## Abstract

We review the general features of particles, waves and solitons in dynamical cavities formed by oscillating cavity mirrors. Considered are the dynamics of classical particles in one-dimensional geometry of a dynamical billiard, taking into account the non-elastic collisions of particles with mirrors, the (quasi-energy) states of a single quantum particle in a potential well with periodically oscillating wells, and nonlinear structures, including nonlinear Rabi oscillations, cavity optical solitons and solitons of Bose–Einstein condensates, in dynamical cavities or traps.

## 1. Introduction

To date, such essentially nonlinear phenomena as solitons have been better studied in optics [1] owing to the availability of high-power laser radiation and the relative simplicity of nonlinear optical schemes. These solitons are of two types: (i) conservative solitons [2], which represent the balance between linear spreading of wave packets and their nonlinear focusing, and (ii) dissipative solitons, which result from the balance between energy input and output in the region of localization [3–6].

The main schemes that support spatial optical dissipative solitons—the cavity solitons—are shown in figure 1*a*,*b*. In high-quality cavities with multiple radiation trips, it is possible to achieve a high concentration of radiation energy and, correspondingly, high medium nonlinearity under resonance conditions. For the scheme of a driven nonlinear interferometer (figure 1*a*), the medium inside the cavity can be passive (without gain). Energy gain is due to external coherent holding radiation transmitted by the partially transparent mirror. Resonances occur when the holding radiation frequency is close to the cavity eigenfrequencies. In laser schemes (figure 1*b*), the external holding radiation is not necessary, and energy supply is due to coherent or incoherent pumping resulting in intracavity medium laser gain. The cavity mirrors can be non-transparent. More recently, owing to progress in the formation of such macroscopic quantum objects as Bose–Einstein condensates (BECs) [7], similar schemes have been studied for solitons of combined light–matter (polariton) waves [8,9].

One difficulty for realization of cavity solitons of pure matter waves is the absence of effective semitransparent mirrors. This can be resolved using cavities with oscillating mirrors, as in figure 1*c*. In this case, the energy gain for the field inside the cavity is due to the kinetic energy of the mirrors, which can be non-transparent. Similar schemes are known for electromagnetic radiation [10,11], including current optomechanics [12,13], as well as for nonlinear acoustics [14]. The analysis of nonlinear structures in the scheme of such a dynamical billiard needs clarification of the general nature of its nonlinear dynamics. At this point, it is useful to revisit the problem of Fermi [15] with stochastic acceleration of particles colliding with moving bodies and that of Ulam [16] with particles bouncing between one motionless and one oscillating wall.

Below we review the following aspects of the problem. In §2, we present the nonlinear dynamics of a classical particle in a one-dimensional dynamical billiard, revealing stationary, with conserved particle kinetic energy, and quasi-chaotic regimes. Compared are the dynamics with elastic and inelastic collisions of particles with the walls. In §3, studied are states, mainly quasi-energetic ones, of a single quantum particle in a potential well with periodically oscillating walls. Section 4 is devoted to nonlinear structures, including nonlinear Rabi oscillations and cavity solitons (§4*a*) and ‘longitudinal’ (§4*b*) and ‘transverse’ solitons in dynamical traps. A general conclusion is given in §5.

## 2. Classical particles in a dynamical billiard

Let us consider the one-dimensional motion of a classical particle in a ‘dynamical billiard’ or cavity formed by two barriers: a motionless wall, coordinate *z*=0, and an oscillating wall, coordinate *z*=*z*_{w}=*L*(*t*), where *t* is time (in the scheme of figure 1*c*, the left mirror is motionless and the right mirror oscillates). For the sake of definiteness, we fix
2.1
The modulation depth *μ* is assumed to be small, *μ*^{2}≪1. Introducing dimensionless time *Ωt*→*t*, one can set *Ω*=1. This problem was considered first by Ulam [16] in connection with the Fermi acceleration effect [15]; subsequent results are reviewed by Lichtenberg & Lieberman [17] and by Loskutov [18]; a recent study of the Ulam problem with stochastic wall motion was performed by Gelfreich *et al*. [19]. Note that Ulam [16] considered a sawtooth modulation of the wall's position, and the prevailing approximation in subsequent papers is neglecting shifts of the wall's position when calculating the time of collisions. These two assumptions do not allow one to describe properly the stability of periodic and quasi-periodic regimes of particle reflections from the walls. For the present review, it is important also to compare the dynamics of classical particles for elastic and inelastic collisions with the walls [20] and that of solitons in similar schemes [21].

