Josephson oscillations of chirality and identity in two-dimensional solitons in spin-orbit-coupled condensates

We investigate dynamics of two-dimensional chiral solitons of semi-vortex (SV) and mixed-mode (MM) types in spin-orbit-coupled Bose-Einstein condensates with the Manakov nonlinearity, loaded in a dual-core (double-layer) trap. The system supports two novel manifestations of Josephson phenomenology: one in the form of persistent oscillations between SVs or MMs with opposite chiralities in the two cores, and another one demonstrating robust periodic switching (identity oscillations) between SV in one core and MM in the other, provided that the strength of the inter-core coupling exceeds a threshold value. Below the threshold, the system creates composite states, which are asymmetric with respect to the two cores, or suffer the collapse. Robustness of the chirality and identity oscillations against deviations from the Manakov nonlinearity is investigated too. These dynamical regimes are possible only in the nonlinear system. In the linear one, exact stationary and dynamical solutions for SVs and MMs of the Bessel type are found. They sustain Josephson self-oscillations in different modes, with no interconversion between them.

In its linear form, the Josephson effect only gives rise to plasma waves in superconducting junctions.The most significant modes in the junctions are topological solitons (fluxons) supported by the nonlinear Josephson relation between the tunneling current and phase difference across the junction.Both classical [7]- [10] and quantum [11] fluxons, as well as superfluxons, i.e., kink excitations in fluxon chains [12], were studied theoretically and experimentally.
New possibilities for realization of dynamical effects in BEC are offered by binary (pseudo-spinor) condensates with spin-orbit coupling (SOC) between their components [45]- [48].In particular, SOC affects the Josephson tunneling between annular condensates [49].Stable 2D solitons, which include vortex components, were predicted in the SOC system of the Rashba type [50] with attractive nonlinear interactions [53].In terms of optics, these interactions are represented by self-and crossphase-modulation (SPM and XPM) terms in the underlying system of coupled Gross-Pitaevskii equations.While in the absence of SOC, attractive cubic terms in 2D equations always lead to collapse [52] (i.e., the system does not have a ground state), the interplay of the attractive SPM and XPM terms with SOC gives rise to ground states, in the form of solitons of the semi-vortex (SV, alias halfvortex [54]) or mixed-mode (MM) types, provided that, respectively, the SPM nonlinearity is stronger or weaker than its XPM counterpart.SVs are composed of a vortex in one component and zero-vorticity soliton in the other, while MMs combine vortical and zero-vorticity terms in both components.In the experiment, such states can be created by using helical laser beams that resonantly couple to one component only, and thus transfer the angular momentum onto that component, without exciting the other one [55][56][57].Two or several solitons in the layers' plane can be created too, by means of an initially applied cellular in-plane potential, which cuts the condensate into lateral fragments, and is lifted afterwards [58].
If the SV soliton is the ground state, the MM is an unstable excited one, and vice versa.Similar results were reported [59] for more general SOC systems, of the Rashba-Dresselhaus [60] type.Two-dimensional SV solitons exist in the form of two different chiral isomers ( right-and left-handed ones), with vorticity sets in their components being, respectively, (S + , S − ) = (0, 1) or (−1, 0) (mirror images of each other).Similarly, there exist two different chiral forms of MMs, which are introduced below in Eq. (22).
In the experimental realization of SOC in binary BEC, two components are represented by different hyperfine states of the same atom, with nearly equal strengths of the SPM and XPM interactions, suggesting one to consider the Manakov nonlinearity [61], with equal SPM and XPM coefficients.In such a case, the system is invariant with respect to rotation of the pseudo-spinor wave function in the plane of its two components.The 2D system combining SOC and the Manakov nonlinearity gives rise to an additional soliton family, which embeds the SV and MM solitons into a continuous set of intermediate states [53].The family is a degenerate one, in the sense that all solitons with a fixed total norm share common values of the energy and chemical potential.The family is dynamically stable against small perturbations, but structurally unstable, as a deviation from the SPM = XPM condition breaks the intermediate states, keeping only the SV and MM solitons as robust modes.
The subject of this work is a junction formed by a pair of Josephson-coupled 2D layers (cores), each one carrying spin-orbit-coupled BEC.In the experiment, a two-layer setup can be realized by loading the condensate in two adjacent valleys of a deep 1D optical lattice, which illuminates the setting in the perpendicular direction (see, e.g., Ref. [62]).In a general form, a dual-core 2D BEC system was introduced in Ref. [63].In that work, 2D solitons, built of components in the two layers, were stabilized by a spatially-periodic in-plane potential, while SOC was not considered.The problem addressed in Ref. [63] was spontaneous breaking of the inter-core symmetry in the solitons, but not Josephson oscillations.Here, we consider oscillations between Josephson-coupled 2D SOC condensates, of both SV and MM types, in the dual-core system.In its linear version, Bessel-shaped exact solutions are found in an analytical form, for stationary states and Josephson oscillatory ones alike.These solutions demonstrate solely intrinsic oscillations in the two components of the condensate, which do not mix states of the SV and MM types, nor different isomers in each type.In the full nonlinear system, solutions for selftrapped states are obtained in a numerical form.They readily demonstrate robust periodic oscillations between components of the 2D solitons with opposite chiralities in the two layers.Furthermore, Josephson-type identity oscillations, i.e., periodic mutual interconversion between solitons' components of the SV and MM types in the two layers, are reported too.
The rest of the paper is structured as follows.The model is formulated in Section II, in which basic types of the considered states are introduced too, including new solutions for the Bessel-shaped SV and MM modes in the linear system.Results for the Josephson oscillations are collected in Section III.It includes exact solutions obtained in the linear system, and a summary of the systematic numerical analysis of Josephson oscillations in 2D solitons of different types in the full nonlinear system.The numerical results make it possible to identify robust regimes of oscillations of the chirality (right left) and identity (SV MM) between the coupled layers.The paper is summarized in Section IV, where we also give estimates of the predicted effects in physical units, and discuss directions for the further work; one of them may be the use of beyond-mean-field effects [51] for the stabilization of the 2D system against the critical collapse [52], when its norm exceeds the respective threshold value.

