Emergence of oscillons in kink-impurity interactions
Mariya Lizunova, Jasper Kager, Stan de Lange, Jasper van Wezel
EEmergence of oscillons in kink-impurity interactions
Mariya Lizunova,
1, 2
Jasper Kager, Stan de Lange, and Jasper van Wezel ∗ Institute for Theoretical Physics, Utrecht University,Princetonplein 5, 3584 CC Utrecht, The Netherlands Institute for Theoretical Physics Amsterdam, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The Netherlands (Dated: December 15, 2020)The (1 + 1)-dimensional classical ϕ theory contains stable, topological excitations in the form ofsolitary waves or kinks, as well as stable but non-topological solutions, such as the oscillon. Both areused in effective descriptions of excitations throughout myriad fields of physics. The oscillon is well-known to be a coherent, particle-like solution when introduced as an Ansatz in the ϕ theory. Here,we show that oscillons also arise naturally in the dynamics of the theory, in particular as the resultof kink-antikink collisions in the presence of an impurity. We show that in addition to the scatteringof kinks and the formation of a breather, both bound oscillon pairs and propagating oscillons mayemerge from the collision. We discuss their resonances and critical velocity as a function of impuritystrength and highlight the role played by the impurity in the scattering process. The classical ϕ theory emerges as an effective descrip-tion of processes throughout fields of physics, going wellbeyond its original introduction as a phenomenologicaltheory of second-order phase transitions [1]. In particu-lar, the solitary wave solutions of the theory, known askinks, feature in the effective description of domain wallsin molecules, solid materials, and even cosmology [2–5],as well as in various toy models in nuclear physics [6–8]and biophysics [9, 10]. The kink is a stable, particle-likeexcitation that is protected from decay by its topologicalcharge. In addition, the ϕ theory contains well-knownnon-topological excitations, in the form of breathers andoscillons. Both are quasi-long-lived, oscillating, stable so-lutions with applications as effective descriptions of ob-jects throughout physics.Unlike solitons in the integrable sine-Gordonmodel [11], kinks and antikinks cannot just passthrough each other [12], but instead interact andundergo dynamic processes including scattering, theformation of bound states, and resonances [13–17].Breathers are formed naturally as post-collision boundstates in simulations of colliding kink-antikink pairs.In contrast, oscillons do not emerge naturally fromkink-antikink collisions, and besides being explicitly in-troduced as an Ansatz for the field configuration [18, 19],they have only been identified in the dynamics of asinh-deformed (1 + 1)-dimensional ϕ model [20], afterapplying a model deformation [21, 22].In this work, we show that oscillons do emerge natu-rally as a product of kink-antikink collision dynamics inthe (1+1)-dimensional ϕ model, if the collision occurs inthe presence of an impurity or defect. Such impurities areunavoidable in the atomic lattices of condensed matterand quantum chemistry settings, while in particle physicsand cosmology, variations in the potential or backgroundmetric may be modeled by the introduction of impuri-ties [23–29]. Although the interaction of a single kinkwith an impurity is well-studied, interactions between kink-antikink pairs and impurities have been consideredonly in the context of the integrable sine-Gordon model[30]. Here, we show that the kink-impurity-antikink dy-namics not only give rise to excitation of the impuritymode and capture, resonance, and reflection of the kink-antikink pair, but that the presence of an impurity alsocatalyzes the formation of pairs of oscillons at the im-purity site. These are observed as quasi-long-lived solu-tions both in the form of bound oscillon-oscillon pairs andin configurations where the two oscillons independentlypropagate away from the impurity location. Model . We consider a classical real scalar field ϕ ( t, x )in (1 + 1)-dimensional spacetime. The Lagrangian in thepresence of a point-like impurity located at x = 0 yieldsthe equation of motion: ϕ tt − ϕ xx + ( ϕ − ϕ )(1 − (cid:15)γ ( x )) = 0 . Here, | (cid:15) | represents the strength of the impurity potential γ ( x ) centered at x = 0. Note that the field dynamics inthe pristine case (cid:15) = 0 is well-studied [31], and alreadygives rise to a rich phenomenology containing breathersand resonances. Below, we discuss the field dynamics inthe presence of a weak impurity, i.e. for | (cid:15) | <
1. To es-tablish a connection to analytic results for the excitationmodes of isolated Dirac-delta impurities [23], we consideran impurity profile with a very narrow Gaussian shape: γ ( x ) = 1 σ √ π exp (cid:34) − (cid:18) xσ √ (cid:19) (cid:35) . We choose the width of the Gaussian to be σ (cid:39) . a r X i v : . [ c ond - m a t . o t h e r] D ec Figure 1. Different types of bound states for subcritical initialvelocities in kink-impurity-antikink collisions. The dynam-ics transitions from (a) a fast decay into the vacuum state(shown at (cid:15) = 0 . v in = 0 .
2) through (b) a quasi-long-lived breather ( (cid:15) = 0 .
