Bubble nucleation around heterogeneities in ϕ 4 -field theories
BBubble nucleation around heterogeneities in φ -field theories c (cid:13) Juan F. Mar´ın ∗ Departamento de F´ısica, Universidad de Santiago de Chile,Usach, Av. Ecuador 3493, Estaci´on Central, Santiago, Chile. (Dated: August 13, 2020)
Abstract
Localised heterogeneities have been recently discovered to act as bubble-nucleation sites in non-linear field theories. Vacuum decay seeded by black holes is one of the most remarkable applications.This article proposes a simple and exactly solvable φ model exhibiting bubble nucleation aroundlocalised heterogeneities. Bubbles with a rich dynamical behaviour are observed depending on thetopological properties of the heterogeneity. The linear stability analysis of soliton-bubbles predictsthe formation of oscillating bubbles and the insertion of new bubbles inside an expanding pre-cursor bubble. Numerical simulations in 2+1 dimensions are in good agreement with theoreticalpredictions. ∗ [email protected] a r X i v : . [ n li n . PS ] A ug . INTRODUCTION The role of heterogeneities in scalar field theories is relevant in many physical phenomena,ranging from the stability of the Higgs vacuum to phase transitions in the early Universe[40–43]. One of the most intriguing aspects of our current understanding of the Universe isthe possibility that the Universe can be trapped in a meta-stable state of the Higgs potential[44]. From such a meta-stable state – the false vacuum state – a phase transition may occurtowards a global minimum– the true vacuum state . A transition of the universe towards anearby global minimum of the Higgs potential– a phenomenon termed as vacuum decay [45],would lead the Universe towards a state where life as we know it might be impossible [44, 46].The possibility that heterogeneities, such as black holes, may enhance the nucleation of true-vacuum bubbles in the Universe and thus seeding vacuum decay is a fascinating subject thatis gaining increasing interest in gravitation and cosmology [40, 41, 43, 47–52].The idea of modelling a black hole as heterogeneity in scalar field theories has beenrecently proposed by Gregory and co-workers [40, 41, 43]. The fundamental mechanism isthat black holes emit high energy particles that perturbs any scalar field φ coupled to theemitted quanta [53]. In scalar field theories where φ resides in a meta-stable state, suchperturbation can drive the field φ over the potential barrier and activate vacuum decay.This scenario corresponds to the nucleation of true-vacuum bubbles around the black hole.For instance, exploding or evaporating black holes would be surrounded by phase-transitionbubbles [47, 54]. This phenomenon is reminiscent to the intuitive picture of the first-orderphase transition occurring in boiling water, where bubbles of the new phase are nucleated–often around impurities–, and eventually expands. Other works address the validity ofthese ideas in the Higgs model, including the effects of gravity [41], large extra dimensions[52], and deviations from the thin-wall approximation [43].This article introduces an exactly solvable heterogeneous φ field theory exhibiting bub-ble nucleation around the heterogeneity. It is shown how space-dependent heterogeneitiesdeform the energy landscape of the φ system affecting the stability of the vacua. The topo-logical properties of the heterogeneity are described through a single parameter, and as willbe shown, a rich variety of bubble dynamics can be observed according to its value. Bubblescan be nucleated, oscillate, or even expand. Moreover, it is shown how the interaction of thebubble with the heterogeneity may trigger nonlinear instabilities that may enhance vacuum2ecay and the insertion of a new bubble inside a precursor bubble.The outline of the article is as follows. Section II introduces the nonlinear field theory andthe solitonic-model of bubbles. Section III summarises the results from the linear stabilityanalysis of the soliton-bubbles under the influence of the heterogeneity. The first implicationfrom the linear stability analysis, which is the existence of oscillating bubbles, is discussedin section IV. Section V is devoted to conditions where non-expanding true-vacuum bubblescan be sustained by the heterogeneity. For some combinations of parameters, vacuum decaycan occur. The nucleation of bubbles inside a precursor expanding-bubble is demonstratedin Section VI. The robustness of the reported phenomena under thermal fluctuations isbriefly discussed in Section VII. Finally, concluding remarks are given in Section VIII. II. BUBBLES AND TOPOLOGICAL DEFECTS IN φ SYSTEMS
Bubbles in scalar field theories are configurations where the field φ is in one phase ina space filled with another phase. In the context of the qualitative theory of nonlineardynamical systems, bubbles are coexistent heteroclinic trajectories – or kinks, also known as topological defects –joining fixed points of the underlying Higgs potential [55]. Ring solitons[56], kink-antikink pairs [57], and other topologically equivalent solutions [58] are physicallyrelevant models of the so-called soliton-bubbles of the nonlinear field theory. Consider a fieldtheory in D + 1 dimensions with a single real scalar field φ , whose action is S = (cid:90) dt d D r (cid:20)
12 ( ∂ α φ ) − µ φ − η ) + f φ (cid:21) , (1)where α = 0 , , . . . , D , f = f ( r ) is a heterogeneous perturbation, r ∈ R D and µ and η areparameters of the model. The theory turns into the celebrated φ model for f = 0, with twodegenerate vacua at φ = ± η separated by a barrier at φ = 0 [59]. If | f | < / ∀ r , the vacuaof the effective potential U eff ( φ ) = µ (cid:0) φ − η (cid:1) − f φ, (2)becomes non-degenerate with a true- and a false vacuum state, as depicted in Fig. 1(a) fora fixed r . The effect of damping can be taken into account using the phenomenologicalRayleigh dissipation function R = ( γ/ ∂ t φ ) , where γ is the damping constant. Theresulting equation of motion is the driven and damped φ equation ∂ tt φ − ∇ φ + γ∂ t φ − µ (cid:0) η − φ (cid:1) φ = f ( r ) . (3)3 o102 o102 FIG. 1. (a)