II. THE MODEL: COUPLED GROSS-PITAEVSKII EQUATIONS
A. The single-layer spin-orbit-coupled (SOC) system For the single effectively two-dimensional layer, the system of coupled Gross-Pitaevskii equations for components φ ± of the pseudo-spinor wave function, coupled by the spin-orbit interaction of the Rashba type, is written as [53,59] where t and x, y are scaled time and coordinates, the SPM and SOC coefficients (the latter one is the coefficient in front of the first-order spatial derivatives) are set to be 1 by scaling, and γ is the relative XPM strength.We will chiefly address the Manakov nonlinearity, with γ = 1, which is close to the experimentally relevant situation [45]- [48]; effects of deviation from the Manakov's case are considered too.Equations ( 1) and ( 2) do not include an external trapping potential, which is necessarily present in the experiment, but its effect on stable solitons is negligible [53].
Stationary self-trapped states of the pseudo-spinor condensate (solitons) with chemical potential µ < 0 are looked for as with the 2D norm (4) Two isomers of solitons of the SV type, with rightand left-handed chiralities, are defined, respectively, by vorticity sets (S + , S − ) = (0, +1) and S+ , S− = (−1, 0) in the two components.Numerical solutions for them are produced, severally, by the following inputs: which are written in polar coordinates (r, θ), with real A 1,2 and α 1,2 > 0. These inputs are natural, as Eqs.(1) and ( 2) are fully compatible with the substitution of SV ansätze with opposite chiralities [53],  25) and ( 26), are shown as functions of norm ratio R φ [see Eq. ( 24)], for the continuous family of stable SOC solitons, numerically generated by input (20), with fixed norm N φ = 4, in the single-core system.Linear dependence L φ (R φ ) follows Eq. ( 27).The energy and chemical potential of all solutions belonging to the degenerate family are E φ = −2.4 and µ = −0.87.Cross-sections of two components of the 2D solitons, marked in (a), are displayed for the following values of integral quantities ( 24) and ( 25), ( 26): where radial form-factors f (r), g(r) obey equations µ + 1 2 µ + 1 2 More general soliton modes are introduced below.
At the edge of the semi-infinite gap, µ = −1/2, Eq. ( 14) gives a single solution with ρ = 1, instead of two, produced inside the gap at µ > −1/2.However, the loss of one solution is compensated by the existence of an additional one, precisely at the edge of gap.Its components are Exact linear SV states with the opposite chirality are generated from expressions ( 12)-( 16) as per Eq. ( 8).Next, MMs states, as exact solutions of the linear SOC system, are constructed from the SVs pursuant to Eqs. (22) and (23).It is also relevant to mention that, for the Rashba SOC replaced by the Dresselhaus form [60], the exact linear solutions for the SV and MM modes are essentially the same as found here for the Rashba system.
In addition to solution (12), which represents the basic SV state in the linear system, it is possible to construct exact solutions for excited states, obtained by injection of extra integer vorticity ∆S = 1, 2, ... in both components of the SV state.Such solutions are obtained replacing Eqs. ( 12) and ( 15), (16), severally, by while Eqs.( 13) and ( 14) keep the same form as above, as well as the possibility to construct the SV with the opposite chirality and MMs, according to Eqs. ( 8) and ( 22), (23).
It is relevant to mention that similarly defined excited states of SV and MM solitons are unstable in the nonlinear system, unlike the ground states with ∆S = 0 [53].On the other hand, the excited-state solitons may be made stable in a model with self-trapping provided by the local repulsive nonlinearity growing fast enough from the center to periphery [68].

