Breathing modes, quartic nonlinearities and effective resonant systems
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2020), 034, 14 pages Breathing Modes, Quartic Nonlinearitiesand Effective Resonant Systems
Oleg EVNIN †‡†
Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand ‡ Theoretische Natuurkunde, Vrije Universiteit Brussel and the International Solvay Institutes,Brussels, Belgium
E-mail: [email protected]
Received February 20, 2020, in final form April 14, 2020; Published online April 23, 2020https://doi.org/10.3842/SIGMA.2020.034
Abstract.
A breathing mode in a Hamiltonian system is a function on the phase spacewhose evolution is exactly periodic for all solutions of the equations of motion. Suchbreathing modes are familiar from nonlinear dynamics in harmonic traps or anti-de Sit-ter spacetimes, with applications to the physics of cold atomic gases, general relativity andhigh-energy physics. We discuss the implications of breathing modes in weakly nonlinearregimes, assuming that both the Hamiltonian and the breathing mode are linear functionsof a coupling parameter, taken to be small. For a linear system, breathing modes dictateresonant relations between the normal frequencies. These resonant relations imply that ar-bitrarily small nonlinearities may produce large effects over long times. The leading effectsof the nonlinearities in this regime are captured by the corresponding effective resonantsystem. The breathing mode of the original system translates into an exactly conservedquantity of this effective resonant system under simple assumptions that we explicitly spec-ify. If the nonlinearity in the Hamiltonian is quartic in the canonical variables, as is commonin many physically motivated cases, further consequences result from the presence of thebreathing modes, and some nontrivial explicit solutions of the effective resonant system canbe constructed. This structure explains in a uniform fashion a series of results in the re-cent literature where this type of dynamics is realized in specific Hamiltonian systems, andpredicts other situations of interest where it should emerge.
Key words: weak nonlinearity; multiscale dynamics; time-periodic energy transfer
Even in a complicated nonlinear dynamical system, with chaotic trajectories and all, it mayturn out that some specific combinations of the dynamical variables always behave periodicallywith the same period, independent of the initial conditions. In such situations, one is talkingabout breathing modes . Perhaps the simplest example is the separation of the center-of-massmotion in a harmonic potential [2], say, for a system of identical classical particles with arbitrarytranslationally invariant 2-body interactions. The center-of-mass always behaves like a singleindependent particle bound by a harmonic potential, and all of its trajectories are periodicwith the same period, providing a breathing mode. There are less obvious examples, such asthe Pitaevskii–Rosch breathing mode [31, 32] for two-dimensional Bose–Einstein condensates in One should distinguish breathing modes from cases where trajectories themselves are periodic, associated withsuperintegrability, such as the periodic trajectories of the Kepler problem. None of the systems that motivate ourstudy are known to be superintegrable, nor even integrable. Their trajectories may be arbitrarily complicated,and are certainly not exactly periodic, while a breathing mode only gives one specific function of the phase spacevariables that is periodic for all trajectories. a r X i v : . [ m a t h - ph ] A p r O. Evnina harmonic potential, as well as relativistic analogs of these systems involving nonlinear waveequations in anti-de Sitter spacetimes that we shall comment upon below. Recent experimentalwork on cold atomic gases motivated by breathing modes can be found in [34].The most immediate consequence of the breathing modes is in existence of time-dependentsymmetry transformations in the system (kinematic symmetries) akin to Galilean and Lorentzboosts, except that the dependence on time is periodic rather than linear. For systems in har-monic traps, these symmetries have been explored, for instance, in [29, 30]. Such symmetries aregenerated by the breathing modes in the same manner as ordinary time-independent symmetrytransformations are generated by conserved quantities. One can of course use these symmetriesto construct new solutions, as has been done in [23] to produce time-dependent solutions fromstationary configurations. It may seem that there is nothing more profound in this than boostingGalilean-invariant systems from one inertial frame to another, though in practice