Effective potential of Eq. (2) for µ = 1 / η = 1 and f = 0 . φ = φ separating a false-vacuum ( φ = φ ) from a true-vacuum ( φ = φ ). (b) A soliton bubble of radius r o representing a bubble of phase φ (a φ -bubble ) in a space filled with phase φ . (c) A solitonbubble of radius r o representing a bubble of phase φ (a φ -bubble ) in a space filled with phase φ .The normalised energy density of each bubble is shown in dashed lines. Topological defects are heteroclinic solutions interpolating the stable phases of the un-derlying potential [55], namely φ and φ . For D = 1 they are known as kinks [60, 61],whereas for D > line solitons or domain walls [56, 59]. Let φ < φ hereon. Soliton bubble solutions of Eq. (3) are real and rotationally symmetric solutions,i.e. φ B ( r ) = φ B ( r ) where r = ( x + . . . + x D ) / , with the following properties [62, 63]. A φ -bubble is a bubble of phase φ in a space filled with phase φ [see Fig. 1(b)] in the formof a rotationally symmetric solution φ B ( r ) such that lim r →∞ φ B ( r ) = φ and0 ≤ ∂ r φ B (0) < ∂ r φ B ( r ) < ∞ , (4a) φ < φ B (0) < φ B ( r ) < φ , (4b)for r ∈ (0 , ∞ ). Similarly, a φ -bubble is a bubble of phase φ in a space filled with phase φ [see Fig. 1(c)] such that lim r →∞ φ B ( r ) = φ and0 ≤ ∂ r φ B (0) > ∂ r φ B ( r ) > −∞ , (5a) φ > φ B (0) > φ B ( r ) > φ , (5b)4or r ∈ (0 , ∞ ).Phenomenological models with exact or approximate analytical solutions are useful tounderstand physical phenomena in real systems. Some examples are the SSH theory ofnonlinear excitations in polymer chains [64] and the Peyrard-Bishop model of denaturationof the DNA [65], both widely used to answer important questions of chemical and biologicalinterest [60]. In our system, solving an inverse problem, it is possible to give systems of type(3) with exact solutions. Later, in the sense of the qualitative theory of nonlinear dynamicalsystems [55], it is possible to generalise analytical results obtained for systems of type (3)to other solutions that are topologically equivalent [58, 66–70]. For instance, the followingsoliton-bubble φ B ( r ) = tanh[ B ( r − r o )] , (6)is an exact solution to Eq. (3) for µ = 1 / η = 1, and f ( r ) = (cid:20)
12 (4 B −
1) tanh[ B ( r − r o )] − ( D − Br (cid:21) sech [ B ( r − r o )] , (7)where r o is the radius of the bubble and B is a parameter that describes the topologicalproperties of both the bubble and the heterogeneity. Parameter B determines the extremevalues of f ( r ), its decay length, and the value of its derivative at its first-order zero. Thedensity energy of the φ -bubble of Eq. (6) is localised at the wall of the bubble [see figures1(b) and 1(c)], forming a disk of radius r o centred at the origin with decay length (cid:96) := 1 /B .For small (large) values of parameter B , the energy becomes localised in a disk with large(small) width.Numerical simulations of Eq. (3) with the soliton bubble of Eq. (6) as initial conditiondemonstrates that bubbles always collapse towards its centre if f = 0. Indeed, the curvatureof the bubble wall is proportional to the surface tension of the bubble, which under no otherrestrictions enhances its collapse. Figure 2 shows the results from numerical simulationfor D = 2, using homogeneous Neuman boundary conditions for the given values of theparameters. Let x := x and x := y be the space-coordinates in R . For all the numericalsimulations shown in this article, the time integration was performed using a fourth-orderRunge-Kutta scheme with dt = 0 .
001 and finite differences of second-order accuracy with dx = dy = 0 .
25 for the Laplace operator.Figure 2(a) shows the time evolution of the collapsing bubble profile at y = 0. The bubblearea decreases in time and the collapse is completed for t = 28 .
73. Notice that the area of5
IG. 2. (a)
Collapse of a free bubble of radius r o = 15 with µ = 1 / η = 1, γ = 0 . B = 1and f = 0. The insets show the snapshots of the bubble as a function of space for the indicatedvalues of the time. The localised burst appearing after the collapse is highlighted with a box. (b) The heterogeneous force of Eq. (7) as a function of space for B = 0 .
6, exhibiting the shape of adouble ring. (c)
The radius R ( t ) for a free bubble (blue dashed-dotted line) and a driven bubble(red solid line) as a function of time, for r o = 15, µ = 1 / η = 1, and γ = 0 .
01. The results forthe driven and free bubbles corresponds to B = 0 . B = 1, respectively. the bubble describes an oscillating behaviour during its collapse, introducing radiative wavesthat propagate towards the boundaries. These oscillations can be understood from the theoryof linear response, studying the excitation spectra of the soliton bubble (6). Similarly to one-dimensional φ kinks, the bubble spectrum of the unperturbed φ equation is composed of acontinuous spectrum and a discrete spectrum [60]. The discrete spectrum features a localisedtranslational mode –an extension of the notion of the Goldstone mode in translationally-invariant systems. In the gap between the origin of the spectral plane and the continuousspectrum, there is also an internal (localised) shape mode [65, 71, 72]. Such internal mode6s responsible for the oscillations in the bubble width, which are nonlinearly coupled to thescattering (phonon) modes in the continuous spectrum. The resulting radiative waves areobserved as small-amplitude rings with an increasing radius in time, as observed in theinsets of Fig. 2(a). These waves eventually decay in the presence of linear damping. Thesimulation also reveals a space-time localised burst just after the high-energetic collapse ofthe bubble, which is indicated with a box in Fig. 2(a). An emergent small bubble appearsafter the burst. However, it is short-lived and eventually collapses.The heterogeneous force of Eq. (7) is depicted in Fig. (2)(b), and has the shape of adouble ring: a negative ring (or trench) with mean radius r ∗ − (cid:96) and a positive ring (orhill) with mean radius r ∗ + (cid:96) , where r ∗ is the zero of the force. Between both rings, thereis a circumference of radius r ∗ where the force vanishes. Similarly to one-dimensional φ kinks [73], the first-order zeros of f ( r ) are equilibrium positions where the bubble wall canbe trapped by the heterogeneity. For a φ -bubble ( φ -bubble) there is a simple condition ofthe equilibrium position of its wall at r = r ∗ , namely, df ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = r ∗ > ( < ) , (8)provided that there are no more first-order zeros in the region [ r ∗ − (cid:96), r ∗ + (cid:96) ]. The sign of thederivative in Eq. (8) depends on the value of parameter B . In Fig. 2(b), the region r = r ∗ isstable for the wall of a bubble of phase φ = − φ = 1. Thus, suchbubbles may be sustained by the heterogeneity preventing its collapse. This observation isconfirmed in Fig. 2(c), where the radius of the bubble, R ( t ), is depicted as a function oftime for both cases, in the presence and the absence of the heterogeneity. The evolution ofthe radius of the bubble of Eq. (6) under the heterogeneous force of Eq. (7) with B = 0 . r ∗ > r .If the initial radius of the bubble coincides with the first-order zero of the heterogeneity,i.e. r o = r ∗ , there will be no damped oscillations and the bubble will remain stationary. Thisis the case in the numerical simulations shown in Fig. 3(a), where a stable and stationarybubble is observed for the given combination of parameters. The effective potential of thesystem can be written as U eff ( φ ) = U ( φ )+ φf ( r ), where U ( φ ) := ( φ − / φ potentialin its standard form. Figure 3(b) shows the effective potential for the same combination ofparameters as Fig. 3(a). The energy landscape has two slightly non-degenerated minima at7 IG. 3. A stationary bubble for B = 1, r o = 30, µ = 1 / η = 1 and γ = 0 . (a) Spatiotemporaldiagram of the profile of φ ( x, y, t ) for y = 0. Any other profile will exhibit a similar behaviour due tothe symmetry of the system. (b) Energy landscape of the system with two slightly non-degeneratedminima at ( φ , r o ) and ( φ , r o ). the phases φ < φ >
0, both with a radius r = r o = 30 .