C. Soliton families
The Manakov's nonlinearity makes it possible to construct a general family of solitons, including SVs, MMs, and intermediate states.The creation of such states is initiated by a combination of inputs ( 5) and ( 6) with the opposite chiralities, where the weight factor takes values −1 ≤ M ≤ 1.Because integrals of cross-products, generated by ansätze (5) and ( 6) vanish, the norm of input (20), defined as per Eq. ( 4), does not depend on M , hence this input gives rise to a family of solitons with the same norm.
The choice of M = +1/ √ 2 or −1/ √ 2 in Eq. ( 20) corresponds to MMs with opposite chiralities.Stationary MM solitons of the Manakov system are produced by the following exact ansatz, which is compatible with the underlying system of equations ( 1) and (2) [cf.Eq. ( 7), that produces the exact ansatz for SV modes]: Here f and g are the same real functions which are introduced above as solutions to Eqs. ( 9) and (10).Another chiral MM isomer is produced by a different ansatz, with radial form-factors f and g obeying the same equations ( 9) and (10).The exact linear relations between the MM and SVs in Eqs. ( 22) and ( 23) are admitted by the symmetry of the Manakov nonlinearity.These relations entail an equality for the total norms (4): To the best of our knowledge, the exact MM ansätze represented by Eqs. ( 22)- (23) were not reported earlier.
The solitons are characterized by the norm ratio between the components, and by the angular momentum per particle in each component and in the entire system, where the angular-momentum operator is Lz = −i(x∂/∂y − y∂/∂x) ≡ −i∂/∂θ, and the norms are taken as per Eq. ( 4).For the family generated by ansatz (20), a typical relation between L and R φ is displayed in Fig. 1(a).The linear L φ (R φ ) dependence, observed in the figure, is explained by the fact that the input in the form of Eqs. ( 5)- (20), leads, taking into regard the orthogonality of φ + and φ where R (SV) φ is given by Eq. ( 24) for the SV soliton corresponding to M = 1 in Eq. ( 20)) [in Fig. 1(a), a numerically computed value is R Both R φ and L φ vanish for solitons of the MM type, due to the overall symmetry between the components in this state.Terminal points of the curves in Fig. 1 (b-f).In particular, SVs are shown in panels (b) and (f), while (d) displays an MM.The stability of the entire family was established in direct simulations.The solitons of all types exist in ranges 0 < −µ < ∞ and 0 < N φ < N coll ≈ 5.85, where N coll is the well-known threshold for the onset of the critical collapse, predicted by the single Gross-Pitaevskii equation [52].In fact, the stability of the 2D spin-orbitcoupled solitons is predicated on the fact that their total norm takes values N < N coll , thus securing them against the onset of the collapse [53].

D. The Josephson-coupled dual-core system
The pair of Josephson-coupled layers, each carrying pseudo-spinor BEC, are modeled by the system of equations for four components of wave functions, φ ± in one layer and ψ ± in the other: cf. Eqs. ( 1) and ( 2), where κ is the Josephson-coupling strength, which is defined to be positive.Note that the Josephson interaction does not linearly couple components φ ± to ψ ∓ , as ones with opposite subscripts represent two distinct atomic states, and the tunneling between the parallel layers does not include a mechanism which would lead to mutual transformation of different atomic states.
The system of Eqs. ( 28)-( 31) conserves the total norm, N φ + N ψ , where N ψ is a counterpart of the φ norm for the ψ-layer, see Eq. ( 4).Other dynamical invariants of the system are the total energy, and the total angular momentum, N φ L φ +N ψ L ψ , cf.Eq. (26).