the effect ofthe transformations generated by the known breathing modes may be much less evident thanin the Galilean case. Our focus in this article, however, will be precisely on the implications ofbreathing modes for the dynamics that go beyond mere applications of ‘boosts.’As with any symmetries, the presence of breathing modes imposes strong constraints onthe Hamiltonian of the system. For a linear system, it mandates resonant relations betweenthe frequencies of the normal modes, which form evenly spaced ladders. If weak nonlinearitiesare turned on, the resonances between the linearized normal modes result in an enhancement ofnonlinear interactions, so that a nonlinearity of order g (cid:28) /g . A standard way to accurately capture the leading effects in thisregime is the resonant approximation, also known as the multiscale analysis, effective equation ortime-averaging method [26, 28], which simply discards all nonresonant mode couplings, irrelevantat small g . Our key result is that, under a simple condition, breathing modes result in ordinaryconserved quantities within the resonant approximation. We then focus on a situation genericfor interacting field theories, relativistic and non-relativistic, where the leading nonlinearity isquartic in the dynamical variables, that is, a Hamiltonian H = H + gH that admits a breathingmode B = B + gB so that H and B are quadratic in the dynamical variables and H is quartic. Making a simple assumption about the Poisson brackets of B and its complexconjugate, which essentially means that the symmetry algebra closes without generating extraconserved quantities apart from the already known ones, we obtain strong constraints on theresonant system corresponding to H at small g . In this formulation, B becomes an exactconserved quantity of the resonant system, while a family of explicit analytic solutions can beconstructed within the resonant system at the full nonlinear level, accurately approximatingsolutions of the original system on long time scales of order 1 /g . This is an example where thepresence of breathing modes constrains the system and allows one to construct novel analyticsolutions that do not follow from applying the symmetry transformations generated by thebreathing mode to any obvious solutions.The structures outlined above have been observed in the recent literature for a number ofspecial cases, motivated by rather disparate topics in physics. Thus, the analysis of [3, 4, 24, 25]is rooted in the Gross–Pitaevskii equation and the physics of Bose–Einstein condensates, whilethe analysis of [8, 9, 10, 11, 17] originates in studies of nonlinear dynamics in anti-de Sitterspacetimes, which is of interest for mathematical general relativity and high-energy physics.(Similarities between these two classes of systems have been pointed out in [7] and explained viataking nonrelativistic limits in [9, 17].) In the case of Bose–Einstein condensates, the existence ofbreathing modes is well-known [2, 31, 32], though their relation to the weakly nonlinear solutionswithin the resonant approximation has not been duly appreciated in the literature. For the caseof [8], the solutions of the resonant approximation came first, and our present analysis willsupply the corresponding breathing mode responsible for these solutions. The general typeof resonant systems relevant for us here, where the presence of an extra conserved quantityreathing Modes, Quartic Nonlinearities and Effective Resonant Systems 3imposes relations between mode couplings and generates some explicit analytic solutions, hasbeen constructed in [5, 6]. Our present exposition explains the origin of these structures in theunderlying Hamiltonian dynamics from which the resonant approximation originates. Breathingmodes are common in system with dynamical symmetry, where the Hamiltonian is realized asa Cartan generator of the corresponding dynamical symmetry Lie algebra. Likewise, quarticnonlinearities that play a significant role in our treatment are generic leading nonlinearitiesunder the assumption that odd order nonlinearities are prohibited by a reflection symmetry inthe configuration space. The framework we present here gives a recipe to search for explicitweakly nonlinear solutions within this class of systems. Consider a system with the Hamiltonian H ( p, q ) and the usual equations of motiond p i d t = − ∂H∂q i , d q i d t = ∂H∂p i . A function B ( p, q ) on the phase space is called a breathing mode ifd B d t = { H, B } ≡ (cid:88) k (cid:18) ∂H∂p k ∂B∂q k − ∂H∂q k ∂B∂p k (cid:19) = i B. (2.1)This equation is evidently solved by B ( t ) = e i t B (0) , and hence B ( p ( t ) , q ( t )) oscillates