Kinks and realistic impurity models in φ 4 -theory
Mariya A. Lizunova, Jasper Kager, Stan de Lange, Jasper van Wezel
KKinks and realistic impurity models in ϕ -theory Mariya A. Lizunova, a,b
Jasper Kager, b Stan de Lange, b Jasper van Wezel b a Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, TheNetherlands b Institute for Theoretical Physics Amsterdam, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands
E-mail: [email protected] , [email protected] Abstract:
The ϕ -theory is ubiquitous as a low-energy effective description of processesin all fields of physics ranging from cosmology and particle physics to biophysics and con-densed matter theory. The topological defects, or kinks, in this theory describe stable,particle-like excitations. In practice, these excitations will necessarily encounter impuritiesor imperfections in the background potential as they propagate. Here, we describe the inter-action between kinks and various types of realistic impurity models. We find that realisticimpurities behave qualitatively like the well-studied, idealized delta function impurities,but that significant quantitative differences appear in both the characteristics of localizedimpurity modes, and in the collision dynamics. We also identify a particular regime ofkink-impurity interactions, in which kinks loose all of their kinetic energy upon collidingwith an impurity. Keywords:
Field Theories in Lower Dimensions, Solitons Monopoles and Instantons,Effective field theories a r X i v : . [ n li n . PS ] J u l ontents A.1 Integration 9A.2 Frequency and amplitude 9
From its introduction in the context of the Ginzburg-Landau theory for second order phasetransitions [1], ϕ -theory has found applications in the low-energy description of manyphysical processes, including domain wall motion [2], molecular dynamics [3], and chemicalequilibrium [4]. Likewise, the solitary wave solutions of ϕ -theory, known as kinks, featurein the phenomenological theory of domain walls in molecules, solids, and cosmology [5–8],as well as in toy models for nuclear physics [9–11]. Recently, using a gauge field theoryapproach, it was found that the kinks of ϕ -theory may even be useful in the description ofbiologically relevant molecules, where they can describe for example the native structure ofbending angles in complex proteins like Myoglobin [12–14]. Collisions among kinks in ϕ -theory are particularly interesting, because unlike for example solitons in the integrable sine-Gordon model [15], kinks and antikinks cannot just pass through each other. Instead, theyinteract and undergo dynamic processes including scattering, formation of bound states,and even resonances [16–20].Besides kink-antikink collisions, interactions between kinks and impurities are a cen-tral ingredient in the modelling of any realistic system. In chemical and condensed mattersettings, actual impurities in the atomic lattice are an unavoidable fact of life, while inparticle physics and cosmology variations in the potential or background metric act as im-purities [21–27]. Idealized impurities, with a Dirac delta spatial profile, have been shown to– 1 –e capable of scattering or capturing kinks, as well as harbouring a localized impurity modeof their own [21]. Here, we extend these results by including interactions between kinksand more realistic impurity models, based on Gaussian or Lorentzian spatial profiles. Wefind qualitative agreement with the physics of idealized defects, but significant quantitativeeffects of the impurity profile on the shape of the localized impurity mode, the scatteringdynamics, and the possible long-time fate of kinks interacting with strong impurities. We consider a classical, real scalar field ϕ = ϕ ( t, x ) in (1 + 1) -dimensional space-time,described by the Lagrangian density [28, 29]: L = 12 (cid:18) ∂ϕ∂t (cid:19) − (cid:18) ∂ϕ∂x (cid:19) −