According to equation (2.1), the extrema of the oscillating wall's position correspond to the coordinates *z*_{0}=*z*_{w}(0)=*L*_{0}(1+*μ*) (maxima for *μ*>0 and minima for *μ*<0). First, let collisions of the particle with the walls be elastic. Then, if the velocity of the particle colliding with an oscillating wall at the moment *t* is *v*, the velocity of the reflected particle is (the dot over a value denotes its temporal derivative). Correspondingly, depending on the instantaneous wall velocity, the kinetic energy of a particle with *v*>0 can increase () or decrease () due to the collision, and the system considered is open. The dynamics of the particle can be described by recurrence relations for times of collisions *t*_{n} and corresponding velocities *v*_{n}, *n*=1,2,3,… [20].

Fixed points of the system of governing equations correspond to regimes with conserved particle kinetic energy. They are possible if all collisions are with an (instantaneously) motionless wall (at time moments *t*=*πN*,*N*=1,2,…). For the first type of fixed points, the collisions occur at time moments *t*=2*πN*, *N*=1,2,… and the particle velocity is *v*_{0,N}=*z*_{0}/*πN*=*L*_{0}(1+*μ*)/*πN*. This case includes variants of collisions at minimum (*μ*<0) and maximum (*μ*>0) cavity length. The second type of fixed points corresponds to particles with velocity *v*_{0,M}=*L*_{0}/*π*(*M*−1/2), *M*=1,2,…, reflecting alternately from the oscillating wall at its maximum and minimum deviations. The degenerate case corresponds to motionless particles (*v*=0) located in the interval 0<*z*<*L*_{0}(1−|*μ*|).

To perform the linear stability analysis of the periodic regimes, let us introduce small deviations of the velocity *δv*_{n}=*v*_{n}−*v*_{0,N} and collision times *δt*_{n}=*t*_{n}−2*πN*, *n*=1,2,…, from corresponding unperturbed values. For the first type of fixed points, in the linear over *δv*_{n} and *δt*_{n} approximation, one gets the following recurrence relations:
2.2
The eigenvalues λ of the corresponding transformation matrix are determined by the following equation:
2.3
Solutions to this quadratic equation are
2.4
The condition of asymptotic stability |λ_{1,2}|^{2}<1 cannot be fulfilled because λ_{1}λ_{2}=1.

For *μ*<0, the value *a* is negative, the eigenvalues are real and not coincident and the maximum eigenvalue λ_{1}>0. This means aperiodic instability.

For *μ*>0, the value *a*>0. If 0<*a*<4, then the radicand in equation (2.4) is negative, the two eigenvalues are complex conjugates, and
2.5
This means neutral stability. In this case, the linear analysis describes also long-term particle dynamics if the initial deviations from the unperturbed values are small,
2.6
Here the (small) amplitude *w* and phase *φ* are arbitrary.

For *a*>4, both eigenvalues are real, and one of them λ_{1}<−1. Then we have oscillatory instability. The stability boundary is determined by the condition *a*_{cr}=4, therefore
2.7
Then *μ*_{cr,1}=0.113, *μ*_{cr,2}=0.026, *μ*_{cr,3}=0.011 and *μ*_{cr,4}=0.007.

The direct simulation of the dynamics of a particle colliding with oscillating and motionless walls confirms these conclusions. For *μ*=0.01, there are three stable fixed points of the first type (*N*=1,2 and 3). Their neutral stability means that if deviations from the unperturbed values are initially sufficiently small, then they remain small during the next evolution. This is illustrated by figure 2*a* (*N*=1) where one can see quasi-periodic variations of the particle velocity *v* (also a dimensionless value, because we fix *L*_{0}=1). The simulations agree very well with the analytic expression (equation (2.6)). In figure 2*b* is presented the quasi-periodic dynamics near the fixed point of the second type (*M*=1). The modulation depth depends on the initial deviations. However, if the initial deviations are fairly large, the dynamics becomes chaotic (figure 2*c*). This figure shows also that, if the particle initial velocity is small, the mean value of the particle energy increases with time. For overcritical values *μ*>*μ*_{cr,1}, there are no stable fixed points, and the dynamics is chaotic (figure 2*d*). Detailed characterization of deterministic chaos regimes [22] needs special consideration. In the limit of large initial velocity of the particle, there are quasi-periodic variations of the velocity near its initial value.