0. Notice that both potentialwells are relatively narrow and symmetrically placed under reflections with respect to thelines r = 30 and φ = 0. Small perturbations of this bubble configuration will producedamped oscillations inside each potential well. However, the extended bubble-structure willbe stable. Notice that there is a barrier –indicated with arrows in the contour diagram atthe bottom of Fig. 3(b)– that prevents the phase φ to invade the phase φ throughout allspace.In summary, although the soliton bubble of Eq. (6) collapses when left unperturbed, thepresence of the heterogeneity may sustain the bubble and prevent its collapse, enlarging itslifetime. The heterogeneity of Eq. (7) has a singularity at the origin and is able to seedphase transitions, as will be shown later in this article.8 II. LINEAR STABILITY ANALYSIS
The effect of heterogeneities and external perturbations in the dynamics of the bubblesoliton of Eq. (6) can be investigated performing the linear stability analysis [60]. With thispurpose, the driven-damped φ equation (3) can be written as ∂ tt φ − ∇ φ + γ∂ t φ − G ( φ ) = f ( r ) , (9)where G ( φ ) := − dU/dφ . Consider a small-amplitude perturbation χ around the soliton-bubble solution φ B , i.e. φ ( r , t ) = φ B ( r ) + χ ( r, t ) , (10a) χ ( r, t ) := ψ ( r ) e λt , | χ | (cid:28) | φ B |∀ ( r , t ) , (10b)with λ ∈ C . Expanding G ( φ ) with φ ∼ φ B , one obtains G = 12 (cid:2) φ B (1 − φ B ) + χ (1 − φ B ) (cid:3) + O ( χ ) , φ ∼ φ B . (11)After substitution of Eqs. (10a) and (11) in the φ equation (9), one obtains for ψ ( r ) thefollowing spectral problem ˆ Lψ = Γ ψ, (12a)Γ := − λ ( λ + γ ) , (12b)where the linear operator ˆ L is ˆ L := −∇ + V ( r, φ = φ B ) , (13a) V ( r, φ = φ B ) = d Udφ (cid:12)(cid:12)(cid:12)(cid:12) φ = φ B . (13b)Consider the dynamics of bubbles whose shape are far from collapse. Such bubbles canbe stable and preserve their shape in time, or rather become unstable due to some nonlinearinstability. For almost-collapsed bubbles near its centre, the condition Br o (cid:28) Br o (cid:29)
1. In the later limit, r ∗ → r , ∇ (cid:39) d /dr and Eq. (12a) is analogous to the one-dimensional Schr¨odingerequation for a particle in the P¨oschl-Teller potential, V ( r ) = − (cid:0) − [ B ( r − r o )] (cid:1) . (14)9hus, the soliton bubble behaves like a potential well for linear waves. The eigenvaluesand eigenfunctions of the spectral problem (12)–(14) can be obtained exactly [73–76]. Thediscrete spectrum corresponds to soliton-phonon bound states, whose eigenvalues are givenby Γ n = B (Λ + 2Λ n − n ) − , ( n = 0 , , . . . , [Λ] − , (15a)with Λ(Λ + 1) = 32 B , (15b)where [Λ] is the integer part of Λ. Thus, for a fixed value of B , the n -th bound state existsif n < Λ . (16)This includes the translational mode Γ o and the internal shape modes Γ n with n > n -th mode, one has to find conditionsfor which the real part of λ crosses the imaginary axis. From Eq. (12b) follows that Γhas a quadratic dependence on λ with roots { , − γ } and a maximum Γ max = γ / λ max = − γ/
2. The system changes its stability properties when λ crosses the vertical axis.Near the bifurcation point ( λ = 0), the stability of the mode is determined by the sign of Γ.Thus, the n -th mode is stable if Γ n > . (17)From the previous results it is possible to predict the behaviour of the soliton bubblefor different combinations of parameters. Interestingly, the condition of Eq. (16) suggestnot only that the first internal mode of the φ bubble can disappear for some combinationsof parameters, but also that more than one internal mode can appear. Such new internalmodes are also able to store energy from an external perturbation, such as the heterogeneityof Eq. (7), and can move with the bubble wall because they are localised around the solitonbubble. Thus, the soliton bubble can be regarded as a quasi-particle with intrinsic excitationmodes. Moreover, Eq. (17) suggest that such modes can also become unstable and lead toremarkable and complex phenomena, such as those detailed in the following sections.10
20 40 60 80 100 120 140 160 180 20029.929.953030.0530.10.5 0.6 0.7 0.8 0.9 110203040 numericstheory
FIG. 4. Damped oscillations of the bubble-wall with initial radius 30 .
1, for r o = 30 . µ = 1 /
2, and η = 1. (a) Typical outcome of under-damped oscillations in the wall position (blue solid line) for B = 0 .
55 and γ = 0 .
01. The amplitude of the oscillations computed through the Hilbert transform(red dashed line) is in good agreement with an exponential decay R ( t ) = r o exp( σt ) with σ = − γ/ (b) Period of the under-damped oscillationsas a function of parameter B for r o = 30 . γ = 0 .
01. Periods from numerical simulations (+)are in good agreement with the theoretical value of the period of the traslational mode (dashed-dotted line). The theoretical curve was obtained from Eqs. (15a), (15b) and (18a) for n = 1 withno fitted parameters. (c) Position of the bubble-wall in the over-damped regime for γ = 0 . B = 0 .