14 (1 − ϕ ) . (2.1)The final term represents the self-interaction potential of the field ϕ . It has two minima, ϕ (1) v and ϕ (2) v , defining the vacuum manifold of the theory. From the Euler-Lagrange equation,the field ϕ is found to obey the equation of motion: ϕ tt − ϕ xx + ϕ − ϕ = 0 . (2.2)The trivial solutions of this equation are ϕ = 0 , corresponding to an unstable localmaximum of the potential, and ϕ = ϕ (1 , v = ± , which are the stable vacuum solutions.Non-trivial solutions may be found by imposing that the field approaches distinct vacua atopposing spatial infinities. The minimal-energy static solution for any given set of spatialboundary conditions is called a BPS saturated configuration [28, 30, 31]. For the ϕ -theorydefined by eq. (2.1), a non-trivial and non-dissipative static BPS-solution connecting ϕ (1) v and ϕ (2) v is given by: ϕ ( x ) = ± tanh (cid:18) x − a √ (cid:19) . (2.3)This solution is known as a kink ϕ K (or antikink ϕ K ) for the plus (minus) sign. Each hasa center at x = a and characteristic length l K (cid:39) √ . Owing to the Lorentz invariance ofeq. (2.2) the static (anti)kink solution transforms into a dynamical solution under a Lorentzboost: ϕ ( t, x ) = ± tanh (cid:32) x − a + vt (cid:112) − v ) (cid:33) , (2.4)where < v < is a velocity of the moving (anti)kink in units of the speed of light. We introduce a single inhomogeneity γ ( x ) into the potential of eq. (2.1) by writing [21, 22,26]:
14 (1 − ϕ ) −→
14 (1 − ϕ ) (1 − (cid:15)γ ( x − x )) . (3.1)– 2 – igure 1 . The value of the field ϕ ( t, x ) at x = 0 as a function of time, for a repulsive impuritywith (cid:15) = − . and σ (cid:39) . (left of the dashed line) and for an attractive impurity with (cid:15) = +0 . and σ (cid:39) . (right of the dashed line). a) For low initial velocity, v in = 0 . , the kink is reflectedby the impurity. b) For high initial velocity, v in = 0 . , the kink traverses the impurity, at the costof loosing some kinetic energy. c) For v in = 0 . , the kink is captured by the attractive impurity. d) For specific intermediate values such as v in = 0 . , a resonance causes the kink to oscillatearound the impurity before being released again. A few of the resonance windows are sketchedalong the bar of v in values at the bottom. e) For high initial velocity, v in = 0 . , the kink traversesthe impurity, leaving behind an excited impurity mode, and loosing some kinetic energy. Here, | (cid:15) | < and x represent the strength and the center of a weak impurity. For (cid:15) = 0 the clean model is recovered, while (cid:15) < and (cid:15) > correspond to a repulsive barrier andan attractive potential well respectively. Including the impurity, the equation of motionbecomes: ϕ tt − ϕ xx + ( ϕ − ϕ )(1 − (cid:15)γ ( x − x )) = 0 . (3.2)We numerically solve eq. (3.2) using a finite differences method (see appendix for de-tails). The maximal detected fluctuations in energy during time evolution of an initial fieldconfiguration were less than . , indicating the stability and accuracy of the numericalroutine.We consider the initial condition of a single kink ϕ = ϕ K , centered at t = 0 at theposition a = 6 , and moving with some initial velocity v = v in towards an impurity locatedat x = 0 . We explore the influence of the impurity profile γ ( x ) on the kink-impurityscattering by considering three distinct types of impurities. The first is the idealized Diracdelta function originally proposed in [21]: γ ( x − x ) → δ ( x − x ) . (3.3)We then generalise to a more realistic Gaussian form for the impurity: γ ( x − x ) → σ √ π exp (cid:34) − (cid:18) x − x σ √ (cid:19) (cid:35) . (3.4)Here, choosing < σ < corresponds to the impurity width σ being less than the width l K of the kink. We compare the results of the Gaussian impurity model with a final realisticAnsatz, given by a Lorentzian profile: γ ( x − x ) → x − x ) + α . (3.5)– 3 – Kink-impurity interactions
To establish a connection to the known result for the idealised Dirac delta impurity [21],we first consider a very narrow form of the Gaussian profile. We choose the height of theGaussian at its centre to coincide with the numerical value for the delta function heightused in ref. [21]. This results in a width σ (cid:39) . that is smaller than the lattice spacingin our numerical routine, in accordance with the Dirac delta limit.For a repulsive impurity with (cid:15) = − . , we reproduce the two types of processes knownto occur as the value of v in is varied [21]. For low initial velocities, such as v in = 0 . (shownin figure 1a), the kink is reflected by the impurity. For velocities above some critical value v cr , the kink is transmitted, but its kinetic energy (velocity) is reduced in the process andemitted in the form of low-amplitude ripples. This is shown in figure 1b for the initialvelocity v in = 0 . .In the case of an attractive impurity, with (cid:15) = +0 . , three different types of known dy-namics are reproduced [21]. At low initial velocities, such as v in = 0 . (shown in figure 1c),the kink is captured by the impurity. That is, it ends up in a final state in which the kinkcentre oscillates around the central position of the impurity. For