Chaotic solitons in driven sine-Gordon model
IINR-TH-2020-018
Chaotic solitons
D. G. Levkov ∗
1, 2 , V. E. Maslov †
1, 2, 3 , and E. Y. Nugaev ‡ Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia Institute for Theoretical and Mathematical Physics, MSU, Moscow 119991, Russia Department of Particle Physics and Cosmology, Faculty of Physics, MSU, Moscow 119991,Russia
April 29, 2020
Abstract
Profiles of static solitons in one-dimensional scalar field theory satisfy the sameequations as trajectories of a fictitious particle in multidimensional mechanics. Weargue that the structure and properties of the solitons are essentially different if therespective mechanical motions are chaotic. This happens in multifield models andmodels with spatially dependent potential. We illustrate our findings using one-fieldsine-Gordon model in external Dirac comb potential. First, we show that the numberof different “chaotic” solitons grows exponentially with their length, and the growthrate is related to the topological entropy of the mechanical system. Second, the fieldvalues of stable solitons form a fractal; we compute its box-counting dimension. Third,we demonstrate that the distribution of field values in the fractal is related to themetric entropy of the analogous mechanical system.
There exists an amusing mathematical analogy between static solitons in one-dimensionalfield theory and point-particle trajectories in multidimensional mechanics. Indeed, the soli-tonic profiles typically satisfy second-order equations [1, 2, 3] ∂ ϕ i ∂x = ∂V∂ϕ i , (1) ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] A p r here ϕ i ( x ) are the fields of the model and V ( ϕ, x ) is their scalar potential. These equationscoincide with the Newton’s law for the evolution in “time” x of a fictitious particle withcoordinates ϕ i ( x ) in an external potential V mech ≡ − V . Studying the mechanical trajectories,one can investigate the solitons. At x → ±∞ the soliton fields approach the vacua — minimaof the potential V . Thus, the respective mechanical trajectories ϕ i ( x ) lie on the separatrix:they start on the maximum of V mech ( ϕ ) at x → −∞ and climb onto the same or anothermaximum in the infinite “future”.In this paper we argue on the basis of the above analogy that one-dimensional staticsolitons have essentially different properties in models with multiple fields or models withposition-dependent potential V ( ϕ, x ) as compared to the simplest case of a single-field scalartheory. Indeed, mechanical motions are typically chaotic in models with several degrees offreedom. Smooth separatrix in this case is destroyed [4], and the maxima of the potential V mech are connected by an infinite number of different trajectories. Since each trajectoryrepresents the soliton, there exists an infinite number of the latter in the multifield models.Below we investigate such “chaotic” solitons and their distribution in the configuration space.Notably, we find that many of these objects are linearly stable from the viewpoint of fieldtheory: they cannot be destroyed by adding a small perturbation and time-evolving theresulting configuration. The subset of stable solitons is of our primary interest.To be specific, we consider sine-Gordon model [5] with coordinate-dependent potential, ϕ (cid:48)(cid:48) = ∂V∂ϕ , V ( ϕ, x ) = U ( x ) (1 − cos ϕ ) , (2)where the prime represents x –derivative and U ( x ) (cid:62) . This model has vacua ϕ n = 2 πn , where n is integer. If U is a constant, the analogous mechanical motion is one-dimensional, conservative, and therefore integrable. In this case there exist only two types ofstatic solitons: “kink” φ K ( x ) and “antikink” φ A ( x ) interpolating between the neighbouringmaxima of V mech ≡ − V , see Fig. 1a. The profiles of these objects form smooth separatrix(Fig. 1b) in the mechanical “phase space” ( ϕ, ϕ (cid:48) ). Below we will consider nonintegrable casewith spatially dependent U ( x ).It is worth noting that the sine-Gordon equation appears in several diverse setups. Itdescribes relative phase difference between two coupled one-dimensional superfluids at lowenergies [12, 13, 14], rotation angle in classical ferromagnetic spin chain interacting withexternal magnetic field [15, 16, 17, 18], or phase of superconductors in long Josephson junc- Unlike the fictitious particle trajectories which are unstable in the chaotic regime. We study only static solitons in this model, not their dynamics. The latter is also related to chaos,see [6, 7, 8, 9, 10, 11]. πn π ( n + 1) xϕ ( x ) φ K ( x ) φ A ( x ) (a) ϕϕ πn ϕϕ πn (b)(b) 2 π ( n + 1)Figure 1: (a) Profiles of “kink” and “antikink” at constant U . (b) Respective trajectories inthe mechanical “phase space” ( ϕ, ϕ (cid:48) ). The “time” x grows along arrows.tion [19]. In all these cases inhomogeneous potential can be achieved by spatial variation ofparameters: external electric or magnetic fields, or impurities between the superconductors[20].In numerical calculations we use the simplest dependence of the potential (2), U ( x ) = 1 + ε ∞ (cid:88) m = −∞ δ ( x − mD ) , (3)where D = 12 is the period and the parameter ε controls chaoticity of the underlying me-chanical model. Although Eq. (3) may seem bizarre from the viewpoint of some applications,we expect that our results remain qualitatively valid for any periodic modulation. For thepotential (3) the analogous mechanical motion is nearly integrable at ε (cid:46) − . In thisregime Kolmogorov-Arnold-Moser (KAM) theory of quasiperiodic motions [21, 22, 23] is ap-plicable, and the “solitonic” trajectories remain close to the separatrix in Fig. 1b. In fact,they toss erratically from vacuum to vacuum along this separatrix. The respective solitonscan be obtained by matching together the kink and antikink profiles, see Fig. 2a. The partof the “phase space” spanned by these trajectories, however, grows with ε and fills a consid-erable region at ε (cid:38) .