6, using the theoretical model (green dashed-dotted line) and numerical simulations (bluesolid line).
IV. BUBBLE OSCILLATIONS: OVER-DAMPED AND UNDER-DAMPED MODES
The first general consequence of the linear stability analysis is that stable modes canexhibit either under-damped oscillations or over-damped decay, whereas unstable modescan be only over-damped. Indeed, from Eq. (12b) follows that λ = ( − γ ± (cid:112) γ − / γ is small enough. This leads to a transient oscillatory behaviouraround the bubble solution according to Eqs. (10).11uppose that the n -th mode exists and is stable, i.e. Γ n >
0. If γ < λ ∈ C andthe mode will perform under-damped oscillations near the bifurcation point. The frequencyand decay rate of such oscillations are given by ω n = 12 (cid:112) n − γ , (18a) σ = − γ , (18b)respectively. Several numerical simulations in the range 1 / < B < r o = 30 . R ( t ) when initially placed a small distance away from its equilibrium radius.The amplitude of the oscillations is computed by means of the Hilbert transform of R ( t ) − r o and is also shown in Fig. 4(a). These results are in agreement with an exponential decayexp( σt ) with σ given by Eq. (18b). The frequency of oscillations is also computed by meansof the fast Fourier transform of R ( t ). Figure 4(b) shows the numerically calculated periods ofoscillations for different values of B , which are in good agreement with the theoretical value T = 2 π/ω . Notice that T o decays as B increases, and increases indefinitely as B → / γ > n , then λ ∈ R and the mode decays with no oscillations near the bifurcationpoint. This corresponds to an over-damped regime, where ω n = 0 and the decay rate is σ = 12 (cid:16)(cid:112) γ − n − γ (cid:17) . (19)In this latter regime, if the bubble wall is initially placed away from the stable equilibriumpoint, it will approach its equilibrium position asymptotically without oscillations, as de-picted in Fig. 4(c). Finally, if the n -th mode is unstable, i.e. Γ n <
0, then λ ∈ R ∀ γ ∈ R .Thus, the mode grows with no oscillations ( ω n = 0) near the bifurcation point with rate σ = 12 (cid:16)(cid:112) | Γ n | − γ − γ (cid:17) . (20)In this latter case, unstable modes do not exhibit oscillations. V. VACUUM DECAY IN AN EFFECTIVE HETEROGENEOUS POTENTIAL
Section II demonstrated that a localised heterogeneity can sustain a bubble. The hetero-geneity deforms the effective energy landscape and stable phases become non-degenerated,12
IG. 5. (a)
Stable bubble for B = 0 . r o = 15 .
0. The system has two stable bound states.The insets show the snapshots of the bubble for the indicated values of the time. Vertical dashedlines indicate the initial position of the bubble wall for comparison. (b)
Bubble oscillations whenthe fundamental mode is unstable for B = 0 .
4. For both cases µ = 1 / η = 1 and γ = 0 . with a true- and a false-vacuum state. This section shows that such heterogeneities maybecome a source of nonlinear instabilities that enhances the growth of bubbles of true vac-uum. Eventually, these instabilities are strong enough to overcome the surface tension andthe bubble expands indefinitely, thus triggering phase transitions and vacuum decay.The fundamental mode ( n = 0) of the spectral problem (13)-(14) is ψ ( r ) = sech Λ [ B ( r − r o )] , (21)where Λ is given by Eq. (15b) [74–76]. This mode exists ∀ B < + ∞ , following Eq. (17).However, the stability of the fundamental mode depends on parameter B . The criticalcondition Γ = Γ ,c := 0 gives the threshold of instability of the fundamental mode, whichis B (0) c = 12 . (22)The fundamental mode is stable (Γ >
0) if
B > B (0) c , and turns unstable if B < B (0) c .Notice that Λ = 2 at the threshold of instability of the fundamental mode, which is explicitly13iven by ψ ( r ) = sech (cid:20)
12 ( r − r o ) (cid:21) . (23)For r sufficiently large, the system possesses translational invariance and Eq. (23) is theGoldstone mode. Such mode arises directly from the differentiation of bubble solution (6).Given that the derivative operator is the generator of the translation group, this fundamentalmode is a translational mode . In the case of the driven bubbles shown in Figs. 2(c) and 3,there is only one stable bound state for the given combination of parameters. Thus, thebubble is stable. Below the threshold of instability, i.e. for B < B (0) c , the bubble is expectedto expand due to the instability of the translational mode.If B < B (1) ∗ := √ /
2, the first excited mode ( n = 1) exists and is given by [74–76] ψ ( r ) = sinh[ B ( r − r o )]sech Λ [ B ( r − r o )] . (24)The threshold of instability B (1) c of this excited mode is obtained from Eq. (17) with n = 1,and gives B (1) c := (cid:32) − √ (cid:33) / . (25)The excited mode (24) exists and is stable if B (1) c < B < B (1) ∗ , and turns unstable if B < B (1) c . Notice that the stable excited mode can coexist with the stable fundamentalmode, since B (1) c < B (0) c < B (1) ∗ . This is the case of the numerical simulations shown inFig. 5(a), where the bubble wall performs stable oscillations due to the stability of modes n = 0 and n = 1.As B approaches B (0) c , the period of oscillations of the fundamental mode increases to-wards infinity –see Fig. 3(b). Bellow the threshold of instability of the fundamental mode,the radius of the bubble will describe an exponential growth with the rate given by Eq. (20)until nonlinearities become important and compensate the growth. This case is shown inthe numerical simulations of Fig. 5(b), where there is an initial burst in the bubble areaenhanced by the instability of the fundamental mode. This behaviour suggests the existenceof a repulsive bubble-heterogeneity interaction. Indeed, the derivative of f ( r ) at r = r ∗ becomes negative for B < B (0) c . Thus, according to (8), the point r = r ∗ becomes unstablefor the bubble wall. Eventually, the bubble area performs damped oscillations around anew equilibrium point, which is ruled by the compensation of the surface tension and therepulsion from the heterogeneity. 14igure 6 shows how the energy landscape is changed when crossing the threshold ofinstability of the fundamental mode. Compared to the case of Fig. 3(b), the potential wellsare broader and are no longer symmetric. Now there is a true-vacuum state with φ = φ and a false-vacuum state with φ = φ . The barrier that prevents false-vacuum to becometrue-vacuum is now significantly smaller than the barrier of Fig. 3(b). When crossing thethreshold of instability, the false-vacuum potential well passes from the region r ≥ r o [seeFig. 6(a)] to the region r ≤ r o [see Fig. 6(b)]. Conversely, the true-vacuum potential wellpasses from the region r ≤ r o to the region r ≥ r o . Thus, bubbles of the true vacuum statebecome larger below the threshold.Further decreasing the value of B in the interval ( B (1) c , B (0) c ), the true-vacuum potentialwell becomes broader, allowing the existence of bubbles of true-vacuum with increasingradius. Thus, a natural question is whether vacuum decay can be enhanced by the instabilityof the translational mode. Indeed, Fig. 7 shows a true-vacuum bubble expanding throughoutthe entire simulation space for B = 0 .