high initial velocities onthe other hand, like v in = 0 . (shown in figure 1e), the kink is transmitted through theimpurity entirely, loosing some kinetic energy in the process, and leaving behind a local-ized excitation centered at the impurity location. The localized impurity mode consistsof small amplitude periodic oscillations of the field at x = x around its vacuum value ϕ (1) v . The value of the initial velocity above which kinks are always transmitted throughthe impurity is called the critical velocity v cr . For certain particular values of the initialvelocity v in < v cr , such as v in = 0 . (shown in figure 1d), the kink is neither capturednor transmitted. Instead, it oscillates around the impurity center a finite number of timesbefore leaving the impurity in the direction or opposite one it originally came from. Thistype of behavior is known as resonance, and can be understood in terms of the exchange ofkinetic energy with internal excitations of the kink [21, 32].For both attractive and repulsive impurity strengths, our results are in perfect agree-ment with those reported in ref. [21], confirming that the idealized Dirac delta impuritygives a faithful approximation of the more realistic Gaussian profile in the limit of very lowwidth. Notice that the precise value of v cr we find is slightly higher than that reportedbefore. This is a direct consequence of our numerical routine being stable to higher finaltimes than previously achievable. We observe that at these later times, kinks that seeminglyescaped the impurity, still return and are captured. Since we have no way of establishingthe dynamics at even later times, we refrain from reporting any more precise value for v cr , and instead focus below on its qualitative behavior as the strength and width of theimpurity profile are varied.Finally, we consider the limit of a very strong attractive impurity, with (cid:15) > . Theresulting dynamics is presented in figure 2a-c. As the impurity strength increases, thecaptured kink oscillates around the impurity with ever lower amplitude, eventually gettingstuck at the impurity position entirely. This regime, in which the oscillation amplitude of– 4 – igure 2 . a-c) The value of the field ϕ ( t, x ) at x = 0 as a function of time, for an attractiveimpurity with width σ (cid:39) . , and fixed initial velocity of the kink v in = 0 . . a) For impuritystrength (cid:15) = 0 . ; b) with (cid:15) = 1 . ; and c) for a very strongly attractive impurity with (cid:15) = 1 . . Inall cases the kink is captured by the impurity and oscillates around the impurity centre with anamplitude lower than the kink width. d) The dominant frequency ω of oscillations of the capturedkink as a function of impurity strength. a captured kink is smaller than its width, so that the field value at the impurity locationnever returns to either of its vacuum values ϕ (1 , v , may be called a super-capture. figure 2dshows the dependence of the dominant oscillation frequency on impurity strength. For an ideal Dirac delta impurity, the value of v cr is determined entirely by the impuritystrength (cid:15) . For vanishing strength, the critical velocity tends to zero, while strong impuritiescost a lot of kinetic energy to traverse, yielding a large v cr . In the case of a Gaussian impurityprofile, the width σ of the impurity as well as its strength (cid:15) may be expected to play a rolein determining the critical velocity, i.e. v cr = v cr ( (cid:15), σ ) .The qualitative effects of the strength and impurity may be understood from the resultsin table 1. The critical velocity increases in value upon either decreasing the width σ forfixed values of the strength | (cid:15) | , or upon increasing | (cid:15) | for fixed values of σ . These trendscan be understood by comparing eq. (3.2) and eq. (3.4), which show that the height of the (cid:15) σ v in fate of kink0.3 0.5 0.02 transmitted0.3 0.3 0.02 captured0.3 0.3 0.05 transmitted0.3 0.1 0.05 captured-0.6 0.5 0.47 transmitted-0.6 0.3 0.47 reflected-0.6 0.3 0.5 transmitted-0.6 0.1 0.5 reflected (cid:15) σ v in fate of kink0.4 0.3 0.1 transmitted0.6 0.3 0.1 captured0.6 0.3 0.28 transmitted0.8 0.3 0.28 captured-0.1 0.3 0.3 transmitted-0.4 0.3 0.3 reflected-0.4 0.3 0.5 transmitted-0.8 0.3 0.5 reflected Table 1 . Left)
The effect of varying impurity width on the kink-impurity interaction, for fixedvalue of the impurity strength. Larger values of σ are seen to yield lower values of v cr . Right)
