distance R . R = = Arcsinh1R = = - π - π π π ϕ V R (a) potential R = = Arcsinh1R = = - - x - π πϕ K (b) kinks Figure 1.
Potential (2) and kinks (5) from the sector ( − π ; 2 π ) for some values of R .We performed numerical simulations of the kink-antikink scattering. To do this we solvedthe equation of motion (3) numerically using finite-difference method with the initial conditionin the form of kink and antikink which were initially placed at x = − ξ and x = + ξ , respectively,and moving towards each other with the velocities v in in the laboratory frame of reference. (Themoving kinks can be obtained from the static by the Lorentz transformations.) Thus the initialconditions for our simulations were taken from the following expression: φ ( x, t ) = φ k x + ξ − v in t (cid:113) − v + φ ¯k x − ξ + v in t (cid:113) − v − π. (6)n the numerical experiments we observed two different regimes of the collision: 1) at v in < v cr the kink and antikink form a bion — a bound state, which then decays slowly, emitting smallwaves, see figure 2(a); 2) at v in > v cr the kink and antikink pass through each other and escapeto spatial infinities, see figure 2(b). The critical velocity v cr depends on the parameter R , as it (a) formation of a bion, v in = 0 . v in = 0 . Figure 2.
Space-time picture of the scattering for R = 1 . escapewindows — intervals of the initial velocity within which the kinks collide several times and thenescape to infinities instead of forming a bound state, see figure 3. This phenomenon can be (a) two-bounce window, v in = 0 . v in = 0 . Figure 3.
Escape of kinks after two and three collisions, R = 1 . v cr on the model parameter R , seefigure 4. The curve v cr ( R ) has several local maxima. (Note that this result has not been reportedpreviously by other authors.) Such behavior could be a consequence of the complex structure ofthe DSG kink — the colliding kink and antikink can be viewed as composed objects constructedfrom subkinks. Pairwise interaction of subkinks can lead to non-monotonous dependence of v cr on the model parameter, see, e.g., [36].Another interesting and new phenomenon that we have obtained is formation and furtherinteraction of oscillons , see figure 5. These oscillating structures can form bound states (figure5(a)), or even escape to spatial infinities after several collisions (figure 5(b)). . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 50 . 0 00 . 0 40 . 0 80 . 1 20 . 1 60 . 2 00 . 2 4 R v c r Figure 4. R -dependence of the critical velocity. (a) v in = 0 . v in = 0 . Figure 5.
Oscillons in the final state.
3. Conclusion
We have studied the scattering of the kink and antikink of the double sine-Gordon model. Ourmain new results are the following. • We have found that the critical value v cr of the initial velocity depends on the modelparameter R non-monotonously. This fact has not been reported for the DSG model byother authors. We can assume that such behavior of v cr ( R ) could be a consequence ofpairwise interaction of subkinks. • At the initial velocities below the critical value, v in < v cr , we observed new phenomenon —formation of two oscillons in the final state. Their behavior seems to be rather complicated,they can form a bound state or, in some cases, can escape to spatial infinities after severalcollisions.This research opens wide prospects for future work. For example, it would be interesting tostudy multikink collisions within the DSG model and to find maximal energy densities in suchprocesses. Acknowledgments
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