1. The solitons in the latter case appear in a wide variety of forms,see Fig. 2b.In the main text we prove that the number of stable solitons N sol fitting in a finite spatialinterval 0 (cid:54) x (cid:54) L grows exponentially with the interval size, N sol ∝ e h S ( ε ) L/D as L → + ∞ , (4)where the growth rate h S ( ε ) monotonically increases with ε . The law (4) is demonstratednumerically in Figs. 3a, b. Steplike features of h S ( ε ) (arrows in Fig. 3b) result from newtypes of solitons emerging at larger ε . 32 π π ϕ ( x ) x/D (a) − π π ϕ ( x ) x/D (b) Figure 2: Examples of static solitons (a) in the KAM regime at ε = 3 × − ; (b) in thechaotic case at ε = 3. Vertical lines mark positions of δ -functions in Eq. (3).110100 0 2 4 6 N s o l L/D (a) . . . . − − − . h S ε (b) Figure 3: (a) The number of stable solitons N sol ( L ) in a finite spatial box as a function ofthe box size L . Numerical data (points) are fitted with Eq. (4) (line). (b) The logarithmicgrowth rate h S ( ε ) as a function of the chaoticity parameter ε .Growth of the soliton multiplicity with L can be easily explained in the KAM regimewhen the analogous mechanical motion proceeds along the smooth separatrix. In this casethe stable solitons are completely specified by the set { ϕ n } of intermediate vacua. Say, thesolitonic profile in Fig. 2a corresponds to the sequence { ϕ , ϕ , ϕ , ϕ , ϕ } . The number ofpossible sequences grows exponentially with their length L/D , and so does the number ofstable solitons .In the main text we demonstrate that the growth rate h S ( ε ) of stable solitons is bounded At ε (cid:28) h T ( ε ) of the analogous mechanical system [24, 25], h S ( ε ) (cid:54) h T ( ε ) . (5)The latter quantity characterizes complexity of the system i.e. diversity of its motions.It is well-known that distinct classes of trajectories are separated by fractal sets in thephase space of chaotic systems [26, 27, 28, 29, 30]. We show that similarly, the solitons forma fractal in the space of static field configurations ϕ ( x ). To visualize the fractal, we computethe field values ϕ (0), ϕ (cid:48) (0) of all stable solitons at a given spatial point x = +0 and plotthem with dots in Fig. 4a. For example, the point S represents the soliton in Fig. 5a.We find that the set in Fig. 4a is approximately self-similar. Indeed, it can be reproducedby magnifying a tiny region near one of its points, see Fig. 4b. To explain self-similarity, wechoose points 1—3 in Fig. 4a and related points 1 (cid:48) —3 (cid:48) in Fig. 4b, then plot their profiles inFigs. 5b and 5c. Notably, at positive (or negative) x the solutions 1 (cid:48) —3 (cid:48) go along S first,then depart from it at x ≈ ± D and follow the related profile 1, 2, or 3. Now, recall that thetrajectory S is unstable from the mechanical viewpoint. Thus, small variations of its initialdata ϕ (0) and ϕ (cid:48) (0) lead to variations of the new “initial data” at x = 8 D enhanced by afactor e λ S (8 D ) , where λ S ( x ) is related to the Lyapunov exponent of S . One concludes thata small vicinity of every point in Fig. 4a contains the entire set of “solitonic” Cauchy datasqueezed by the Lyapunov factor e − λ S .In Fig. 6 we plot the box–counting dimension [31, 32] d ( ε ) of the “stable solitons” fractalin Fig. 4a at different values of the chaoticity parameter ε . Apparently, d is not integer .Besides, it changes non-monotonically with ε due to two competing effects. First, at larger ε new solitons appear, increasing d . Second, Lyapunov exponents of already existing solitonsgrow with ε , making their field values closer in the ( ϕ, ϕ (cid:48) ) plane. This effect decreases d atlarge ε .In the main text we will demonstrate that at small ε the fractal dimension d ( ε ) is boundedfrom below by the stable solitons growth rate: d ( ε ) (cid:62) h S ( ε ) /D .An important characteristic of chaotic dynamics is the metric (Kolmogorov) entropy K .This quantity reflects divergence of the trajectories or, in other words, information growthrate during evolution [25]. Positive values of K indicate chaos. We suggest field-theoreticanalogue E of this quantity characterizing the distribution of stable soliton field values ϕ (0), ϕ (cid:48) (0) at a given point x = +0. In particular, E = 0 if all stable solitons have the same Recall that the soliton arriving to ϕ n at x → + ∞ lies on the boundary between the solutions with ϕ > ϕ n and ϕ < ϕ n at large x . On the other hand, we will show that the field values ϕ (0) , ϕ (cid:48) (0) of all solitons, both stable and unstable,form a dense set with fractal dimension 2. − − − − − S
12 3 ϕ ( ) ϕ (0) (a) − − − − − − [ ϕ ( ) − ϕ S ( ) ] × F [ ϕ (0) − ϕ S (0)] × F (b) Figure 4: (a) Field values ϕ (0), ϕ (cid:48) (0) of stable solitons at x = +0 in the model with ε = 3 × − . Only the region | ϕ | , | ϕ (cid:48) | < − is shown. (b) Field values in the vicinity ofthe point S in Fig. 4a magnified by a factor F = e λ S (8 D ) , where λ S (8 D ) ≈ . S .( ϕ, ϕ (cid:48) ). Uniform distribution of solitonic field values gives E = h S . In general case E takessome value between these two limits, but it cannot exceed the Kolmogorov entropy of theunderlying mechanical system, E ≤ K . Thus, studying the soliton configurations one caninvestigate dynamical chaos in Eq. (1).The paper is organized as follows. In Sec. 2 we introduce the model (2) and discuss its ap-plications. The procedure of finding the solitons and determining their stability is describedin Sec. 3. Soliton multiplicity and its relation to the topological entropy are considered inSecs. 4 and 5, respectively. The distribution of stable solitons in the configuration space isdiscussed in Sec. 6. The “solitonic” analogue of the metric entropy is suggested in Sec. 7.Section 8 contains concluding remarks. We consider the theory of one-dimensional static scalar field with energy H [ ϕ ] = (cid:90) dx (cid:18)