16, converting false-vacuum into true-vacuum. Thecurvature of the bubble decreases as the radius of the bubbles increases, and the dynamics
FIG. 6. Effective heterogeneous potential for (a) B = 0 . B > B (0) c ) and (b) B = 0 . B < B (0) c ).For both cases r o = 15 . µ = 1 / η = 1.
15f the bubble is eventually dominated by the instability of the translational mode. Due tothe finite computational domain in the numerical simulations of Fig. 7(a) and Fig. 7(b),the bubble eventually collides with the boundaries and some small-amplitude waves reflectstowards the origin. These waves eventually dissipate in the presence of damping, resultingin a flat configuration with the space filled with phase φ .It is important to remark that not all bubbles of phase φ grow for the combination ofparameters shown in Fig. 7. Figure 7(c) also evidences the existence of another potentialwell for bubbles of phase φ around r = 0. Thus, true-vacuum bubbles with a relativelysmall radius ( r o < r ∗ ) fall into this potential well and thus collapses towards the origin. Thisobservation was confirmed by numerical simulations. In such a case, since r o < r ∗ , both therepulsive interaction with the heterogeneity and the surface tension enhances the collapseof the bubble. VI. BUBBLE NUCLEATION INSIDE A PRECURSOR BUBBLE
The first excited mode turns unstable if
B < B (1) c , where B (1) c is given by Eq. (25).As previously discussed, these excited modes are associated with shape-modes, which areintrinsic excitation modes of the bubble. Previous sections demonstrated that the instabilityof the translational mode produces bubble expansion or bubble collapse. However, morecomplex phenomena can occur when energy is stored in the internal (shape) modes of thebubble. Shape modes are responsible for variations in the kink-like profile of the bubble walland may affect the global dynamics of the bubble. A phenomenon termed as shape-modeinstability occurs when a shape mode becomes unstable [58, 68–70, 77]. In this scenario, theshape of the wall is no longer stable and the bubble breaks up.Figure 8 shows the nucleation of a bubble of phase φ inside an expanding bubble of phase φ due to a shape-mode instability. Initially, the φ -bubble expands due to the instabilityof the fundamental mode. However, the first internal mode of the bubble absorbs enoughenergy from the heterogeneity to become unstable and break the structure. Most of theabsorbed energy is used for the nucleation of a new stable bubble of phase φ inside theprecursor φ -bubble. The remaining energy after the nucleation turns into small-amplituderadiative waves that dissipates in the presence of damping. The final configuration of thefield is a bubble of phase φ sustained by the heterogeneity in a space filled with phase φ .16 IG. 7. Vacuum decay for B = 0 . r o = 15 . µ = 1 / η = 1 and γ = 0 . (a) Snapshots fromnumerical simulations taken with steps ∆ t = 25. Time increases from left to right and from topto bottom. The initial and final values of time are indicated in the figure. (b) Time evolution ofthe x profile of the true-vacuum bubble for y = 0. (c) Energy landscape with a true vacuum stateinside a broad potential well with r ≥ r o . Figure 9(a) shows the evolution in time of the x -profile of the bubble breakup process.The oscillations of the internal modes are stronger when compared to previous cases, anda new φ -bubble is generated from the origin. Following Eq. (8), the radius r = r ∗ is astable position for the wall of the newborn φ -bubble, and an attractive interaction betweenits wall and the heterogeneity is on sight. Thus, the φ -bubble expands and performsdamped oscillations around the stable equilibrium radius r = r ∗ . At t → ∞ , the new17 IG. 8. Nucleation of a φ -bubble inside an expanding φ -bubble due to an internal-mode insta-bility. Snapshots from numerical simulations taken with steps ∆ t (cid:39) . B = 0 . µ = 1 / η = 1, and γ = 0 .