15 and v in = 0 .
15) to (c) an oscillon-oscillon bound state ( (cid:15) = − . v in = 0 . moving towards each other: ϕ ( t, x ) = − (cid:32) x + a − v in t (cid:112) − v ) (cid:33) − tanh (cid:32) x − a + v in t (cid:112) − v ) (cid:33) . Here, ± a are centers of localization for the kink and an-tikink at t = 0, and ∓ v in are their initial velocities. Weuse natural units, so that 0 < v in <
1. The equationof motion for the field can be solved numerically usinga method of finite differences for open boundary condi-tions [32], with the continuity equation verified at eachtime step.
Attractive impurity.
We first consider the case of thekink and antikink colliding at the site of an attractive im-purity, with (cid:15) >
0. If the initial velocity | v in | is smallerthan a critical velocity v cr , a breather is formed after thecollision, similar to what occurs in kink-antikink scat-tering without an impurity present. However, in thepresence of the attractive impurity, the breather is nolonger a quasi-long-lived solution and the field configura-tion quickly decays to the vacuum solution ϕ ( x ) = − v in < v cr , at which the kink and the antikinkleave each other after a finite number of impacts. Notethat we observe resonances for various combinations ofimpurity strength and initial velocity, such as (cid:15) = 0 . v in = 0 . (cid:15) = 0 .
15 and v in = 0 . v cr , they leavethe impurity location after a single collision and propa-gate to spatial infinity. They leave behind an oscillat-ing mode, localized at the site of the impurity, x = 0.Similar impurity modes have been observed in numeri-cal simulations of kink-impurity interactions [23, 33], andfound analytically for a Dirac-delta impurity [23], whereits time-dependent profile δϕ ( t, x ) was predicted to be: δϕ ( t, x ) ∝ exp ( − (cid:15) | x | ) cos(Ω t ) , Ω = 2 − (cid:15) . (1)To extract the frequency ˜Ω and amplitude A of the im- Figure 2. (a) The impurity mode frequency ˜Ω and (b) itsamplitude A , as a function of the initial kink and antikinkvelocity v in , for fixed values of impurity strength. (c) Thefrequency and (d) amplitude as a function of the impuritystrength (cid:15) , keeping the initial velocity constant. purity mode oscillations for the case of the kink-antikinkpair interacting with a narrow Gaussian impurity, we usea discrete Fourier transform of ϕ (0 , t ). We only considerfield values at late times ( t > (cid:15) = 0 . . (cid:104) ˜Ω (cid:105) (cid:39) .
41 with amaximal deviation of 0 .
45% for (cid:15) = 0 . (cid:104) ˜Ω (cid:105) (cid:39) . .
10% for (cid:15) = 0 . (cid:15) = 0 . v in = 0 .
4, the fieldconfiguration seems to approach a stable oscillation withfrequency ˜Ω (cid:39) .
30. This is in good agreement with thefrequency Ω (cid:39) .
32 predicted by Eq. (1) for an impuritymode at this value of the impurity strength.
Repulsive impurity.
In the case of the kink and an-tikink colliding at the site of a repulsive impurity, with (cid:15) <
0, we find that for initial velocities greater than v cr the kink and antikink always leave the impurity site andpropagate to infinity after a single collision without ex-citing any impurity mode. For low initial velocities on Figure 3. Moving pair of oscillons observed for (cid:15) = − . v in = 0 .
3. The arrows in panel (a) represent the direction ofmotion of the oscillons, and the dashed line is a fit of the profile to the Ansatz of Eq. (2). Panels (b) and (c) indicate how theprofile evolves over time. the other hand, with v in < v cr , we identify three dis-tinct types of behaviour. First, for special values of theinitial velocity, we encounter resonances, in which thekink and antikink escape to infinity after a finite numberof collisions. This is in stark contrast to kink-impurityinteractions [23, 33], in which resonances are observedonly for attractive impurities. Here, we find them atvarious repulsive impurity strengths and initial velocity,such as (cid:15) = − . v in = 0 . v in = 0 . (cid:15) = − . v in = 0 . v in = 0 . v in = 0 . x = 0, as shown in Fig. 3, or remain close to the im-purity site in an oscillon-oscillon bound state (see Figs. 4and 1c). An oscillon ϕ O ( x ) is a quasi-long-lived boundstate of the ϕ model that periodically oscillates aroundone of the vacua [34]. At the extreme points in its oscil-lation, the oscillon has a Gaussian profile: ϕ O ( x ) = A O exp (cid:2) − ( x − B ) /C (cid:3) . (2)Here, A O , B , and C are the amplitude, center, and widthof the oscillon respectively. This function gives an excel-lent fit to the numerically obtained field configurations ϕ ( t, x ) after kink-impurity-antikink collisions, as shownin Figs. 3 and 4. For (cid:15) = − . v in = 0 .
30, thebest fit is obtained with A O = 0 .
58 and C = 3 .