12 ( ∂ x ϕ ) + (1 − cos ϕ ) U ( x ) (cid:19) , (6)62 π π π − − ϕ ( x ) x/D (a) − π − π π π − − ϕ ( x ) x/D (b) π π π − − − ϕ ( x ) x/D (c) Figure 5: (a) The soliton with the Cauchy data S at x = +0, see Fig. 4a. (b) The solitons1—3 corresponding to empty circles in Fig. 4a. (c) The related solitons 1 (cid:48) —3 (cid:48) with Cauchydata in Fig. 4b.where U ( x ) is given by Eq. (3). By definition, the solitonic profiles extremize this energy atits finite values, i.e. satisfy Eq. (2). Stable solitons, in addition, correspond to local minimaof H [ ϕ ]. They cannot be destroyed by adding a small perturbation and time-evolving thefield in an energy-conserving way.Let us describe several situations where Eqs. (6) and (2) appear. First, (1+1)-dimensionalrelativistic scalar field ϕ ( t, x ) satisfies equation ∂ t ϕ − ∂ x ϕ = − ∂V /∂ϕ that reduces to (2)in the static case for a particular choice of the potential V , see [1, 2]. The function U ( x ) isthen a time-independent external field.Second, one can consider Bose-Einstein condensate in the double well potential [12, 13, 14]forming two valleys stretched along the x direction, see Fig. 7a. The condensates in thewells interact via tunneling through the potential barrier. It can be shown [12, 13, 14]7 . . . − − − . d ε Figure 6: Box-counting dimension of the fractal in Fig. 4 at different ε .that the relative phase difference ϕ ( x ) = arg ψ − arg ψ of the condensate wavefunctions ψ , ψ , satisfies (2), where U characterizes coupling between the condensates. The spatialmodulation can be introduced into this equation by periodically changing the barrier betweenthe wells. Then the chaoticity parameter ε in (2) is related to the modulation amplitude.The third example is a classical ferromagnetic spin chain arranged along the x axis [15, 16]and interacting with the external magnetic field B , as depicted in Fig. 7b. Long-rangedynamics of this chain can be described [17, 18] by Eq. (2), where ϕ ( x ) is a rotation angleof spins in the y − z plane. In this case U ∝ B/ ( J s ∆ ), with ∆ representing the interspindistance and J s characterizing interaction between the neighboring spins. Introducing spatialinhomogeneity into the magnetic field B = B ( x ), we again obtain (2). x y U ( x ) y V (a) B x (b)
Figure 7: (a) Bose-Einstein condensate (thick solid lines) in a double-well potential withmodulated coupling U ( x ). (b) Ferromagnetic spin chain in a spatially inhomogenous mag-netic field B .There are many other applications of the static sine-Gordon equation (e.g. [19, 20]), where8patial modulation of the potential can be achieved by variation of external parameters.Keeping these applications in mind, below we consider general properties of static solitonsin the model (2), (3) with D = 12 and different ε .The trivial solutions to Eq. (2) are the vacua ϕ n = 2 πn, n ∈ Z corresponding to theabsolute minima of energy (6) at all ε . In what follows we consider finite-energy solitonsapproaching these vacua at x → ±∞ .To illustrate chaos in Eq. (2), we build the Poincar´e sections [33] for the analogousmechanical system at different ε . To this end we consider a generic solution ϕ ( x ) startingfrom the vacuum ϕ → x → −∞ . Numerically evolving the solution, we plot the valuesof ϕ, ϕ (cid:48) at x = mD + 0, i.e. after every period of the external potential, and obtain Fig. 8.At ε = 10 − (Fig. 8a) the solution remains close to the smooth curve — the separatrix inFig. 1b. In this case the analogous mechanical motion is nearly integrable. At ε > − (Fig. 8b) the values of ( ϕ, ϕ (cid:48) ) already form a sizeable “chaotic” region near the destroyedseparatrix. At even larger ε in Fig. 8c the solution ϕ ( x ) tosses randomly inside the “phasespace” covering a substantial part of it. Below we see how this chaos affects the solitonsolutions in field theory. −
303 0 π π ϕ ϕ mod 2 π (a) ε = 10 − −
303 0 π π ϕ ϕ mod 2 π (b) ε = 10 − −
303 0 π π ϕ ϕ mod 2 π (c) ε = 0 . ε . The values of ϕ are taken modulo 2 π . In this section we describe numerical method to find the solitons and determine their stability.Since all vacua are equivalent, we consider only the solutions starting from ϕ = 0 at x → −∞ . Near the vacuum Eq. (2) becomes a linear Schr¨odinger equation in the periodic Note that D is greater than the kink width in the pure sine-Gordon model, so the kink fits into a singleperiod of U ( x ). In particular, spatially inhomogeneous vacua do not exist. U ( x ) [34]. General solution of this equation includes exponentially growing anddecreasing parts, ϕ ≈ A e λ v x f A ( x ) + B e − λ v x f B ( x ) , (7)where A, B are arbitrary constants, λ v = 1 + O ( ε ) > f A,B ( x ) are periodic; we normalize them by f A,B (0) = 1. We explicitly find λ v and f A,B in Appendix A. Clearly, all solitons starting from the vacuum at x → −∞ have B = 0, ϕ ( x ) → A e λ v x f A ( x ) as x → −∞ , (8)and we parametrize them with the shooting parameter A .Analogously, the soliton profile arrives to some vacuum ϕ n at x → + ∞ , and its deviationfrom this vacuum is described by Eq. (7) with coefficients A (cid:48) ≡ B (cid:48) . Takingthe derivative of (7), we obtain the boundary condition ϕ (cid:48) ( x ) = (cid:18) − λ v + f (cid:48) B ( x ) f B ( x ) (cid:19) ( ϕ − ϕ n ) at x → + ∞ . (9)In what follows we solve Eq. (2) with boundary conditions (8), (9).We strongly rely on the shooting method. Imposing the boundary condition (8) at x = 0,we numerically solve Eq. (2) for every A . Then we tune the value of A to satisfy Eq. (9) atlarge x = L . Once this is done, we have all solitons localized inside the interval < x < L .In the chaotic regime our solutions are exponentially sensitive to the initial data. Thus,we need an efficient and extraordinary precise numerical method to solve Eq. (2). Our designof this method essentially relies on the simplified form of U ( x ) in Eq. (3). Namely, in theregions between the δ -functions mD < x < ( m + 1) D the potential U is constant and Eq. (2)can be solved explicitly in terms of elliptic functions, see Appendix B. At x = mD oneobtains matching conditions ϕ ( mD + 0) = ϕ ( mD − , ϕ (cid:48) ( mD + 0) − ϕ (cid:48) ( mD −
0) = ε sin ϕ ( mD ) . (10)As a result, our algorithm acts sequentially. Starting from ϕ and ϕ (cid:48) at x = mD + 0, itevolves them to x = ( m + 1) D − In practice, ln A is more convenient at A >