Now, let the collisions of the particles with the oscillating wall be not exactly elastic, with the velocities of the incident *v* and reflected *v*^{′} particle connected by the relation . The value 1−*q*^{2} is the measure of dissipation (inelasticity); the previous purely elastic case corresponds to *q*=1. In this case, there are also stable periodic regimes, and now they are asymptotically stable (stable attractors, small initial deviations decay with time) (figure 3*a*,*b*). For large initial deviations, chaotic dynamics takes place, as in the previous case (figure 3*c*).

The results presented in this section indicate that classical particles in the dynamical billiard with elastic collisions have dynamics intermediate between conservative and dissipative ones. The neutral, in contrast to asymptotic, stability of fixed points is a feature of conservative systems. On the other hand, the increase of average energy for evolution with small initial energy is characteristic for dissipative systems. The dynamics of the particle resembles the dynamics of conservative systems [22], though the particle energy is not conserved after collisions. This could be explained by the fact that this energy can both increase and decrease due to collisions, and the energy is conserved at the average over the period of wall position modulation. For non-elastic collisions, we have asymptotic stability of periodic regimes, but chaotic dynamics takes place again for large deviations of the initial values from the steady-state values.

## 3. A single quantum particle in a dynamical trap

For quantum particles in a trap, essential are their wave nature and their energy spectrum discreteness. If a classical particle can be motionless with an arbitrary position between oscillating walls, a quantum particle cannot be at rest, and its wave function should oscillate because of the walls' oscillations. Additionally, owing to the wave features and corresponding ‘dispersion’ and ‘diffraction’, the wave packet of an initially localized quantum particle diffuses with time. This diffusion can be compensated if we consider, not a single particle, but a large number of interacting particles under critical temperature—the BEC [7]; however, the BEC is the subject of §4. The diffusion and motion of atomic wave packets in a trap with oscillating walls were studied by Steane *et al*. [23] and Saif *et al*. [24]. Here, we are interested in a different, purely quantum, aspect connected with the discreteness of the quantum object's energy spectrum. Below we review the results of our paper [25] on quasi-energy states of a single quantum particle in a dynamical trap.

The wave function *ψ* of a single quantum particle in a one-dimensional trap obeys the Schrödinger equation,
3.1
with the coordinate *z*, time *t*, the reduced Planck constant , the particle mass *m*_{p} and the trap potential *U*. For an infinite potential well with oscillating barriers, this equation is applied for *L*_{left}(*t*)<*z*<*L*_{right}(*t*), where *U*=0, and the boundary conditions are
3.2
In the case of motionless walls (*L*_{left}=0, *L*_{right}=*L*_{0}=const., modulation depth *μ*=0), solutions to equations (3.1) and (3.2) are represented by the discrete energy spectrum
3.3
For periodic (harmonic) oscillations of barriers with the same period, *T*=2*π*/*Ω*, for the left and right barriers, there is a set of states with definite quasi-energy *ε*,
3.4
Periodic in time functions *u*_{ε}(*x*, *t*) can be decomposed into Fourier series,
3.5
Then functions *χ*_{ε,l}(*z*) obey the ordinary differential equations
3.6
with the boundary conditions (3.2). Owing to problem linearity, the general solution is given by a linear superposition of partial solutions corresponding to states with different quasi-energies.

For small modulation depth, the quasi-energy states can be found by perturbation theory, and the lowest-order solution is given by equations (3.3). Owing to the non-parabolic (rectangular) shape of the trap potential, the energy spectrum is highly non-equidistant, . Therefore, if the modulation frequency is close to the frequency of transition between the two levels *n* and *m*,
3.7
then only these two levels are subjected to the excitation due to modulation, and the scheme is reduced to the two-level one [26,27]. The amplitudes of the resonance states *a*_{n} and *a*_{m} obey the following equations:
3.8
For pure quasi-energy states, the temporal dependence of the amplitudes is . Under the resonance conditions (equation (3.7)), there are splitting of quasi-energies and Rabi oscillations with periodic exchange of the resonance levels' populations. An additional feature that is beyond the two-level approximation is the possibility of resonances of higher orders corresponding to ‘multiphoton’ transitions. More detail can be found in [25,28].