01. Time increases from left to right and from top to bottom. IG. 9. Numerical simulations for B = 0 . µ = 1 / η = 1, and γ = 0 . (a) Spatiotemporaldiagram at y = 0 and (b) heterogeneous effective potential for the bubble breakup shown in Fig. 8. bubble becomes stationary. The corresponding heterogeneous potential landscape is shownin Fig. 9(b). There is also a potential well for bubbles of phase φ around r = 0, and nowthe stable phases φ and φ are slightly non-degenerated.Further decreasing the value of B below the threshold B (1) c , more excited modes canappear and store energy from the heterogeneity. Such modes can also become unstable for B sufficiently small. The dynamics of the bubble become more complex in the presence ofa mixture of stable and unstable internal modes. VII. BUBBLE NUCLEATION UNDER THERMAL NOISE
Numerical simulations shown in Sections IV, V and VI demonstrated bubble dynamicsand nucleation using the φ -bubble of Eq. (6) as the initial condition. The purpose ofusing such initial condition was to reproduce the conditions assumed in the linear stabilityanalysis of Section III, whose predictions have been confirmed by numerical results. Finally,this section shows that these results are robust to changes in the initial conditions, even in19he presence of thermal fluctuations.Consider the following driven-damped φ system ∂ tt φ − ∇ φ + γ∂ t φ − (cid:0) − φ (cid:1) φ = f ( r ) + ˜ ξ ( r , t ) , (26)where f ( r ) is given by Eq. (7), ˜ ξ ( r , t ) is a space-time white noise source with (cid:104) ˜ ξ ( r , t ) (cid:105) = 0and (cid:104) ˜ ξ ( r , t ) ˜ ξ ( r (cid:48) , t (cid:48) ) (cid:105) = 2 k B T δ ( t − t (cid:48) ) δ ( r − r (cid:48) ). Here, T has the interpretation of temperatureand k B is the Boltzmann’s constant [78]. The φ potential U ( φ ) has two degenerated phases φ = − φ = 1 separated by a barrier at φ = 0. Both the heterogeneity f ( r ) andthe thermal noise can deform such potential and break the degeneracy of the minima of U ( φ ). To check the robustness of the results of this article on initial conditions, considerthe following initial configurations of the field φ given by a random distribution of values inspace, φ ( r , t = 0) = φ i + δ ˜ (cid:15) ( r ) , (27)where i = 0 , , U ( φ ), ˜ (cid:15) is a delta-correlated spacedependent random variable with (cid:104) ˜ (cid:15) (cid:105) = 0, and δ = 0 . i = 0 and the given valuesof parameters. In this case, the initial condition of Eq. (27) is a small-amplitude randomdistribution of values around the unstable equilibrium state φ . After a transient of wavesemitted from the heterogeneity and reflected from the boundaries, the system finally decaysto the potential well of phase φ throughout all space. This is expected given that φ is thetrue-vacuum state. At t = 300 the field is in a noisy configuration inside the potential wellcorresponding to the phase φ . Notice that the linear response of the field to the ring-shapedheterogeneity is appreciable at the end of the simulation.For the case i = 1, Eq. (27) is a small-amplitude random distribution of values aroundthe stable equilibrium phase φ . Given that the noise strength is small and unable to inducea phase transition by itself, the field φ remains in such potential well, as demonstrated inFig. 10(b). Noise dissipates fast in this case, and the final configuration is the linear responseof the field to the heterogeneity, almost free of noise.For the case i = 2, the initial condition is a random distribution around the meta-stablephase φ , and the outcome is different. The heterogeneity seeds a phase transition, and abubble of phase φ is nucleated around the heterogeneity. Figure 10(c) shows how the areaof the φ -bubble initially grows and performs damped oscillations around the equilibrium20 IG. 10. Evolution of random initial conditions under thermal fluctuations for δ = 0 . B = 0 . µ = 1 / η = 1, and γ = 0 .
01. Initial conditions for each case are shown in the left column. Thecentral column shows the spatiotemporal evolution of the x -profile of φ at y = 0. The right columnshows the field configuration at the end of the simulation. (a) If the initial condition is a noisydistribution around the barrier of the φ potential, the system decays towards the phase φ . (b) If the initial condition is noisy around the phase φ , the system remains in the correspondingpotential well. (c) If the initial condition is noisy around the phase φ , a φ -bubble is nucleatedaround the heterogeneity. radius. In conclusion, a heterogeneity under the effects of thermal noise can nucleate bubblesin this two-dimensional φ system. 21 III. CONCLUSIONS
In summary, this article investigated a two-dimensional φ model exhibiting bubble nu-cleation around heterogeneities. Bubbles are studied as radially symmetric heteroclinicsolutions interpolating two stable phases of the underlying potential. Through the solutionof an inverse problem, a nonlinear field theory with exact solutions is proposed. The dy-namics and stability of soliton-bubbles in such a theory is investigated under the influence ofthe heterogeneity. Numerical and analytical results have shown that bubbles can be nucle-ated and sustained by the heterogeneity, even in the presence of thermal noise. The modelpredicts the formation of oscillating bubbles, vacuum decay, and the nucleation of bubblesinside a precursor expanding-bubble, depending on the combination of parameters associ-ated with topological properties of the heterogeneity. Numerical simulations have showngreat agreement with analytical results. These results may be useful in the study of simplemodels of vacuum decay seeded by heterogeneities, as well as the nucleation and stability oflocalised structures in the Universe with a long life-time containing new physics. ACKNOWLEDGMENTS
The author thanks M. Ahumada for her advice on the article. This work was partiallysupported by Universidad de Santiago de Chile through the POSTDOC DICYT projectnumber 042031GZ POSTDOC and by ANID FONDECYT/POSTDOCTORADO/3200499. [1] R. Gregory, I. G. Moss, and B. Withers, “Black holes as bubble nucleation sites,”
J. HighEnergy Phys. , vol. 2014, p. 81, Mar 2014.[2] P. Burda, R. Gregory, and I. G. Moss, “Vacuum metastability with black holes,”
Journal ofHigh Energy Physics , vol. 2015, no. 8, p. 114, 2015.[3] P. Burda, R. Gregory, and I. G. Moss, “Gravity and the stability of the higgs vacuum,”
Phys.Rev. Lett. , vol. 115, p. 071303, Aug 2015.[4] P. Burda, R. Gregory, and I. G. Moss, “The fate of the higgs vacuum,”
Journal of High EnergyPhysics , vol. 2016, p. 085017, 2016.