0; solitons with negative A are then obtained by reflection ϕ → − ϕ . To this end we change A in small steps. We check that no solution is lost by changing the size of thesesteps. Expressions (8), (9) are more accurate if ϕ (0) and ϕ ( L ) are closer to the vacua. To increase precision,we perform computations on the larger interval − D < x < L + D , and then select solutions staying close tothe vacua at x < x > L . ÷
200 digits. This gives us correct chaotic solutions of Eq. (2) of arbitrary complexity. Theexamples of these solutions are shown in Figs. 2 and 5. To study their statistical properties,below we obtain thousands of solitons of different forms and lengths.We are mainly interested in stable static solitons. They correspond to local minima ofenergy (6). Adding small variation θ ( x ) to the solution ϕ ( x ), one finds H [ ϕ + θ ] = H [ ϕ ] + 12 (cid:90) dx θ ( x ) ˆ L ϕ ( x ) θ ( x ) , (11)where ˆ L ϕ ( x ) = − ∂ x + cos ϕ ( x ) U ( x ). Thus, the soliton ϕ ( x ) is stable if the operator ˆ L ϕ ispositive-definite in the space of perturbations θ ( x ) vanishing at x → ±∞ .Numerically, we determine stability of solitons from the standard oscillation theorem.Namely, consider the perturbation θ ( x ) = ∂ϕ ( x ) ∂A , (12)where ϕ ( x ) is a solution of (2) with the initial data (8). By construction, θ ( x ) satisfiesˆ L ϕ θ = 0 and vanishes at x → −∞ . Then by the oscillation theorem the number of itszeros equals the number of negative eigenvalues of the operator ˆ L ϕ . In numerical code wecompute θ ( x ) for each soliton and count the number of its roots. The soliton is stable if θ ( x ) is positive-definite. In Appendix C we explain how this calculation can be convenientlyperformed within our shooting procedure.02 π π ϕ ( x ) x/D (a) x r l n | θ ( x ) | x/D (b) Figure 9: (a) Stable soliton (solid line) and unstable soliton (dashed line) at ε = 4 . × − .(b) Perturbations θ ( x ) of these solitons. Sharp cusp in the plot of unstable soliton pertur-bation at x r ∼ . D is its root.Figure 9 shows the example of stable soliton (solid line), unstable soliton (dashed line)and their perturbations θ ( x ). Below we focus on stable solitons and prove that their numberis infinite. 11 Multiplicity of solitons
There are only two static solitons in the pure sine-Gordon model: kink φ K ( x ) = 4 arctan e x (13)and antikink φ A ( x ) = − φ K ( x ). The most general soliton solution includes spatial shifts ofthese two and a choice of the left vacuum: ϕ = φ K ( x − x K )+2 πn . It is impossible to combinekinks and antikinks in a static chain of solitons, since they interact with energy E int ( s , s , R ) = 32 s s e − R . (14)where s α = +1 for a kink, − R (cid:29) ε (cid:28)
1) does not change the kink andantikink profiles, but affects weak forces between them. First, consider a single kink. Sub-stituting ϕ k ( x ) = φ K ( x − x k ) into Eq. (6), we obtain a periodic potential E δ ( x k ) ≡ H [ ϕ k ] − M k = 2 ε (cid:88) m ∈ Z ( mD − x k ) (15)which pulls the kink towards the equilibrium positions at x k = D ( m + 1 / M k = (cid:82) dx [( ∂ x ϕ k ) / − cos ϕ k ] = 8. Note that at D (cid:29) δ -functions of the external potential. In particular, the potential energy of the kinkcentered at 0 < x k < D approximately equals E δ ( x k ) ≈ ε (cid:18) ( x k ) + 1cosh ( D − x k ) (cid:19) (16)with equilibrium at x k = D/ lD < x (cid:48) k < ( l + 1) D . The totalinteraction energy of the soliton pair is now E ( x k , x (cid:48) k ) = E δ ( x k )+ E δ ( x (cid:48) k ) ± − ( x (cid:48) k − x k ) . It isclear that if l is large enough, the interaction between the (anti)kinks is exponentially small, We take l >
1, so that interaction energy of each soliton with the δ -functions is not affected by anothersoliton. x k = D/ x (cid:48) k = ( l + 1 / D .At small l interaction between the solitons pulls them out of their potential wells, destabiliz-ing the pair. Direct minimization of E ( x k , x (cid:48) k ) shows that the kink-kink and kink-antikinkpairs exist at l > D ln ε − l > D ln ε , (17)respectively. Recall that the soliton pairs are not static in the original sine-Gordon model,so this is a new property that already can be traced back to the nonintegrability of theanalogous mechanical system. Appearance of soliton pairs produces steps in the exponentialgrowth rate of stable soliton multiplicity, see Fig. 3b. In particular, the leftmost arrow inthis figure corresponds to the threshold ε = 2e − D for the existence of kink-antikink pair with l = 2.Let us demonstrate exponential growth of the stable soliton multiplicity with their lengthat small ε . To this end consider configuration of N (anti)kinks, ϕ = N (cid:88) α =1 s α ϕ k ( x − x α ) , (18)where s α = ± j α D < x α < ( j α + 1) D . We will assume that distances between the adjacent(anti)kinks are large enough, so that the condition j α +1 − j α > p, with p = − D ln ε
32 + 1 (19)is satisfied. In this case the interaction energy (14) between the adjacent (anti)kinks | E int | (cid:54) − ( p − D is at least twice smaller than their potential wells produced by the δ -functions. This implies that the total energy of the chain has a local minimum with respectto the position of every kink, i.e. the stable equilibrium exists. Thus, the solitons can bearbitrarily added to the chain at distances exceeding pD .We denote the number of the above “sparse” solitonic chains of length lD or smaller by N p ( l ), where l is an integer. In the larger interval of length ( l + p ) D one can add a kink, anantikink or none of them to the chain. Thus, N p ( l + p ) (cid:62) N p ( l ) . Using N p (1) = 3 as theinitial condition, we find that N p ( l ) (cid:62) ( l + p − /p , i.e. the multiplicity of solitons grows atleast exponentially with their length lD The number of sparse solitonic chains inside the interval of length ( l + p ) D is, in fact, greater than 3 N p .Indeed, one can start with a chain of length smaller than lD and add an (anti)kink at various positions.Taking this effect into account, one obtains more accurate recurrence relation: N p ( l ) − N p ( l −
1) = 2 N p ( l − p )with an exponentially growing solution for N p ( l ). However, the latter approach also considers “sparse”solitonic chains and therefore significantly underestimates the exponential growth rate h S .