## 4. The Bose–Einstein condensate in a dynamical trap

The BEC represents a macroscopic object that can be characterized by a single wave function *ψ* obeying the nonlinear Gross–Pitaevskii equation [7]:
4.1
The boundary conditions to equation (4.1) have the form of equations (3.2) in our case. This equation is valid for weakly non-ideal atomic gases at sufficiently low temperature. The nonlinearity parameter *U*_{0} depends on the external magnetic field and can be either positive or negative. A similar mean-field equation for exciton or polariton condensates in semiconductors is known as the Keldysh equation [29].

### (a) Two-level scheme and nonlinear Rabi oscillations

For the resonance conditions (equation (3.7)), and neglecting transverse effects, the consideration can also be reduced to the two-level scheme [25,28]. In this case, the governing equations for the resonance states' amplitudes have the following form:
4.2
In the linear case, *U*_{0}=0, they coincide with equations (3.8). It is possible to solve equations (4.2) analytically [28]. The main results are illustrated by figure 4*a*–*c* (exact resonance) and *d*–*f* (non-zero frequency detuning). The real amplitudes of the resonance states, *A*_{n} and *A*_{m}, are periodic functions of time, with the ‘Rabi period’ depending on the initial conditions (cf. figure 4*b* with 4*c*, and also figure 4*e* with 4*f*). As for the phase difference of the resonance states' amplitudes *Φ*, it can be either a periodic (figure 4*b*,*c*,*f*) or a monotonic (figure 4*e*) function of time.

It is convenient to analyse the solutions to equations (4.2) with the help of the phase plane of this system where the solutions are represented by closed lines (*A*_{n}, *Φ*) and to treat *A*_{n} as the polar radius and *Φ* as the polar angle of the phase plane. For normalization, the amplitudes of the resonance levels are connected by the relation ; therefore, the state is characterized fully by the values *A*_{n} and *Φ*, neglecting an inessential constant shift of the phase. Trajectories pass through each point of the phase plane inside the circle with radius *A*_{n}=1. At *A*_{n}=1 (only the *n*th level is occupied) and *A*_{n}=0 (only the *m*th level is occupied), there are singularities of corresponding equations.

Fixed points of the phase plane can be found when equalizing to zero the derivatives in equations (4.2). They correspond to the quasi-energy states. In the case of *exact resonance*, one can see from figure 4*a* that the phase plane is divided by separatrix *D* into two cells. In each cell, trajectories are closed lines disposed concentrically around the corresponding fixed point: *B* in the right cell and *C* in the left cell. The trajectories correspond to periodic oscillations with time of both amplitude *A*_{0}(*t*) and phase difference *Φ*(*t*), as shown in figure 4*b*,*c*. The period of the oscillations (the Rabi period) depends on the initial conditions.

For *non-zero detuning*, the phase plane has a more complicated structure. In figure 4*d*–*f*, shown are results for fairly large detuning *δω*=1.5. Now separatrix *D* does not include the coordinate origin 0, and separatrix *E* appears that passes through 0. The temporal dependence of the phase difference, *Φ*(*t*), can be of two types, depending on the initial conditions: (i) periodic, as in the previous case, and (ii) monotonic, which can be decomposed into a sum of a periodic function and a component linear in time. The phase plane is divided into three cells (figure 4*d*). The left cell—a ‘half-moon’—is bounded by the left semicircle and the separatrix *D*. It is of the same type as in the previous case, i.e. it consists of closed trajectories wrapping around the fixed point *C*; for trajectories in this cell, both amplitude *A*_{n} and phase difference *Φ* vary periodically with time (the first type of trajectories). The same are features of trajectories inside the second separatrix *E*. However, in the cell bounded by separatrices *D* and *E* and the right semicircle *C*, trajectories wrap on the beginning of coordinates 0 and correspond to periodic temporal variation of amplitude *A*_{n} and monotonic variation of phase difference *Φ* (the second type of trajectories). These two types of trajectories are illustrated in figure 4*e*,*f*. They are similar to trajectories of a classical pendulum with angle periodic variation for small initial velocities and monotonic variation for large velocities. Beyond the two-level approximation, the direct numerical solution of the one-dimensional Gross–Pitaevskii equation (equation (4.1)) reveals some additional phenomena studied in [28].