The effect of varying impurity strengt, for fixed value of the impurity width. Larger values of | (cid:15) | are seen to yield higher values of v cr . – 5 – igure 3 . a) The spatial profile of the impurity mode, excited by a kink traversing the impurity site x = x = 0 and travelling to the right. b) The value of the field ϕ ( t, x ) at x = x = 0 as a functionof time, for an attractive impurity with width σ = 0 . and strength (cid:15) = 0 . . An impurity modeis excited by a kink with initial velocity v in = 0 . passing x = x at approximately t ≈ . Thehorizontal dashed lines in the inset are guides to the eye highlighting the quasi-long-lived characterof the impurity mode. c) Comparison of the numerically obtained values for the field ϕ ( t, x ) aftera kink traversed the impurity site (blue dots), with the analytical prediction of eq. (4.2) (red line).The impurity potential used is a narrow Gaussian with strength (cid:15) = 0 . and width σ (cid:39) . , whilethe incoming kink had initial velocity v in = 0 . . d) The same comparison using a wide Gaussianimpurity potential with strength (cid:15) = 0 . and width σ (cid:39) . . impurity potential at x is proportional to the ratio (cid:15)/σ .To further confirm that the interaction of kinks with a wide impurity differs onlyquantitatively from that with a narrow impurity, we confirm that resonances still occur forrealistic impurity widths. For the values (cid:15) = 0 . and σ (cid:39) . there is a resonance around v in = 0 . . Upon increasing the impurity width to σ = 0 . , we instead find a resonanceat v in = 0 . . At (cid:15) = 0 . and σ = 0 . we observe a resonance at the same value of v in . Thequalitative behavior of having resonance windows below v cr for attractive impurities thussurvives also for more realistic impurity profiles of non-zero width. When a kink traverses an attractive impurity it leaves behind an oscillating mode, localizedat the site of the impurity (see figure 3a). This mode oscillates at x = x with near-constantamplitude and frequency, as shown in figure 3b. It is a quasi-long-lived mode, analogous tothe wobbling kink [33], and bion solutions found in kink-antikink collisions [15].For the case of the idealised Dirac delta impurity, the approximate shape of the impuritymode profile can be found analytically [21]. Since the mode is localized at the impuritysite and excited by the passing of a kink, we can write it as a small deviation δϕ fromthe vacuum solution. Substituting ϕ ( t, x ) = ϕ (1) v + δϕ ( t, x ) in eq. (3.2) and eq. (3.3) andkeeping only terms up to linear order in δϕ yields the equation of motion for the impuritymode: δϕ tt − δϕ xx + 2 (1 − (cid:15)δ ( x − x )) δϕ = 0 . (4.1)Using the Ansatz that the impurity mode oscillates with a fixed spatial profile and constantfrequency, δϕ ( t, x ) ∝ (cid:60) χ ( x ) exp ( − i Ω t ) , it is found to obey [21]: δϕ ( t, x ) ∝ (cid:60) exp ( − (cid:15) | x | ) exp ( − i Ω t ) , Ω = 2 − (cid:15) . (4.2)– 6 – igure 4 . a) The impurity mode frequency ˜Ω and b) its amplitude A , as a function of theinitial kink velocity v in , for fixed values of impurity strength and width. c) The frequency and d) amplitude as a function of impurity strength (cid:15) , keeping the initial velocity and impurity widthconstant. e) Frequency and f ) amplitude as a function of the the impurity width σ , for fixedimpurity strength and initial velocity. The numerically obtained field configuration in figure 3c indicates that for a very narrowimpurity potential, (cid:15) = 0 . and σ (cid:39) . , an incoming kink with v in = 0 . excites animpurity mode whose shape closely matches the prediction of eq. (4.2). The numericalresults in figure 3d on the the other hand, show that for wider impurity potential, (cid:15) = 0 . and σ (cid:39) . , the analytic solution no longer gives an accurate prediction for the impuritymode profile. The spatially smooth impurity potential in this case does not allow for anydiscontinuities in the field configuration or its derivatives, forcing the impurity mode profileto remain smooth around the impurity location.Besides the spatial profile of the impurity mode, its amplitude A and frequency ˜Ω mayalso vary with impurity width, and deviate from the analytic prediction of eq. (4.2). Fornarrow impurity potential ( σ (cid:39) . ) we find (cid:104) ˜Ω (cid:105) (cid:39) . for (cid:15) = 0 . and (cid:104) ˜Ω (cid:105) (cid:39) . for (cid:15) = 0 . , both in excellent agreement with the analytic prediction of eq. (4.2). Forwider impurities, the approximations underlying the analytic solution break down, and