13n the other hand, the total number of solitons is bounded from above by the number3 l of all possible soliton equilibrium positions with (anti)kinks occupying individual periodsinside the interval lD . As a consequence, the number of stable solitons grows exponentially,see Eq. (4), and the growth rate h S is bounded byln 3 p (cid:54) h S (cid:54) ln 3 , (20)where p was introduced in Eq. (19).We numerically computed the number of stable solitons N sol ( l ) within the interval oflength lD , see Fig. 3a. The multiplicity indeed grows exponentially, although the growthrate h S ( ε ) is much higher than our lower bound (20), see Figs. 3b and 10a.0 . . . . − − − . h S ε (a) π π ϕ ( x ) x/D (b) Figure 10: (a) Exponential growth rate h S ( ε ) of stable solitons. Points with errorbars areobtained by counting the number of numerically computed solitons. The solid line is foundby minimizing the energy (21). (b) An example of the soliton at ε = 10 − which is notrepresented by the ansatz (18).The next step is to consider larger ε corresponding to mostly chaotic dynamics of theanalogous mechanical system, see Fig. 8b. In this case we use general expression for theenergy of the solitonic chain, E N ( x , . . . , x N ) = N − (cid:88) α =1 E int ( s α , s α +1 , x α +1 − x α ) + ε (cid:88) m (cos ( ϕ ( mD )) − , (21)because when two (anti)kinks occupy adjacent periods of U ( x ), they can affect each other’sinteraction with the external potential. Minimizing (21) numerically with the conjugategradient method, we determine whether a stable soliton chain exists for a given { s α , j α } ,where j α D < x α < ( j α + 1) D . The exponential growth rate obtained from numerical14inimization of energy is shown by the solid line in Fig. 10a. It coincides with the exactgraph at small ε , but starts to deviate from it at ε ∼ − . This is due to the new types ofsolitons appearing in the system, with two or more (anti)kinks squeezed into one period of U ( x ). The examples of such solitons are presented in Figs. 2b and 10b; the ansatz (18) isnot valid for them. Not surprisingly, appearance of these solitons coincides with transitionto chaos in the corresponding mechanical system, cf. Figs. 8b,c. An important quantity characterizing complexity of a dynamical system is the topologicalentropy [24]. In this section we define this quantity for the analogous mechanical system ,then use it to constrain the soliton growth rate h S ( ε ).Consider the solutions starting from ϕ → x → −∞ . Let us split the field values intosegments − π + 2 πn (cid:54) ϕ (cid:54) π + 2 πn. We characterize every solution ϕ ( x ) in the segment of length lD with the sequence of regions( n , . . . , n l ) it visits after every period of U ( x ), i.e. at x = mD + 0. One can argue that thenumber of different sequences N seq ( l ) grows exponentially with the length of the interval lD .We therefore call h T = lim l →∞ ln N seq ( l ) l (22)the topological entropy of the analogous mechanical system.The value of h T is an indicator of chaos. The above definition gives h T = 0 for ε = 0.Indeed, solutions approaching the vacuum at x → −∞ include the vacuum itself and(anti)kinks at different positions, resulting in 2 l + 1 sequences of length l . At small nonzero ε the quantity h T is positive and bounded from below by the exponential growth rate h S of the soliton multiplicity. Indeed, every stable soliton corresponds to a unique sequence ofvisited vacua ( n , . . . , n l ). Thus, N sol ( l ) (cid:54) N seq ( l ), implying (5). In this section we study the set of values ( ϕ (0) , ϕ (cid:48) (+0)) taken by the solitonic fields at x = +0.We consider a small vicinity of vacuum | ϕ (0) | , | ϕ (cid:48) (+0) | (cid:28)
1. In this case the decomposition The original topological entropy was defined in systems with compact phase space. We generalize it in astraightforward way considering a particular set of trajectories and a particular sampling of the phase space. x ≈ +0, where the first and second terms vanish exponentially at negative andpositive x , respectively. Then the complete nonlinear solution can be represented as a sum ϕ ( x ) ≈ ϕ L ( x ) + ϕ R ( x ) of “left” and “right” parts vanishing at x → + ∞ and x → −∞ . Inwhat follows we study only the “right” sector of solitons, with “left” solutions obtained byreflection x → − x . In particular, if { ϕ α ( x ) } is the set of “right” solitonic field values, theentire fractal in Fig. 4a consists of points ϕ αβ (0) = ϕ α (0) + ϕ β (0) , ϕ (cid:48) αβ (+0) = ϕ (cid:48) α (+0) − ϕ (cid:48) β ( − . (23)Details on computing the set of “right” solitons { ( ϕ α (0) , ϕ (cid:48) α (+0)) } are given in Appendix E.Let us explain self-similarity of the fractal in Fig. 4. Suppose the “right” soliton ϕ S hasparameter A = A S in Eq. (8) and length lD . Solution in its tiny vicinity can be representedas ϕ A ( x ) = ϕ S ( x ) + ( A − A S ) θ ( S )0 ( x ) , where θ ( S )0 is the perturbation (12) in the background of ϕ S ( x ). Taking A − A S = (cid:104) θ ( S )0 ( l (cid:48) D ) (cid:105) − with l (cid:48) > l , one obtains the solution ϕ A ( x ) staying close to ϕ S ( x ), arriving to the same vac-uum ϕ n and then departing from it at x > l (cid:48) D . At x ≈ l (cid:48) D the solution has the form ϕ A ( x ) ≈ ( A − A S ) θ ( S )0 ( l (cid:48) D ) f A ( x ) e x − l (cid:48) D + ϕ n , where the asymptotics of θ ( S )0 ( x ) at x → + ∞ was used. Thus, at x = l (cid:48) D the boundary condition (8) is satisfied, with ( A − A S ) θ ( S )0 ( l (cid:48) D )playing the role of the new parameter A . This mechanism is illustrated in Fig. 4b and theright parts of Figs. 5b, c. Note that the function λ S ( x ) ≡ ln (cid:12)(cid:12)(cid:12) θ ( S )0 ( x ) (cid:12)(cid:12)(cid:12) describes exponentialgrowth of the perturbation and therefore is related to the Lyapunov exponent of the soliton ϕ S ( x ).Now, let us compute the box-counting dimension of the fractal formed by the field valuesof solitons. To warm up, consider the entire set of solitons from the “right” sector, stableand unstable. This set is dense in the chaotic region, and its fractal dimension is 1. Indeed,consider two close solutions ϕ and ϕ parametrized by A and A . If the chaos is on, theydiverge exponentially, with | ϕ ( x ) − ϕ ( x ) | > π at sufficiently large x . Then by continuitythere exists a trajectory with parameter A S ∈ ( A , A ) that arrives precisely to the vacuumbetween ϕ ( x ) and ϕ ( x ). This trajectory is a soliton, which proves the statement.The parameters A of stable “right” solitons, however, form a Cantor-like set with fractaldimension less than 1. Indeed, consider the soliton ϕ S ( x ) with A = A S . We already arguedthat it contains the entire set of solitons in its arbitrarily small vicinity | A − A S | (cid:28)
1, and,in particular, unstable solitons. However, the solutions near the unstable solitons are also We assume that l (cid:48) is large enough for | θ ( x ) | to reach maximum at the rightmost point x = l (cid:48) D of theinterval. θ ( x ). Thus, the field values of stable soliton do not form adense set, as their vicinities | A − A S | (cid:28) a = ln Aλ v D . (24)instead of
A >
0. Since A ( a ) is a smooth function, this does not alter fractal dimension.Transformation A → A e λ v D trivially shifts the solution by one period of the external potentialand changes a → a + 1. Thus, the fractal is periodic in a ; in what follows we consider onlythe segment a ∈ [0 , δ , we count the number N box ( δ ) of boxes with stable soliton parameters { a S } inside. The details on this procedureare given in Appendix E. The box-counting fractal dimension d R then can be extracted fromthe asymptotics ln N box ( δ ) → − d R ln δ as δ → . (25)The function N box ( δ ) is shown in Fig. 11.024 0 4 8 l n N b o x − ln δ dataEq. (25) Figure 11: Number of boxes with stable solitons versus the box size at ε = 3 × − . Fitwith Eq. (25) (line) gives box-counting dimension d R = 0 . ± . lD or smaller. Perturbations θ ( x ) in their backgroundsgrow with x at a slower rate than the vacuum perturbations, | θ ( x ) | (cid:54) e λ v x | f A ( x ) | , simplybecause cos ϕ ( x ) in the equation ˆ L ϕ θ = 0 is maximal at ϕ = 2 πn . As a consequence, thesolitons cannot have A parameters at a distance closer than δA l = [max θ ( x )] − (cid:62) e − λ v lD , (26)17here the maximum is taken within the interval 0 (cid:54) x (cid:54) lD . If Eq. (26) is not satisfied, thesolitons would coincide in the entire interval. This gives the typical distance between the a parameters of the solitons, δa l (cid:38) e − λ v lD (27)for A (cid:46) O (1). Breaking the a -interval into the boxes (27), one obtains d R (cid:62) lim l →∞ ln N sol ( l ) − ln δa l (cid:62) h S λ v D , (28)where we used Eq. (4). Note that this bound is a serious underestimation: for ε = 3 × − it gives d R (cid:38) .