### (b) ‘Longitudinal’ solitons

In this section, we follow [21]; we do not use here the resonance approximation. In one-dimensional geometry, equation (4.1) takes the form
4.3
For *U*_{0}=0, it coincides with the Schrödinger equation (equation (3.1)). Mathematically, equation (4.3) is the well-known nonlinear Schrödinger equation [2]. In infinite space, without a trap, known are solutions to equation (4.3) describing modulational instability, cnoidal waves, bright and dark solitons, and oscillating localized structures—breathers. For a dynamical trap with finite length (equations (3.2)), it is possible to find quasi-energy states, as in §3 [25]; however, owing to the problem nonlinearity, the superposition principle is not applicable in this case.

When a moving bright soliton collides with an ideal motionless mirror, its kinetic energy does not change. In fact, one can replace this problem by the collision of a soliton with its antiphase mirror image, the summed field at the mirror location being zero; this problem is solved by the inverse scattering transform method [2]. If the mirror moves with some constant velocity, the problem is reduced to the previous one using Galilean transformation symmetry; depending on the sign of the mirror velocity, the soliton can be accelerated or decelerated. For periodic oscillations of mirrors, increase or decrease of soliton kinetic energy depends periodically on the oscillation phase at the moment of collision, as for point classical particles (we consider below the case of narrow solitons with dimensionless width *w*≪1 and oscillations with small frequency *Ω*≪*w*^{−2}). Additionally, as well as for classical particles, sufficiently slow solitons can collide with the same mirror several times repeatedly before they move to the cavity centre. Results of simulations of soliton collisions with a single oscillating mirror are illustrated in figure 5. For large initial velocity, *V* _{i}=1, the dependence is very close to sinusoidal. With decrease of the initial velocity, this dependence deforms. The dip near *δt*=0.5 for *V* _{i}=0.04 is because, for these conditions, the soliton collides with the mirror not a single time, but twice before it moves away from the mirror.

Now let us consider the case of a two-mirror cavity. Note that classical particles can be at rest in any place inside the cavity not available for oscillating mirrors, and for any non-zero initial velocity they travel along the whole cavity. Opposite are features of bright solitons: even with motionless mirrors, a soliton's position necessarily oscillates, if it is not disposed in the cavity centre with zero velocity; generally, a soliton with small initial velocity oscillates in the vicinity of the cavity centre (figure 6). This is due to the interaction of the soliton's tails with the mirrors even when the soliton width is much less than the cavity length.

Simulations show that, in traps with oscillating barriers, the solitons survive even after a large number of reflections from barriers. Similar to the classical case, there are stable periodic and quasi-periodic regimes of soliton reflections from oscillating mirrors (figure 7*a*). The velocity modulation depth depends on the soliton initial velocity, as well as for elastic reflections of classical particles. It is interesting that, for small initial velocity, the soliton dynamics is chaotic (figure 7*b*), as well as for a classical particle (see figure 2*c*,*d*).

### (c) ‘Transverse’ solitons

Generalization of the resonance approach governing equations (equations (4.2)) to the (2+1)D-geometry (two transverse coordinates and time) gives [30]
4.4
Here, Δ_{⊥}=∂^{2}/∂*x*^{2}+∂^{2}/∂*y*^{2} is the transverse Laplacian, and *x* and *y* are the transverse coordinates. According to equations (4.4), the total number of particles is conserved for localized structures,
4.5
Next, equations (4.4) have the Galilean symmetry: if functions *A*_{n,m}(*x*,*y*,*t*) give a solution to equations (4.4), then there is a family of solutions with an arbitrary transverse velocity *V* ,
4.6
Evidently, equations (4.4) are invariant to a phase shift of both amplitudes, , *δΦ*=const., and to shifts of transverse coordinates, (*x*,*y*)→(*x*+*δx*,*y*+*δy*). For exact resonance, *δΩ*=0, equations (4.4) are also invariant to the replacement *n*→*m*.

In the case of exact resonance, there are solutions of equations (4.4) with equal populations of the two resonance levels, *a*_{m}=±*a*_{n}≡*a*. Then, after replacement *a*=*b* exp[±i(−1)^{m−n}*t*] and using dimensionless values, equations (4.4) are reduced to the standard (two-dimensional) nonlinear Schrödinger equation
4.7
A wider class of solitons described by equations (4.4) was studied in [30].

In one-dimensional geometry, well known are bright, sech-type solitons and their collisions [2]. However, even in the resonance case (equation (4.7)), we deal not with scalar but with vector solitons, because both in-phase, *a*_{m}=*a*_{n}, and antiphase, *a*_{m}=−*a*_{n}, solitons are described by this equation. Below we present results of computer simulation of these solitons' collisions [31]. More exactly, we will consider here only the case of exact resonance and collisions of a soliton moving with velocity *V* with an initially motionless in-phase soliton; the colliding solitons have different widths and, correspondingly, different maximum amplitudes.