Chirality and identity oscillations
In the framework of the full nonlinear system, we first address the evolution initiated by an input composed of a stable single-layer SV soliton, with vorticities (S + , S − ) = (0, 1), in one core, and its chiral (left-handed) isomer with (S + , S − ) = (−1, 0) in the other, which are created, at t = 0, as per ansätze presented in Eqs. ( 7) and (8).The respective stationary single-layer solutions were produced by means of the accelerated imaginarytime-evolution method [64], and their stability was then tested by simulations of Eqs. ( 1) and ( 2) in real time.Simulations of Eqs. ( 28)- (31) with such an input produce a robust regime of periodic transmutations displayed in Fig. 2(a-d).Each quarter period, SVs in both cores switch into their chiral counterparts, returning to the initial state, but with the opposite sign, each half period.Thus, chiralities of the two SVs periodically flip, remaining opposite in the two cores, while the corresponding density patterns oscillate.Distributions of densities of the four involved components, displayed at t = 0 in panels (e-f), and at t = 10 in (i-l) (the latter time is close to 0.8 periods of the Josephson oscillations) help to understand the evolution of the density patterns in the course of the oscillations.Further, Fig. 2(m) demonstrates Josephson oscillations of the angular momentum in the same dynamical state, with the momentum per particle in the cores oscillating between L φ,ψ = ±0.4 and ∓0.4.
Note that the chirality-switching Josephson-oscillation mode is possible only in the nonlinear system, as the exact linear Josephson-oscillation solution (35) admits solely oscillations that do not mix different chiralities.Indeed, the chirality flipping implies generation of new angular harmonics, which is not possible without nonlinearity.
Next, we consider Josephson oscillations between rightand left-handed chiral isomers of the MM type, which are created, at t = 0, in the two layers, as per ansätze (22) and (23).A typical numerical solution, displayed in Fig. 3, features stable periodic switching between the MM isomers in each layer.It also demonstrates spatiotemporal helicity, i.e., clockwise rotation of the density distributions (unlike nonrotating SV patterns in Fig. 2).The input with the opposite relative sign between the MM isomers in two layers gives rise to a similar stable dynamical state, but with rotation of the density profiles in the opposite, counter-clockwise, direction.Again, it is relevant to stress that flipping of the MM chirality in the course of the Josephson oscillations is only possible in the nonlinear system.
Recall that the single-core SOC system with the Manakov nonlinearity maintains the family of 2D solitons which are intermediate between the SV and MM states.Such states correspond to in Eq. ( 20), |R φ | = 0.2 or 0 in Fig. 1, and accordingly, eigenvalues L φ different from ±0.4 and 0, see Eq. ( 26).Systematic simulations of Josephson oscillations in pairs of chiral isomers of the intermediate states also reveal robust periodic chirality oscillations, in a combination with the rotation of the density profiles (the spatiotemporal helicity), similar to what is displayed above in Figs. 2  and 3 for the states of the SV and MM types.The results for the intermediate states are not displayed in a separate figure, as the overall pictures seem quite close to ones in Figs. 2 and 3).Furthermore, at all values of inter-core coupling κ the frequency of the oscillations and rotation is exactly equal to κ, cf. the exact solution of the linear system given by Eqs. ( 28)- (31).Although, as said above, the chirality oscillations represent a nonlinear effect, the Josephson frequency is not affected by the nonlinearity, due to its "isotopic" invariance with respect to the rotation of the pseudo-spinor wave function in the plane of its two components.The next step is to consider Josephson oscillations between inputs corresponding to SV in one core and MM, with the same norm, in the other.In this case, the identity of the 2D solitons periodically switches between the SV and MM types in each core [see Figs.4(a-d) and (m)], therefore this stable dynamical state may be naturally categorized as identity oscillations, coupled to the clockwise helicity rotation, cf.Figs. 2 and 3.This regime, which is again possible solely in the nonlinear system, cf.exact solution (35) obtained in the linear limit (which actually corresponds to the system strongly dominated by the inter-core coupling), takes place if the strength of the Josephson interaction exceeds a certain threshold value, κ > κ thr .The dependence of the threshold on the total norm in one layer, N , is shown in Fig. 4(n); recall that the establishment of the robust chirality oscillations in the dual-core SV and MM solitons, considered above, does not require κ to be larger than any finite threshold value.
To explain the latter feature, we note that, as it follows from Eqs. ( 28)-( 31) and (32), the interplay of the kinetic energy, nonlinearity, SOC, and Josephson interaction between the cores predicts scaling κ thr ∼ N at small N .On the other hand, the above-mentioned onset of the collapse in the single-core system at N = N coll implies divergence of the system's sensitivity to perturbations at N → N coll , hence κ thr also diverges in this limit.These qualitative arguments are corroborated by the numerically found dependence κ thr (N ) in Fig. 4(n).
At κ < κ thr , the inter-core coupling is too weak to prevent the occurrence of spontaneous symmetry breaking between the cores.Accordingly, additional numerical results demonstrate, for values of N which are relatively small in comparison with N coll (namely, in an interval of N < N crit < N coll ) the existence of asymmetric composite solitons, with different amplitudes in the two cores.In the adjacent interval of N crit < N < N coll , the collapse takes place at κ < κ thr .On the other hand, if the Josephson coupling is strong enough, with κ > κ thr , it forestalls the onset of the collapse at all values of N smaller than N coll , by keeping the total norm equally distributed between the two cores (for instance, the identityoscillation dynamical regime, displayed in Fig. 4, is stable at N = 4 < N coll ≈ 5.85).