06, an order of magnitude smaller than the actual fractal dimension. Never-theless, it proves that the dimension of our fractal is nonzero.Since the fractal in Fig. 4 is a direct sum of “left” and “right” fractals, its dimension is d ( ε ) = 2 d R ( ε ), see Fig. 6. Metric (Kolmogorov-Sinai) entropy [25] is an important quantity indicating whether thedynamical system is chaotic or not. It was originally introduced for systems with compactphase space. Since our analogous mechanical system does not have this property [35], wefirst modify the entropy definition as follows.We again restrict ourselves to the “right” solutions of length L = lD starting from ϕ ≈ x = 0. Besides, we consider only a finite interval | A | (cid:54) A of their shooting parameter.We divide the phase space into strips:2 πν (cid:54) ϕ + ϕ (cid:48) (cid:18) λ v − f (cid:48) B (+0) f B (0) (cid:19) − < π ( ν + 1) , (29)cf. Eq. (9). For every solution ϕ ( x ) of length lD we construct the sequence ω = ( ν , ν , . . . , ν l )of visited regions at the start of every period x = mD + 0. This divides the interval − A (cid:54) A (cid:54) A of solution parameters into the regions T ω corresponding to certain sequences.The solitons belong to the boundaries of T ω due to Eq. (9). We define the metric entropy K as K l = − (cid:88) ω ∆ A ( T ω )2 A ln (cid:18) ∆ A ( T ω )2 A (cid:19) , and K = lim l → + ∞ K l l , (30)where ∆ A ( T ω ) is the total length of T ω . The only difference from the original Kolmogorov-Sinai construction is that we considered a selected set of trajectories and a particular sam-pling (29). 18ote that at ε = 0 only two non-trivial “right” solutions exist, the kink and the antikink,which belong to the regions with ν = 0 and 1, respectively, at every x . We obtain only twosequences. Hence, K = 0, as it should be in the integrable case.Let us introduce the quantity analogous to the metric entropy considering the stable“right” solitons of length L < lD . Indeed, their shooting parameters divide the segment − A (cid:54) A (cid:54) A into multiple intervals R α . We therefore define E l = − (cid:88) α ∆ A ( R α )2 A ln (cid:18) ∆ A ( R α )2 A (cid:19) and E = lim l →∞ E l l , (31)cf. (30). Clearly, E characterizes (in)homogeneity of distribution of the stable soliton shoot-ing parameters. If all solitons have the same A , then E = 0. If they are evenly distributed,then E l = ln N sol ( l ) and E = h S , see Eq. (4).Since the boundaries of R α are also the boundaries of T ω , splitting { R α } is a coarse-graining of { T ω } obtained by merging some of the regions together. However, if two regionsof lengths ∆ A and ∆ A are merged into one, − (∆ A + ∆ A ) ln (cid:18) ∆ A + ∆ A A (cid:19) (cid:54) − ∆ A ln (cid:18) ∆ A A (cid:19) − ∆ A ln (cid:18) ∆ A A (cid:19) . This proves that E l (cid:54) K l , and therefore E (cid:54) K . In Fig. 12 we demonstrate the values of E l , K l (points) and their linear fits (lines).123 2 4 6 8 10 K l E l K l , E l l Figure 12: Values of E l and K l computed at ε = 3 × − and A ≈ . × − . Linear fitgives K = 0 . ± .
01 and E = 0 . ± . Generalization
Our results show that the systems obeying driven sine-Gordon equation have an infinite set ofstable static solitons, and the field values of these objects form a fractal in the configurationspace. The box-counting dimension of the fractal is non-integer.We do not want to leave an impression, however, that our model is special in someregard. Similar “chaotic” solitons should exist in many one-dimensional theories with non-integrable static equations, cf. [36, 37]. Indeed, consider scalar fields ϕ and ϕ , with theenergy functional H = Γ H [ ϕ ] + H [ ϕ , ϕ ] , (32)where Γ is a constant. Equations for static solitons are δH [ ϕ ] δϕ + 1Γ δH [ ϕ , ϕ ] δϕ = 0 , δH [ ϕ , ϕ ] δϕ = 0 . (33)At Γ (cid:29) ϕ satisfies an independent equation, while ϕ ( x ) evolves in the externalpotential ϕ ( x ). Thus, our results can be extended at least to a certain class of two-fieldmodels.In general, one expects to find an infinite total number of solitons in non-integrable case.On the other hand, the distribution of stable solitons may be model-dependent. Indeed, thesesolutions are the local minima of the static energy H [ ϕ ] which coincides with the classicalaction in the mechanical analogy. Presently, there is no general classification of mechanicaltrajectories locally minimizing the action, though some works in this direction appear [38].We hope that the instruments developed in this paper — metric and topological entropies,and fractals formed by field values — will be useful for studies of chaotic solitons in differenttheories. Acknowledgements.
We are grateful to V. A. Rubakov for comments and criticism. Thiswork was supported by the grant RSF 16-12-10494. Numerical calculations were performedon the Computational Cluster of the Theoretical Division of INR RAS.