wenumerically find the behavior displayed in figure 4.The frequencies in figure 4b do not vary much with the initial velocities. Taking theaverage over all points shown yields a value of (cid:104) ˜Ω (cid:105) (cid:39) . for (cid:15) = σ = 0 . , and (cid:104) ˜Ω (cid:105) (cid:39) . for (cid:15) = σ = 0 . . The decrease of the frequency with increasing impurity strength indicatedin figure 4d is in line with the analytic prediction of eq. (4.2) for ideal Dirac delta impurities,but is now seen to also depends on the width σ of the more realistic Gaussian profile. Thefrequency increases with increasing width, as shown in figure 4f, in agreement with the factthat increased width in eq. (3.4) implies a lower value of the impurity potential at x = x .– 7 – .4 Lorentzian profile To see whether the qualitative effects of a wide impurity profile, rather than an idealisedDirac delta form, are generic to more realistic impurity models, we next compare the resultsof the Gaussian case to interactions between a kink and a Lorentzian impurity. We repeatthe analysis of section 4.1 using the impurity profile of eq. (3.5), with (cid:15) = ± . . To ensurethat the peak height at x = 0 matches that of the Gaussian considered before, we take α = 0 . .For the repulsive Lorentzian impurity, we find that the kink is reflected for all initialvelocities up to v in = 0 . . However, at v in = 0 . the kink traverses the impurity, showingthat a critical velocity exists in the range . < v in < . . Reversing the sign of (cid:15) andconsidering an attractive Lorentzian impurity, we observe that for initial velocities below v in = 0 . , the kink is always captured by the impurity.Although the capturing and reflecting of kinks is qualitatively similar to the behaviorin the presence of a Gaussian impurity, the values of the critical velocities are much higherfor a Lorentzian profile than for a Gaussian with the same maximum value. We thereforealso compare Lorentzian and Gaussian profiles with the same integrated strength, by usingthe scaled function: α/π ( x − x ) + α . (4.3)We again take (cid:15) = ± . and α = 0 . .In this case, the kink is reflected by a repulsive impurity for initial velocities up to v in = 0 . , and traverses the impurity site for velocities v in = 0 . and above. Likewise,the attractive impurity captures the kink at v in = 0 . , while it is transmitted for initialvelocities above v in = 0 . .These results indicate a qualitative agreement between the dynamics of kink-impurityinteractions for Gaussian and Lorentzian impurity profiles. We leave more detailed analysesfor future study, looking for example for resonances around the Lorentzian impurity, orinvestigating the precise influence of the parameter α on the critical velocity. In conclusion, we have shown that the behavior of kink-impurity interactions in (1 +1) − dimensional ϕ -theory is qualitatively the same for idealised Dirac delta impurity pro-files and more realistic Gaussian or Lorentzian shapes. In all cases, repulsive impuritiescan either reflect or transmit an incoming kink, depending on its initial velocity. Attractiveimpurities on the other hand either capture the kink, release after a few oscillations in aresonant process, or transmit it immediately. In the latter case, an impurity mode is excitedat the impurity location by the passing kink. We also observe that a kink impinging onparticularly strong impurities may lead to an extreme form of capture, in which the kinklooses all of its kinetic energy and remains localized at the impurity site.The Gaussian profile reduces to precisely a Dirac delta impurity in the limit of vanishingwidth, and all observed properties of the kink-impurity interaction agree with the knownresults for ideal impurities in that limit. For wider impurity profiles, the results differ– 8 –uantitatively from the ideal case. The amplitude and frequency of the impurity modeobtain a dependence on the impurity width, the values of critical velocities are affected,and the kink velocity at which resonances appear is altered. Similar quantitative effects areobserved for interactions between a kink and a Lorentzian impurity. Acknowledgments
The authors are very grateful for discussions with Dario Bercioux, Alexander Kudryavtsev,and Cristiane Morais Smith. This work was done within the Delta Institute for Theoret-ical Physics (DITP) consortium, a program of the Netherlands Organization for ScientificResearch (NWO) that is funded by the Dutch Ministry of Education, Culture and Science(OCW).