5] M. S. Turner and F. Wilczek, “Is our vacuum metastable?,”
Nature , vol. 298, no. 5875, p. 633,1982.[6] E. J. Weinberg,
Classical solutions in quantum field theory: Solitons and Instantons in HighEnergy Physics . Cambridge University Press, 2012.[7] T. Kibble, “Some implications of a cosmological phase transition,”
Physics Reports , vol. 67,no. 1, pp. 183 – 199, 1980.[8] I. G. Moss, “Black-hole bubbles,”
Phys. Rev. D , vol. 32, pp. 1333–1344, Sep 1985.[9] W. A. Hiscock, “Can black holes nucleate vacuum phase transitions?,”
Phys. Rev. D , vol. 35,pp. 1161–1170, Feb 1987.[10] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, “Black holes initiate false-vacuum decay,”
Phys. Rev. D , vol. 43, pp. R3112–R3116, May 1991.[11] J. A. Gonz´alez, A. Bellor´ın, M. A. Garc´ıa- ˜Nustes, L. Guerrero, S. Jim´enez, J. F. Mar´ın,and L. V´azquez, “Fate of the true-vacuum bubbles,”
Journal of Cosmology and AstroparticlePhysics , vol. 2018, pp. 033–033, jun 2018.[12] R. Gregory, K. M. Marshall, F. Michel, and I. G. Moss, “Negative modes of Coleman–De Luccia and black hole bubbles,”
Phys. Rev. D , vol. 98, p. 085017, Oct 2018.[13] L. Cuspinera, R. Gregory, K. M. Marshall, and I. G. Moss, “Higgs vacuum decay from particlecollisions?,”
Phys. Rev. D , vol. 99, p. 024046, Jan 2019.[14] C. Cheung and S. Leichenauer, “Limits on new physics from black holes,”
Phys. Rev. D ,vol. 89, p. 104035, May 2014.[15] M. Nach, “Hawking’s radiation of sine-Gordon black holes in two dimensions,”
Int. J. Mod.Phys. A , vol. 34, no. 16, p. 1950086, 2019.[16] J. Guckenheimer and P. Holmes,
Nonlinear Oscillations, Dynamical Systems, and Bifurcationsof Vector Fields . New York: Springer-Verlag, 1986.[17] P. L. Christiansen and P. S. Lomdahl, “Numerical study of 2+1 dimensional sine-Gordonsolitons,”
Physica D , vol. 2, no. 3, pp. 482–494, 1981.[18] J. Gonz´alez, A. Marcano, B. Mello, and L. Trujillo, “Controlled transport of solitons andbubbles using external perturbations,”
Chaos Soliton Fract. , vol. 28, no. 3, pp. 804–821, 2006.[19] A. G. Castro-Montes, J. F. Marn, D. Teca-Wellmann, J. A. Gonzlez, and M. A. Garca-ustes, “Stability of bubble-like fluxons in disk-shaped josephson junctions in the presence ofa coaxial dipole current,”
Chaos: An Interdisciplinary Journal of Nonlinear Science , vol. 30, o. 6, p. 063132, 2020.[20] T. Vachaspati, Kinks and domain walls: An introduction to classical and quantum solitons .Cambridge University Press, 2006.[21] M. Peyrard and T. Dauxois,
Physique des Solitons . Savoirs actuels, 2004.[22] N. Manton and P. Sutcliffe,
Topological solitons . Cambridge University Press, 2004.[23] I. Barashenkov, A. Gocheva, V. Makhankov, and I. Puzynin, “Stability of the soliton-likebubbles,”
Physica D , vol. 34, no. 1, pp. 240–254, 1989.[24] I. V. Barashenkov and E. Y. Panova, “Stability and evolution of the quiescent and travellingsolitonic bubbles,”
Physica D , vol. 69, no. 1, pp. 114–134, 1993.[25] A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, “Solitons in conducting polymers,”
Rev. Mod. Phys. , vol. 60, pp. 781–850, Jul 1988.[26] M. Peyrard and A. R. Bishop, “Statistical mechanics of a nonlinear model for dna denatura-tion,”
Phys. Rev. Lett. , vol. 62, pp. 2755–2758, Jun 1989.[27] J. A. Gonz´alez and B. de A Mello, “Kink catastrophes,”
Physica Scripta , vol. 54, pp. 14–20,jul 1996.[28] J. A. Gonz´alez and F. A. Oliveira, “Nucleation theory, the escaping processes, and nonlinearstability,”
Phys. Rev. B , vol. 59, pp. 6100–6105, Mar 1999.[29] J. Gonzlez, A. Bellor´ın, and L. Guerrero, “How to excite the internal modes of sine-gordonsolitons,”
Chaos, Solitons and Fractals , vol. 17, no. 5, pp. 907 – 919, 2003.[30] M. A. Garc´ıa- ˜Nustes, J. F. Mar´ın, and J. A. Gonz´alez, “Bubblelike structures generated byactivation of internal shape modes in two-dimensional sine-gordon line solitons,”
Phys. Rev.E , vol. 95, p. 032222, Mar 2017.[31] J. F. Mar´ın, “Generation of soliton bubbles in a sine-gordon system with localised inhomo-geneities,”
Journal of Physics: Conference Series , vol. 1043, p. 012001, jun 2018.[32] M. Chirilus-Bruckner, C. Chong, J. Cuevas-Maraver, and P. G. Kevrekidis,
Sine-Gordon Equa-tion: From Discrete to Continuum , pp. 31–57. Cham: Springer International Publishing, 2014.[33] P. G. Kevrekidis and J. Cuevas-Maraver,
A Dynamical Perspective on the ϕ model: Past,present and future. Springer, 2019.[34] J. A. Gonz´alez and J. A. Ho(cid:32)lyst, “Behavior of ϕ kinks in the presence of external forces,” Phys. Rev. B , vol. 45, pp. 10338–10343, May 1992.[35] S. Fl¨ugge,
Practical quantum mechanics . Springer Science & Business Media, 2012.
36] J. A. Ho(cid:32)lyst and H. Benner, “Universal family of kink-bearing models reconstructed from aP¨oschl-Teller scattering potential,”
Phys. Rev. B , vol. 43, pp. 11190–11196, May 1991.[37] J. A. Gonz´alez, M. A. Garc´ıa- ˜Nustes, A. S´anchez, and P. V. E. McClintock, “Hawking-likeemission in kink-soliton escape from a potential well,”
New J. Phys. , vol. 10, p. 113015, nov2008.[38] J. A. Gonz´alez, A. Bellor´ın, and L. E. Guerrero, “Internal modes of sine-gordon solitons inthe presence of spatiotemporal perturbations,”
Phys. Rev. E , vol. 65, p. 065601, Jun 2002.[39] G. Lythe,
Stochastic Dynamics of φ Kinks: Numerics and Analysis , pp. 93–110. Cham:Springer International Publishing, 2019.[40] R. Gregory, I. G. Moss, and B. Withers, “Black holes as bubble nucleation sites,”
J. HighEnergy Phys. , vol. 2014, p. 81, Mar 2014.[41] P. Burda, R. Gregory, and I. G. Moss, “Vacuum metastability with black holes,”
Journal ofHigh Energy Physics , vol. 2015, no. 8, p. 114, 2015.[42] P. Burda, R. Gregory, and I. G. Moss, “Gravity and the stability of the higgs vacuum,”
Phys.Rev. Lett. , vol. 115, p. 071303, Aug 2015.[43] P. Burda, R. Gregory, and I. G. Moss, “The fate of the higgs vacuum,”
Journal of High EnergyPhysics , vol. 2016, p. 085017, 2016.[44] M. S. Turner and F. Wilczek, “Is our vacuum metastable?,”
Nature , vol. 298, no. 5875, p. 633,1982.[45] E. J. Weinberg,
Classical solutions in quantum field theory: Solitons and Instantons in HighEnergy Physics . Cambridge University Press, 2012.[46] T. Kibble, “Some implications of a cosmological phase transition,”
Physics Reports , vol. 67,no. 1, pp. 183 – 199, 1980.[47] I. G. Moss, “Black-hole bubbles,”
Phys. Rev. D , vol. 32, pp. 1333–1344, Sep 1985.[48] W. A. Hiscock, “Can black holes nucleate vacuum phase transitions?,”
Phys. Rev. D , vol. 35,pp. 1161–1170, Feb 1987.[49] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, “Black holes initiate false-vacuum decay,”
Phys. Rev. D , vol. 43, pp. R3112–R3116, May 1991.[50] J. A. Gonz´alez, A. Bellor´ın, M. A. Garc´ıa- ˜Nustes, L. Guerrero, S. Jim´enez, J. F. Mar´ın,and L. V´azquez, “Fate of the true-vacuum bubbles,”
Journal of Cosmology and AstroparticlePhysics , vol. 2018, pp. 033–033, jun 2018.