07, whilethe case (cid:15) = − . v in = 0 .
43 results in A O = 0 . C = 3 .
02. Both values for the width of the oscil-lon fall within the range 2 . ≤ C ≤ .
54 that has beenshown to yield the longest-lived oscillons in (3+1) dimen-sions [34]. While the frequency of oscillations is close to ω O (cid:39) .
26 for both cases considered, the final velocitiesof the separating oscillons differ significantly ( v f = 0 . v f = 0 .
16, respectively). The impurity strength andinitial velocity thus mostly influence the amplitudes andfinal velocities of the oscillons, while having only a mi-nor influence on their widths or frequencies. As shownin Fig. 4, a similarly excellent fit to Eq. (2) can be foundfor the bound state of two oscillons.
The critical velocity.
Figure 5 shows the critical veloc-ity as a function of the impurity strength. For repulsiveimpurities ( (cid:15) <
0) colliding kink-antikink pairs with suffi-ciently low initial velocities form oscillons at the impuritysite. Increasing the impurity strength | (cid:15) | results in kink-antikink pairs with ever higher initial velocities formingoscillons. For large positive impurity strength, (cid:15) (cid:38) . (cid:15) , however, quasi-long-lived breathers are formed, andthe critical velocity goes up with decreasing impuritystrength.The fact that V cr is a smooth function of (cid:15) across thepoint (cid:15) = 0 and has a minimum at a non-zero positivevalue of the impurity strength is suggestive of the ideathat the field configuration of a bound oscillon pair maybe smoothly connected to that of a breather. The re-gion of low but positive (cid:15) then contains quasi-long-livedbreathers which obtain part of their stability relative tothe bound kink-antikink pairs at high values of the im-purity strength from the presence of some oscillon char-acter. This interpretation is consistent with the qualita-tive observation that oscillon bound states can resonantlydecay into propagating kink-antikink pairs within reso-nance windows for negative (cid:15) , as well as with the quanti-tative observation that the oscillon frequency ω O (cid:39) . ω = (cid:112) / (cid:39) . ϕ model without animpurity [20]. There, kink-antikink collisions resulted inpropagating oscillons rather than bound oscillon-oscillonstates for specific values of the initial velocity, whichwas explained by the resonant energy exchange betweentranslational and vibrational modes of the deformed kink Figure 4. Oscillon-oscillon bound state observed for (cid:15) = − . v in = 0 .
1. The dashed line in panel (a) is a fit of the profileto the Ansatz of Eq. (2). Panels (b) and (c) indicate how the profile evolves over time. and antikink. In the current kink-impurity-antikink evo-lution the oscillons appear naturally without deforma-tions, but the matching frequencies between kink andoscillon, and the minimum of v cr at non-zero (cid:15) suggesta similar exchange between oscillon formation and kinkexcitation. Conclusions.
Studying the collision of a kink and an-tikink in (1 + 1)-dimensional classical ϕ theory in thepresence of a single point-like impurity allows us to makeseveral observations about the phenomenology encoun-tered in the ϕ model.First of all, we showed that an internal mode of theimpurity could be excited at high initial velocities for anattractive impurity. The observed values for the impuritymode frequency match theoretical and numerical resultsreported earlier in the study of a single kink interactingwith a localized impurity [23, 33], which shows the excita-tion to be a property of the impurity that is independentof its excitation conditions.For small initial velocities of the kink-antikink pair anda repulsive impurity, we observe the emergence of a low-amplitude, quasi-long-lived, oscillating solution, whichwe identify to be an oscillon. In contrast to the breather,whose exact structure is still unknown, the oscillon hasa Gaussian form at the extreme moments of its oscilla- ● ● ● ● ● ● ● ● ● ●●●●●●●●● - - - - ϵ v cr Figure 5. The dependence of the critical velocity v cr on theimpurity strength (cid:15) . tions. Originally, oscillons were introduced in (3 + 1)-dimensional ϕ theories [34]. They were then introducedinto simulations in lower dimensions [35], as well as inother models, such as the signum-Gordon model [36],and models of cosmic inflation [37]. In all these cases,the oscillons are introduced as an Ansatz for the initialfield configuration [18, 19], rather than emerging fromphysical processes. Such a natural way for oscillons to ap-pear was recently discovered in the sinh-deformed (1+1)-dimensional ϕ model [20], after the application of modeldeformations [21, 22]. Here, we show that oscillons alsoemerge naturally from collisions of kinks and antikinks inthe classical (1 + 1)-dimensional ϕ model without anydeformations. Finally, we observe that the critical veloc-ity as a function of impurity strength has a minimum atnon-zero positive values of the impurity strength. This isindicative of a smooth transition between bound states ofkink-antikink pairs and oscillon-oscillon pairs, with someoscillon character already being present in the breatherformed without an impurity present. Such a crossoveris consistent with the match between oscillon frequenciesand internal kink excitations, which suggests a resonancemay arise between the two. Acknowledgments.
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