The collision scenario depends strongly on the (relative) velocity of approach of the solitons. Below, two limiting cases are considered with fairly small (figures 8 and 9, *V* =0.1) and large (figures 10 and 11, *V* =1) velocities.

For small velocities (*V* =0.1), the solitons initially approach, reaching a minimum distance Δ*x*=5.3, and then move away (figure 8*a*). It is possible to say that the second soliton is reflected from the first soliton pushing it. As figure 8*b* shows, the total population of the resonance levels of the two solitons changes after the collision, increasing for the first soliton and decreasing for the second soliton. The populations of the first (antiphase) soliton separate levels begin to oscillate; it transforms into a long-living breather or oscillon (figure 8*c*). These oscillations are not so pronounced for the second (in-phase) soliton (figure 8*c*). Temporal evolution of the profiles of the population, both lower (|*a*_{n}|^{2}) and upper (|*a*_{m}|^{2}), is illustrated in figure 9. To distinguish between in-phase and antiphase solitons, we present here also profiles of value *B*=−|*a*_{n}−*a*_{m}|^{2}/4; for in-phase solitons *B*=0.

For large velocity, *V* =1, the collision scenario is different (figures 10 and 11). First, the initially moving soliton traverses the motionless soliton practically without change of its velocity; the initially motionless soliton moves in the same direction with small velocity (figure 10*a*). Second, after collision, both solitons begin to oscillate, transforming to breathers (figure 10*c*).

## 5. Conclusion

The results presented confirm the efficiency of excitation of various nonlinear structures inside dynamical billiards—cavities or traps with oscillating mirrors (barriers). In such schemes, the power supply is due to kinetic energy of the mirrors, which can be non-transparent.

A single classical particle in a one-dimensional dynamical billiard with periodic modulation of the barriers' position has, in a certain range of parameters, one or a number of stable regimes with conserved kinetic energy. With increase of modulation depth, these regimes become unstable, and the particle dynamics becomes chaotic.

The billiard serves as a trap for single quantum particles, and simultaneously its oscillations excite the particle to higher energy levels. For periodic oscillations, there is a discrete set of particle quasi-energies. When the oscillation frequency is close to a frequency of transition between quasi-energy levels, resonance occurs, with strong Rabi oscillations of resonance levels population.

An atomic BEC under the resonance conditions has two types of dynamics: (i) with periodic variation of populations and resonance states phase difference, and (ii) with periodic variation only of populations and monotonic variation of the phase difference. The period of Rabi oscillations depends strongly on the initial distribution of populations. Neglecting the transverse distribution, ‘longitudinal’ Schrödinger-like solitons exist in the dynamical trap, and their dynamics can be regular or chaotic, as in the case of classical particles. For transversely distributed schemes, various types of solitons exist, including vector spatial solitons. Their collisions can change soliton type, transforming, for example, a stationary soliton to a breather.

The solitons presented are related to localized structures found in vibrated granular media [32,33]. Their nature in the cases considered is intermediate between conservative and dissipative solitons, because the energy input for objects inside the billiard is nearly balanced by their energy losses on average over the modulation period. Definitely dissipative structures can be found for active polariton condensates where pumping compensates dissipation [9]. Owing to mathematical equivalence, the results for BEC structures can be applied to optical beams and pulses in a planar waveguide with Kerr nonlinearity of the medium and periodic oscillations of the reflecting boundaries.

Many related topics, including quantum manifestation of classical dynamical chaos and multidimensional solitons, deserve separate consideration. Similar problems in electrodynamics [10,11] and acoustics [14,34] are also of much current interest.

## Funding statement

This work was partially financially supported by the Programme of the Russian Academy of Sciences ‘Fundamental problems of nonlinear dynamics in mathematical and physical sciences’ and by the Government of the Russian Federation, grant no. 074-U01. The current stage of our research is supported by the Russian Scientific Foundation, grant no. 14-12-00894.

## Acknowledgements

The authors are grateful to Yu.V. Rozhdestvenskii for helpful discussions.

## Footnotes

One contribution of 19 to a Theme Issue ‘Localized structures in dissipative media: from optics to plant ecology’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.