51] R. Gregory, K. M. Marshall, F. Michel, and I. G. Moss, “Negative modes of Coleman–De Luccia and black hole bubbles,”
Phys. Rev. D , vol. 98, p. 085017, Oct 2018.[52] L. Cuspinera, R. Gregory, K. M. Marshall, and I. G. Moss, “Higgs vacuum decay from particlecollisions?,”
Phys. Rev. D , vol. 99, p. 024046, Jan 2019.[53] C. Cheung and S. Leichenauer, “Limits on new physics from black holes,”
Phys. Rev. D ,vol. 89, p. 104035, May 2014.[54] M. Nach, “Hawking’s radiation of sine-Gordon black holes in two dimensions,”
Int. J. Mod.Phys. A , vol. 34, no. 16, p. 1950086, 2019.[55] J. Guckenheimer and P. Holmes,
Nonlinear Oscillations, Dynamical Systems, and Bifurcationsof Vector Fields . New York: Springer-Verlag, 1986.[56] P. L. Christiansen and P. S. Lomdahl, “Numerical study of 2+1 dimensional sine-Gordonsolitons,”
Physica D , vol. 2, no. 3, pp. 482–494, 1981.[57] J. Gonz´alez, A. Marcano, B. Mello, and L. Trujillo, “Controlled transport of solitons andbubbles using external perturbations,”
Chaos Soliton Fract. , vol. 28, no. 3, pp. 804–821, 2006.[58] A. G. Castro-Montes, J. F. Marn, D. Teca-Wellmann, J. A. Gonzlez, and M. A. Garca-ustes, “Stability of bubble-like fluxons in disk-shaped josephson junctions in the presence ofa coaxial dipole current,”
Chaos: An Interdisciplinary Journal of Nonlinear Science , vol. 30,no. 6, p. 063132, 2020.[59] T. Vachaspati,
Kinks and domain walls: An introduction to classical and quantum solitons .Cambridge University Press, 2006.[60] M. Peyrard and T. Dauxois,
Physique des Solitons . Savoirs actuels, 2004.[61] N. Manton and P. Sutcliffe,
Topological solitons . Cambridge University Press, 2004.[62] I. Barashenkov, A. Gocheva, V. Makhankov, and I. Puzynin, “Stability of the soliton-likebubbles,”
Physica D , vol. 34, no. 1, pp. 240–254, 1989.[63] I. V. Barashenkov and E. Y. Panova, “Stability and evolution of the quiescent and travellingsolitonic bubbles,”
Physica D , vol. 69, no. 1, pp. 114–134, 1993.[64] A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, “Solitons in conducting polymers,”
Rev. Mod. Phys. , vol. 60, pp. 781–850, Jul 1988.[65] M. Peyrard and A. R. Bishop, “Statistical mechanics of a nonlinear model for dna denatura-tion,”
Phys. Rev. Lett. , vol. 62, pp. 2755–2758, Jun 1989.
66] J. A. Gonz´alez and B. de A Mello, “Kink catastrophes,”
Physica Scripta , vol. 54, pp. 14–20,jul 1996.[67] J. A. Gonz´alez and F. A. Oliveira, “Nucleation theory, the escaping processes, and nonlinearstability,”
Phys. Rev. B , vol. 59, pp. 6100–6105, Mar 1999.[68] J. Gonzlez, A. Bellor´ın, and L. Guerrero, “How to excite the internal modes of sine-gordonsolitons,”
Chaos, Solitons and Fractals , vol. 17, no. 5, pp. 907 – 919, 2003.[69] M. A. Garc´ıa- ˜Nustes, J. F. Mar´ın, and J. A. Gonz´alez, “Bubblelike structures generated byactivation of internal shape modes in two-dimensional sine-gordon line solitons,”
Phys. Rev.E , vol. 95, p. 032222, Mar 2017.[70] J. F. Mar´ın, “Generation of soliton bubbles in a sine-gordon system with localised inhomo-geneities,”
Journal of Physics: Conference Series , vol. 1043, p. 012001, jun 2018.[71] M. Chirilus-Bruckner, C. Chong, J. Cuevas-Maraver, and P. G. Kevrekidis,
Sine-Gordon Equa-tion: From Discrete to Continuum , pp. 31–57. Cham: Springer International Publishing, 2014.[72] P. G. Kevrekidis and J. Cuevas-Maraver,
A Dynamical Perspective on the ϕ model: Past,present and future. Springer, 2019.[73] J. A. Gonz´alez and J. A. Ho(cid:32)lyst, “Behavior of ϕ kinks in the presence of external forces,” Phys. Rev. B , vol. 45, pp. 10338–10343, May 1992.[74] S. Fl¨ugge,
Practical quantum mechanics . Springer Science & Business Media, 2012.[75] J. A. Ho(cid:32)lyst and H. Benner, “Universal family of kink-bearing models reconstructed from aP¨oschl-Teller scattering potential,”
Phys. Rev. B , vol. 43, pp. 11190–11196, May 1991.[76] J. A. Gonz´alez, M. A. Garc´ıa- ˜Nustes, A. S´anchez, and P. V. E. McClintock, “Hawking-likeemission in kink-soliton escape from a potential well,”
New J. Phys. , vol. 10, p. 113015, nov2008.[77] J. A. Gonz´alez, A. Bellor´ın, and L. E. Guerrero, “Internal modes of sine-gordon solitons inthe presence of spatiotemporal perturbations,”
Phys. Rev. E , vol. 65, p. 065601, Jun 2002.[78] G. Lythe,
Stochastic Dynamics of φ Kinks: Numerics and Analysis , pp. 93–110. Cham:Springer International Publishing, 2019., pp. 93–110. Cham:Springer International Publishing, 2019.