A Numerical methods
A.1 Integration
To integrate the equation of motion in eq. (3.2), we use finite differences on a discretizedlattice in both space and time: ϕ k +1 j = 2 ϕ kj − ϕ k − j + τ h ( ϕ kj +1 − ϕ kj + ϕ kj − ) − τ (cid:0) ϕ − ϕ (cid:1) kj (1 − (cid:15)γ ( x j − x )) . (A.1)Here, ϕ kj equals ϕ ( x, t ) at the discrete positions x j = − L + jh and times t k = kτ , with τ and h the sizes of the discrete space and time steps. We demand τ < h to ensure stabilityof the numerical integration.All calculations reported in this article use < t < t f with t f = 350 , and − L < x < L with L = 730 . To avoid any effect of the spatial boundaries on the reported results, wepresent only results within the interval − L (cid:48) < x < L (cid:48) , where L (cid:48) = L − t f h/τ .As a measure of the accuracy of the numerical integration, we check whether energy isconserved in time, taking into account the flow of energy through the borders x = ± L (cid:48) ateach time step t k : E [ ϕ ( t = t k )] − t k (cid:90) ∂ϕ∂t ∂ϕ∂x (cid:12)(cid:12)(cid:12)(cid:12) L (cid:48) − L (cid:48) dt = E [ ϕ ( t = 0)] . (A.2)In this expression, the instantaneous energy is found from the spatial integral of the La-grangian, E [ ϕ ] = (cid:82) L [ ϕ ] dx . Appropriate values for the steps τ and h are determinedempirically by demanding that energy is conserved, and all results are independent of thechosen step values. Here, we use τ = 0 . and h = 0 . , and the maximal detected deviationin energy is less . . A.2 Frequency and amplitude
To extract the frequency and amplitude of the impurity mode oscillations from the nu-merically obtained field profile, we can either use a discrete Fourier transform of ϕ ( x , t ) ,– 9 –r directly average the distances between its observed minima and maxima. In either ap-proach, we only consider the field values for t > , to ensure that the kink has entirelypassed by the impurity centre. The accuracy of both methods is limited by the discretesampling of the continuous field ϕ ( x , t ) , and the values obtained for A and ˜Ω with the twomethods do not differ significantly. References [1] V. N. Ginzburg and L. D. Landau,
On the theory of superconductivity , Zh. Exsp. Teor. Fiz. (1950) 1064.[2] Y. Wada and J. R. Schrieffer, Brownian motion of a domain wall and the diffusion constants , Phys. Rev. B (1978) 3897–3912.[3] T. Schneider and E. Stoll, Molecular-dynamics study of a three-dimensional one-componentmodel for distortive phase transitions , Phys. Rev. B (1978) 1302–1322.[4] S. Aubry, A unified approach to the interpretation of displacive and order–disorder systems.ii. displacive systems , J. Chem. Phys. (1976) 3392.[5] A. R. Bishop and T. Schneider, Solitons and condensed matter physics , in
Proceedings of theSymposium on Nonlinear (Soliton) Structure and Dynamics in Condensed Matter , (Oxford),1978.[6] A. R. Bishop,
Defect states in polyacetylene and polydiacetylene , Solid State Commun. (1980) 955.[7] M. J. Rice and E. J. Mele, Phenomenological theory of soliton formation in lightly-dopedpolyacetylene , Solid State Commun. (1980) 487.[8] A. Friedland, H. Murayama and M. Perelstein, Domain walls as dark energy , Phys. Rev. D (2003) 043519 [ astro-ph/0205520 ].[9] D. K. Campbell and Y.-T. Liao, Semiclassical analysis of bound states in thetwo-dimensional σ model , Phys. Rev. D (1976) 2093.[10] T. D. Lee and G. C. Wick, Vacuum stability and vacuum excitation in a spin-0 field theory , Phys. Rev. D (1974) 2291.[11] J. Boguta, Abnormal nuclei , Phys. Lett. B (1983) 19.[12] S. Hu, M. Lundgren and A. J. Niemi,