Structural stability of the oscillation regimes
The fact that the Manakov's form of the nonlinearity is approximate in experimental settings makes it necessary to test effects of deviation from it [γ = 1 in Eqs.(1) and (2)], i.e., structural stability of the chirality-and identityoscillation dynamical regimes.The result, illustrated by Figs.5(a,b) for γ = 1 ± 0.05, is that the chirality oscillations of the SV soliton persist in the course of dozens of periods (which is sufficient for the experimental observation), and then gradually detune from the regular regime.
Additional simulations (not shown here in detail) suggest similar conclusions concerning the effect of the deviation from the Manakov's case on chirality and identity oscillations of MM solitons.As concerns the intermediate states, defined above as per Eq. ( 36), the deviation from the Manakov's case for them is more disruptive, as, even in the single-core system, taking γ = 1 causes evolution of such inputs towards SVs or MMs, for γ < 1 or γ > 1, respectively [53].
Stability of the Josephson oscillations against variations of the initial conditions was tested too, by taking the input in the form of a single-core SV soliton in one layer, and its chiral isomer in the other, but with different norms.As shown in Fig. 5(c), dephasing of the resulting chirality oscillations sets in at times exceeding ∼ 10 periods.

IV. CONCLUSION
The objective of this work is to introduce and theoretically elaborate a setting for the study of Josephson oscillations of the chirality and species type (identity) of 2D solitons in the double-layer trap, loaded with a spin-orbit-coupled condensate, including self-and crossattractive nonlinearity (SPM and XPM) in its components.It is known that, in the Manakov's case of SPM = XPM, the single-layer system supports a broad family of composite 2D solitons, which includes dynamically and structurally stable ones of the SV (semi-vortex) and MM (mixed-mode) types, each existing in two chiral isomers, left-and right-handed ones.In addition to that, the Manakov's nonlinearity admits the existence of the intermediate family of solitons, between the SV and MM states, which are dynamically stable but structurally unstable against deviation from the condition SPM = XPM.All these states exist with the norm in each component taking values smaller than the well-known threshold value necessary for the onset of the critical collapse, N < N coll , We consider Josephson oscillations initiated by inputs in the form of SV, MM, or intermediate-type solitons with equal norms and opposite chiralities, originally created in the coupled cores (layers), or SV in one core and MM in the other.In the former case, the system features robust chirality oscillations at all values of the inter-core coupling, κ.In the latter case, persistent periodic switching between the SV and MM species (identity oscillations) occur at κ > κ thr , where κ thr increases with the growth of the condensate's norm, while at κ < κ thr the system develops states with unequal amplitudes in the cores at relatively small values of N , or collapses at larger N .In all cases, the oscillation frequency is equal to κ. Deviation of the nonlinearity from the Manakov's form, as well as a difference in the norms of the inputs in the two cores, leads to slowly developing dephasing of the Josephson oscillations.These regimes are nonlinear ones, while the exact solutions reported here for Josephson oscillations in the linear system admit solely self-oscillations of SVs and MMs modes with the Bessel spatial shape, without switching between different chiralities or mode types.In fact, the exact Bessel-shaped linear modes are new stationary solutions for the singlelayer system too.