for all solutions of the equations of motion with the same periodequal 2 π . Note that whenever { H, B } is proportional to i B , we can always set it equal to i B ,as in (2.1), by rescaling H , and this is the normalization of the Hamiltonian we shall assumebelow without loss of generality.Existence of breathing modes of the form (2.1) is a strong restriction on the system (that weintend to exploit), but at the same time the algebraic structure of (2.1) is completely generic forsystems with dynamical symmetries. Indeed, if { H, ·} is a generator of a dynamical Lie grouplying in the Cartan subalgebra and { B, ·} is a generator corresponding to a positive root, one getsa relation of the sort (2.1). In such situations, many breathing modes can be present on the samefooting, corresponding to different generators of the dynamical symmetry group, as is indeed thecase for the systems that motivate our current study [3, 4, 8, 9, 10, 11, 17, 24, 25]. Nonetheless,one can often construct consistent dynamical truncations of such systems to a subset of degreesof freedom, either at the level of the full system or at the level of the resonant approximationin the weakly nonlinear regime, so that only one breathing mode is relevant in each truncation,which is again what happens in [3, 4, 8, 9, 10, 11, 17, 24, 25]. We shall therefore focus here onsystematically exploring the consequences of having one breathing mode, while keeping in mindthat in cases with many breathing modes some extra work may have to be done to make ourresults applicable.The breathing mode generates a kinematic symmetry given by q i → q i + η ∂B∂p i + ¯ η ∂ ¯ B∂p i , p i → p i − η ∂B∂q i − ¯ η ∂ ¯ B∂q i , (2.2)where η is a complex-valued infinitesimal parameter, and bars denote complex conjugation, hereand for the rest of our treatment. Unlike the case of ordinary symmetries (whose generators havevanishing Poisson brackets with the Hamiltonian), these transformations do not commute with O. Evninthe evolution, but rather induce very simple, predictable changes in the dynamical trajectory.Namely, if one applies (2.2) at t = 0, the subsequent trajectory is transformed as q i ( t ) → q i ( t ) + e i t η ∂B∂p i + e − i t ¯ η ∂ ¯ B∂p i , p i ( t ) → p i ( t ) − e i t η ∂B∂q i − e − i t ¯ η ∂ ¯ B∂q i . For the simplest case of systems in harmonic potentials, such transformations are discussed in[23, 29, 30].Our main focus in this article will be on weakly nonlinear systems with a breathing mode,so that H = H + gH , B = B + gB , (2.3)with g (cid:28)
1. We shall assume that H is quadratic in the dynamical variables (which simplymeans that g = 0 corresponds to a linear system), and so is B (which means that the corre-sponding kinematic symmetry for this linear system at g = 0 is linearly realized). Substitutingthese expressions in (2.1) and equating the coefficients of different powers of g , we obtain { H , B } = i B , { H , B } + { H , B } = i B , { H , B } = 0 . (2.4)In the later parts of our analysis, we shall also be assuming that H is quartic in the dynamicalvariables, which is a generic situation for classical field systems with a field-reflection symmetry(and corresponds to generic two-body interactions in the quantum case).We conclude this section with a few examples of breathing modes in relativistic and non-relativistic field systems that fit the above framework: • Consider a classical complex nonrelativistic field in D spatial dimensions with the Hamil-tonian H = 12 (cid:90) d D x (cid:20) ∂ k ¯Ψ( x ) ∂ k Ψ( x ) + x k x k | Ψ | ( x )+ g | Ψ | ( x ) (cid:90) d D y V ( x − y ) | Ψ | ( y ) (cid:21) . (2.5)The momenta conjugate to Ψ( x ) are understood to be i ¯Ψ( x ), so that the Hamiltonianequations of motion take the form of a nonlinear Schr¨odinger equation. (The first twoterms of the Hamiltonian may of course be equivalently rewritten in the vector notation as |∇ Ψ | + x | Ψ | .) Quantization of this Hamiltonian (which we do not consider here) wouldhave led to a system of identical bosons in an external harmonic potential interacting viatranslationally invariant two-body interactions given by V ( x − y ), which is a standard sub-ject in the physics of cold atomic gases. The classical Hamiltonian given above describes,from this perspective, the regime in which the trapped bosons undergo Bose–Einstein con-densation. There is a set of breathing modes associated to the center-of-mass motion in D spatial dimenstions: B n = (cid:90) d D x (cid:0) x n | Ψ | − ¯Ψ ∂ n Ψ (cid:1) . (2.6)Of particular importance in our context are combinations of these modes in the form B x + i B y (not necessarily in two dimensions) that play a role in the dynamics