Discrete frenet frame, inflection point solitons, andcurve visualization with applications to folded proteins , Phys. Rev. E (2011) 061908[ ].[13] A. Molochkov, A. Begun and A. Niemi, Gauge symmetries and structure of proteins , EPJWeb of Conferences (2017) 04004 [ ].[14] S. Hu, A. Krokhotin, A. J. Niemi and X. Peng,
Towards quantitative classification of foldedproteins in terms of elementary functions , Phys. Rev. E (2011) 041907 [ ].[15] T. I. Belova and A. E. Kudryavtsev, Solitons and their interactions in classical field theory , Usp. Fiz. Nauk (1997) 377 [Sov. Phys. Usp. , 359 (1997)].[16] M. J. Ablowitz, M. D. Kruskal and J. F. Ladik, Solitary wave collisions , SIAM J. Appl.Math. (1979) 428–437. – 10 –
17] P. Anninos, S. Oliveira and R. A. Matzner,
Fractal structure in the scalar λ ( ϕ − theory , Phys. Rev. D (1991) 1147–1160.[18] R. Goodman and R. Haberman, Kink-antikink collisions in the φ equation: The n -bounceresonance and the separatrix map , SIAM J. Appl. Math. Dyn. Syst. (2005) 1195.[19] V. A. Gani, A. E. Kudryavtsev and M. A. Lizunova, Kink interactions in the (1 + 1) -dimensional ϕ model , Phys. Rev. D (2014) 125009 [ ].[20] V. A. Gani, V. Lensky and M. A. Lizunova, Kink excitation spectra in the (1+1)-dimensional ϕ model , JHEP (2015) 147 [ ].[21] Z. Fei, Y. S. Kivshar and L. Vazquez, Resonant kink-impurity interactions in the φ model , Phys. Rev. A (1992) 5214.[22] Z. Fei, V. V. Konotop, M. Peyrard and L. Vazquez, Kink dynamics in the periodicallymodulated φ model , Phys. Rev. E (1993) 548.[23] K. Javidan, Interaction of topological solitons with defects: Using a nontrivial metric , J.Phys. A: Math. Gen. (2006) 10565 [ hep-th/0604062 ].[24] E. Hakimi and K. Javidan, Soliton-potential interaction in the ϕ model , Phys. Rev. E (2009) 016606.[25] A. Ghahraman and K. Javidan, Analytical formulation for ϕ field potential dynamics , Brazilian J. Phys. (2011) 171 [ ].[26] K. Javidan, Analytical formulation for soliton-potential dynamics , Phys. Rev. E (2008)046607 [ ].[27] A. Askari, D. Saadatmand and K. Javidan, Collective coordinate system in (2+1)dimensions: cp lumps-potential interaction , Waves in Random and Complex Media (2018) 368.[28] D. Bazeia, Defect structures in field theory , in
Proceedings of XIII J.A. Swieca SummerSchool on Particles and Fields , SP, 2005, hep-th/0507188 .[29] R. Rajaraman,
Some non-perturbative semi-classical methods in quantum field theory , Phys.Rep. C (1975) 227.[30] E. B. Bogomolny, Stability of classical solutions , Yad. Fiz. (1976) 861 [Sov. J. Nucl. Phys. , 449 (1976)].[31] M. K. Prasad and C. M. Sommerfield, Exact classical solution for the ’t hooft monopole andthe julia-zee dyon , Phys. Rev. Lett. (1975) 760.[32] D. K. Campbell, J. S. Schonfeld and C. A. Wingate, Resonance structure in kink-antikinkinteractions in ϕ theory , Phys. D (1983) 1.[33] A. E. Kudryavtsev and M. A. Lizunova, Search for long-living topological solutions of thenonlinear ϕ field theory , Phys. Rev. D (2017) 056009 [ ].].