With typical values of the relevant physical parameters (see, e.g., Ref. [70]), such as the scattering length of the attractive inter-atomic interactions ∼ −0.1 nm, transverse trapping frequency ∼ 300 Hz, and the gap of width ∼ 5 µm separating two layers, the expected number of atoms in the stable solitons, which are capable to feature the Josephson oscillations, is in the range of N atom ∼ 10 3 − 10 4 , and the oscillation frequency ∼ 20 Hz (while the experiment time may extend up to 1 s, making the observation of many cycles of the oscillations quite feasible).At essentially larger values of N atom , the scaled norm will exceed value N coll , which will trigger the onset of the collapse.On the other hand, it was recently predicted [51,71], and demonstrated experimentally in various setups [72]- [76], that 3D and quasi-2D self-trapped matter-wave states may be stabilized, in the form of soliton-like quantum droplets, by beyond-meanfield (Lee-Huang-Yang) effects of quantum fluctuations around the mean-field states.In the present context, this possibility suggests to consider the dual-core SOC system in the case of N > N coll , that may be stabilized by the Lee-Huang-Yang effect.In this connection, it is relevant to mention that the stabilization of 2D spin-orbit-coupled solitons by this effect in the single-layer system was analyzed in Ref. [77].
As another extension of the present work, it may be interesting to consider Josephson oscillations in 2D spatiotemporal optical solitons carried by dual-core planar waveguides and stabilized by a SOC-emulating mechanism, viz., temporal dispersion of the inter-core coupling coefficient [78,79].Another potentially interesting point is the study of intrinsic vibrations of the composite solitons, initiated by a separation of their components in the two layers.

FIG. 2 .
FIG. 2. (Color online) Numerically simulated Josephson oscillations of chirality (with random-noise perturbation at the 1% amplitude level, added for testing stability of the dynamical regime), initiated by the input composed of a stable singlecore SV in one layer, and its left-handed counterpart in the other, see Eqs. (7) and (8).Panels (a,b) and (c,d) display the evolution of the pseudo-spinor components in the first and second cores, respectively.The norm of the input in each layer is N = 4 (hence the total norm in the two-layer system is 8), and the Josephson coupling constant is κ = 0.5.Panel (e)-(h) and (i)-(l) exhibit 2D density distributions of the components at, respectively, t = 0 and t = 10.(m) The respective oscillations of angular momenta per particle in both layers , L φ and L ψ , see Eq. (26).

yFIG. 3 .
FIG. 3. (Color online) The same as in Fig. 2 but for the input in the form of MMs with opposite chiralities in the two cores, and equal norms in them, N = 4.The angular momentum per particle remains equal to zero in each core, due to the symmetry between the components of the pseudo-spinor wave function in the MM states.In addition to the Josephson oscillations of the MM chirality, the figure also demonstrates the spatiotemporal helicity in the oscillating soliton, see the text.Panel (e)-(h) and (i)-(l) exhibit 2D density distributions of the components at, respectively, t = 0 and t = 10.

FIG. 4 .
FIG. 4. (Color online) Panels (a-m): the same as in Fig. 2, but for identity oscillations, initiated by the input in the form of SV and MM solitons with equal norms, N = 4, in the coupled cores.(n) The minimum (threshold) value of the inter-core coupling κ, which is necessary for the stability of the identity oscillations, vs. the norm of the 2D soliton in each core, N .

FIG. 5 .
FIG. 5. (Color online) Simulations similar to those displayed for the SV soliton in Fig. 2, but with γ = 1.05 in (a) and 0.95 in (b).The chirality oscillations remain robust in the course of 40 and 20 oscillation periods, respectively.(c) The chirality oscillations of the SV solitons, with different initial norms in the two cores, N ψ = 0.95N φ = 3.8.The oscillations stay undisturbed in the course of 10 periods.