A Deriving asymptotic conditions
Near the vacuum ϕ = 0 Eq. (2) takes the form of a Schr¨odinger equation ϕ (cid:48)(cid:48) ( x ) = ϕ ( x ) U ( x ) (34)20n the periodic potential U ( x ), Eq. (3). The particular solutions of this equation coincidewith the eigenfunctions of the shift operator x → x + D . This suggests the ansatz ϕ ( x ) = e λx f ( x ) , (35)where f ( x ) has period D . Solving Eq. (34) inside the interval 0 < x < D , one obtains, f ( x ) = e − λx ( C + e x + C − e − x ) . (36)The linearized matching conditions (10) at the endpoints of this interval take the form f (0) = f ( D ) , f (cid:48) ( D ) + εf (0) = f (cid:48) (0) , (37)where we recalled that f is periodic. Substituting Eq. (36) into Eq. (37), we arrive to thehomogeneous linear system (cid:32) e (1 − λ ) D − − (1+ λ ) D − ε + λ − − λ )e (1 − λ ) D ε + λ + 1 − (1 + λ )e − (1+ λ ) D (cid:33) (cid:32) C + C − (cid:33) = 0 , (38)which has nontrivial solutions only if the matrix has zero determinant. This gives two roots λ = ± D ln (cid:16) σ + √ σ − (cid:17) ≡ ± λ v , σ = cosh D + ε D . (39)We obtained Eq. (7), where the particular solutions f A , f B are given by Eq. (36) with λ = ± λ v and coefficients C + , C − representing the eigenvectors in Eq. (38). We normalizethe solutions by C + + C − = 1. B General solution of the static sine-Gordon equation
In the regions between the δ -functions Eqs. (2), (3) reduce to the equation for physicalpendulum: ϕ (cid:48)(cid:48) = sin ϕ . The motion of the latter system depends on whether its mechanicalenergy E m = ϕ (cid:48) ϕ − E m = 0 or not [39]. Periodic motions at E m < ϕ ( x ) = 2 arccos( ± k sn( x − x π , k )) + 2 πn , n ∈ Z , (41)while the “rotating” solutions at E m > ϕ ( x ) = π ± (cid:18) k ( x − x π ) , k (cid:19) . (42)21ere sn and am denote the elliptic sine and Jacobi amplitude, respectively [40]. The solutions(41), (42) have two integration constants: k = (cid:112) E m / x π . Signs ± in these equations discriminate two branches of solutions with opposite signs of ϕ (cid:48) ( x ).Numerically, we use Eqs. (41), (42) as follows. Starting with the values of ϕ and ϕ (cid:48) at x = mD + 0, we compute E m , k , and determine the relevant branch of the general solution.Inverting Eq. (41) or (42), we find x π and hence — values of ϕ and ϕ (cid:48) within the entireinterval mD < x < ( m + 1) D . Using the matching conditions (10), we proceed with thenext interval. C Linear stability
Let us describe a practical way to study soliton stability within the shooting approach. Tothis end we count zeros of the perturbation θ ( x ) in Eq. (12).As the shooting parameter A changes, these zeros cannot disappear or emerge sporadicallyinside the interval 0 < x < L . Indeed, one can regularize the δ -functions in Eq. (3), making θ a smooth function of x and A . After that the roots of θ can appear at the real axis ordisappear from it only in pairs at points x ∗ such that θ ( x ∗ ) = θ (cid:48) ( x ∗ ) = 0. However, θ ( x ) isnon-trivial and satisfies the second-order linear equation ˆ L ϕ θ = 0. It cannot vanish togetherwith its first derivative at any x .As a consequence, the number of θ roots inside the interval 0 < x < L changes by ± L . This happens at certain values A = A ∗ of the shooting parametersatisfying ∂ϕ ( L ) /∂A = 0. Thus, we just need to find out whether the root x r ( A ) comes in orgoes out of the interval. Differentiating the equality θ ( x r ( A ) , A ) = 0, we obtain the change∆ N r in the number of roots,∆ N r = − sgn dx r dA (cid:12)(cid:12)(cid:12)(cid:12) A ∗ = sgn ∂ A θ ( L ) θ (cid:48) ( L ) (cid:12)(cid:12)(cid:12)(cid:12) A ∗ . (43)Expression (43) can be simplified using the mechanical energy E m , Eq. (40), which does notdepend on x inside the finite intervals mD < x < ( m + 1) D . Using Eq. (12), one finds ∂ x θ ( L ) = ∂ x ∂ A ϕ ( L ) = ∂ A E m /ϕ (cid:48) ( L ) at A = A ∗ . Therefore,∆ N r = sgn (cid:0) ∂ A E m · ∂ A ϕ ( L ) · ϕ (cid:48) ( L ) (cid:1)(cid:12)(cid:12) A ∗ . (44)The factors in Eq. (44) can be computed numerically using the values of ϕ and ϕ (cid:48) at x = L and different A . The latter are provided by the shooting method.We apply Eq. (44) as follows. At small A the solutions ϕ A ( x ) remain close to the vacuum,and θ ( x ) does not have zeros, N r = 0. We change A in small steps and determine the values22 ∗ corresponding to ∂ A ϕ ( L ) = 0. At these points we change the number of θ roots accordingto Eq. (44). Once all solitons are obtained, we select the stable ones, i.e. those with N r = 0.We tested the above procedure by explicitly solving the equation ˆ L ϕ θ = 0 via thesequential algorithm, cf. Appendix B, arriving to the same result for the number of θ zeros. D Interaction energy of a soliton pair
Consider a kink and an (anti)kink in the pure sine-Gordon model with centers separated bydistance R (cid:29)
1. This field configuration is approximated by a sum ϕ ( x ) ≡ x + R/ ± x − R/ = ϕ l ( x ) ± ϕ r ( x ) , (45)where plus and minus signs correspond to kink and antikink at x = R/
2, respectively.Substituting (45) into the energy (6) at ε = 0, we find, H = − + ∞ (cid:90) −∞ dx (1 ∓ cosh 2 x )(cosh 2 x + cosh R ) + const = ± − R + O (e − R ) + const , (46)where the constant includes all R -independent terms. In the last equality we computed theintegrals and extracted the asymptotics R → ∞ , reproducing the well-known result [3]. E Finding the fractal to a given precision
In a nutshell, our numerical procedure for computing the fractal of “right” solitonic values(Sec. 6) is straightforward. We split the values of a ∈ [0 ,
1] into small boxes of size δ ,search for stable solitons within each box by the shooting method, and plot their field valueswith points in Figs. 4a,b. Changing the box size δ , we obtain Fig. 11 and the coefficient inEq. (25).To calculate the fractal with resolution δ , however, we need to search for the solitons ofdifferent length L = lD in different a -boxes. We estimate L by recalling that the differencebetween any two solutions grows exponentially with x : ∆ ϕ ∼ ∆ A · θ ( x ) . Thus, for a givensoliton length L we take the interval ∆ A = [ θ max ( L )] − , where θ max ( L ) is the maximum of | θ ( x ) | at 0 < x < L . Solutions within this interval satisfy ∆ ϕ (cid:46)
1, so it contains O (1)stable solitons of length L , if they exist. Inversely, for a given ∆ a = δ we take large enough L satisfying θ max ( L ) (cid:62) A (cid:62) e − λ v D · a δ · λ v D , (47)23here in the last expression we converted ∆ A into ∆ a = δ and used Eq. (24).In practice we compute θ ( x ) for the solution in the center of each a -box to the point x = L where Eq. (47) is already satisfied, then search for solitons of length L within thisbox. References [1] R. Rajaraman, “Solitons and Instantons. An Introduction to Solitons and Instantons inQuantum Field Theory,” Amsterdam: North-Holland (1982)[2] V. A. Rubakov, “Classical theory of gauge fields,” Princeton Univ. Press, 2002.[3] N. S. Manton and P. Sutcliffe, “Topological solitons,” Cambridge University Press, 2004.[4] G. M. Zaslavsky, “The physics of chaos in the Hamiltonian systems,” 2nd ed., ImperialCollege Press, 2007.[5] J. Cuevas-Maraver, P. G. Kevrekidis, F. Williams (ed.), “The Sine-Gordon Model andits Applications: From Pendula and Josephson Junctions to Gravity and High-EnergyPhysics,” Springer, 2014.[6] M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Classical Particle-Like Behavior of Sine-Gordon Solitons in Scattering Potentials and Applied Fields,”Phys. Rev. Lett. , 1411 (1976) [ Erratum-ibid , 314 (1976)].[7] M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B , 1578 (1977).[8] J. F. Currie, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Numerical simulationof sine-Gordon soliton dynamics in the presence of perturbations,” Phys. Rev. B ,5567 (1977).[9] R. Scharf, Y. S. Kivshar, A. S´anchez, A. R. Bishop, “Sine-Gordon kink-antikink gener-ation on spatially periodic potentials,” Phys. Rev. A , R5369 (1992).[10] A. S´anchez, R. Scharf, A. R. Bishop, and L. V´azquez, “Sine-Gordon breathers on spa-tially periodic potentials,” Phys. Rev. A , 6031 (1992).[11] W. Hai, Z. Zhang, and J. Fang, “Chaotic solitons in Sine-Gordon system,” Eur. Phys.J. B , 103 (2001). 2412] N. K. Whitlock and I. Bouchoule, “Relative phase fluctuations of two coupled one-dimensional condensates,” Phys. Rev. A , 053609 (2003).[13] V. Gritsev, A. Polkovnikov, and E. Demler, “Linear response theory for a pair of coupledone-dimensional condensates of interacting atoms,” Phys. Rev. B , 174511 (2007).[14] T. Schweigler et al. , “Experimental characterization of a quantum many–body systemvia higher–order correlations,” Nature , 323 (2017).[15] P. Kumar, V. K. Samalam, “Solitons in an easy-plane ferromagnetic chain,” J. Appl.Phys. , 1856 (1982).[16] G. Wysin, A. R. Bishop, and P. Kumar, “Soliton Dynamics on an Easy-Plane Ferro-magnetic Chain,” J. Phys. C , 5975 (1984).[17] T. Kawasaki, “Dynamics Of Solitons In An Easy Plane Ferromagnetic Chain: A DiscreteLattice Model,” Prog. Theor. Phys. , 534 (1986).[18] H. J. Mikeska, “Solitons in a one-dimensional magnet with an easy plane,” J. Phys. C , L29 (1978).[19] S. A. Vasenko, K. K. Likharev, and V. K. Semenov, “Static properties of distributedinhomogeneous Josephson junctions,” Sov. Phys. JETP , 766 (1981).[20] D. W. McLaughlin and A. C. Scott, “Perturbation analysis of fluxon dynamics,” Phys.Rev. A , 1652 (1978).[21] V. I. Arnol’d, “Proof of a Theorem of A. N. Kolmogorov on the Preservation of Condi-tionally Periodic Motions under a Small Perturbation of the Hamiltonian,” Russ. Math.Surv. , 9 (1963).[22] V. I. Arnol’d, “Mathematical Methods of Classical Mechanics,” Springer, 1989.[23] J. Moser, “On Invariant Curves of Area-Preserving Mappings of an Annulus,” Nachr.Akad. Wiss. G¨ottingen Math.-Phys. Kl. II, 1 (1962).[24] R. L. Adler, A. G. Konheim, and M. H. McAndrew, “Topological Entropy,” Trans.Amer. Math. Soc. , 309 (1965).[25] E. Ott, “Chaos in Dynamical Systems,” Cambridge University Press, 2002.2526] E. Ott, “Strange attractors and chaotic motions of dynamical systems,” Rev. Mod.Phys. , 655 (1981).[27] D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, “Resonance structure in kink-antikink interactions in φ theory,” Physica D , 1 (1983).[28] P. Anninos, S. Oliveira, and R. A. Matzner, “Fractal structure in the scalar λ ( φ − theory,” Phys. Rev. D , 1147 (1991).[29] D. G. Levkov, A. G. Panin and S. M. Sibiryakov, “Complex trajectories in chaoticdynamical tunneling,” Phys. Rev. E , 046209 (2007) [ nlin/0701063 ].[30] T. Roma´nczukiewicz and Y. Shnir, “Oscillon resonances and creation of kinks in particlecollisions,” Phys. Rev. Lett. , 081601 (2010) [ arXiv:1002.4484 ].[31] J. Theiler, “Estimating fractal dimension,” J. Opt. Soc. Am. A , 1055 (1990).[32] N. Sarkar, B. B. Chaudhuri, “An efficient differential box-counting approach to computefractal dimension of image,” IEEE Trans. Systems, Man, and Cybernetics , 115(1994).[33] S. H. Strogatz, “Nonlinear Dynamics and Chaos,” Perseus Books, 1994.[34] V. I. Arnold, “Ordinary Differential Equations,” MIT Press, 1978.[35] S. F. Singer, “Symmetry in Mechanics: A Gentle, Modern Introduction,” Birkh¨auser,Boston, MA, 2004.[36] R. Scharf, A. R. Bishop, “Soliton chaos in the nonlinear Schrodinger equation withspatially periodic perturbations,” Phys. Rev. A , 2973 (1992).[37] M. Yamashita, “Multisolitons and Soliton Lattices in Sine-Gordon System with VariableAmplitude,” Prog. Theor. Phys. , 622 (1985).[38] C. G. Gray, E. F. Taylor, “When action is not least,” Am. J. Phys.75