of theLandau level truncations [3, 4, 24] of the evolution corresponding to (2.5). There is some similarity between these expressions and the ‘integrable matrix theory’ of [35, 36, 37], whereHamiltonians and symmetry generators depending linearly on a coupling parameter are considered for quantum-mechanical systems with finite-dimensional Hilbert spaces. Our classical phase space functions are naturallyreplaced by matrices, and commutators take the place of our Poisson brackets. The only substantial differenceis that the right-hand sides of (2.4) would be zero in the framework of [35, 36, 37], since one is dealing withconserved quantities rather than breathing modes. reathing Modes, Quartic Nonlinearities and Effective Resonant Systems 5 • In two spatial dimensions and for the case of contact interactions, the symmetries of (2.5)get enhanced [30]. The corresponding Hamiltonian is H = 12 (cid:90) d x d y (cid:0) | ∂ x Ψ | + | ∂ y Ψ | + (cid:0) x + y (cid:1) | Ψ | + g | Ψ | (cid:1) . This is known to possess the Pitaevskii–Rosch breathing mode [31, 32], which manifestsitself in perfectly periodic evolution of I = (cid:90) d x d y (cid:0) x + y (cid:1) | Ψ | . To recast this mode in our standard form (2.1), one introduces B = ( I − H ) − (cid:90) d x d y (cid:2) x (cid:0) ¯Ψ ∂ x Ψ − Ψ ∂ x ¯Ψ (cid:1) + y (cid:0) ¯Ψ ∂ y Ψ − Ψ ∂ y ¯Ψ (cid:1)(cid:3) , (2.7)which satisfies d B/ d t = 2i B . (This can be changed to d B/ d t = i B to literally match ourdefinition (2.1) by a simple rescaling of the Hamiltonian, as per our general discussion.) • We now formulate relativistic analogs of the above two cases. An analog of the harmonicpotential is provided by anti-de Sitter (AdS) spacetimes (maximally symmetric spacetimesof constant negative curvature) that play the same role for relativistic wave equations asthe harmonic potential does for nonlinear Schr¨odinger equations.For d spatial dimensions, we denote the corresponding AdS space as AdS d +1 . It can berealized as a hyperboloid in an auxiliary flat pseudo-Euclidean space of dimension d + 2parametrized by (cid:0) X, Y, X k (cid:1) with the line element d s = − d X − d Y +d X k d X k , defined by − X − Y + X k X k = − . (2.8)One can parametrize this embedded hyperboloid by X k and t so that the two remainingembedding coordinates are given by X = (cid:112) X k X k cos t, Y = (cid:112) X k X k sin t. The AdS metric can be the extracted as [21]d s = − (cid:0) X k X k (cid:1) d t + (cid:18) δ ij − X i X j X k X k (cid:19) d X i d X j . (Note that t runs from 0 to 2 π in the embedding (2.8), but as the AdS metric does notdepend on t , it can be straightforwardly extended to run from −∞ to ∞ , which is how theAdS space is normally understood.) One can now define a relativistic field theory in thisspace, which shares many properties of the nonlinear Schr¨odinger equation in a harmonictrap. We shall use a real scalar field φ ( X , t ) and its conjugate momentum π φ ( X , t ) = ∂ t φ/ (cid:0) X k X k (cid:1) , though a complex field could easily be employed if more contact withnonrelativistic theories is needed. The Hamiltonian is then H = (cid:82) d X h ( X ; π φ , φ ) with h ( X ; π φ , φ ) = 12 (cid:104)(cid:0) X k X k (cid:1) π φ + ∂ k φ∂ k φ + (cid:0) X k ∂ k φ (cid:1) + m φ + g φ (cid:105) , (2.9)as derived from the standard action S = − (cid:82) d d +1 x √− g (cid:0) g µν ∂ µ φ ∂ ν φ + m φ + gφ / (cid:1) .Just like for a harmonic trap, the center-of-mass motion separates for any self-interactionsrespecting the AdS isometries (in particular, the φ interactions in the Hamiltonian above)and performs independent oscillations described by the breathing modes B n = (cid:90) d X (cid:18) X n h √ X k X k + i (cid:112) X k X k π φ ∂ n φ (cid:19) . (2.10) O. Evnin • Finally, there is a relativistic analog of the Pitaevskii–Rosch breathing mode that can bemade manifest by considering the systems defined by (2.9) in three spatial dimensions andwith the value of m corresponding to a conformally coupled scalar [8]. In fact, it is moreconvenient to consider the same type of scalar field on a spatial 3-sphere, which is relatedto the above AdS setup by a conformal transformation. We refer the reader to [8] fordetailed analysis of the corresponding equations. The important point for us here is that,restricting to scalar fields that only depend on time t and the polar angle x on the 3-sphere,and introducing v ( x, t ) = φ ( x, t ) sin x , one obtains the following nonlinear wave equation ∂ t v − ∂ x v + gv sin x = 0with the boundary conditions v (0 , t ) = v ( π, t ) = 0. This equation possesses a breathingmode of the form B = (cid:90) d x (cid:20) cos x (cid:18) ( ∂ t v ) + ( ∂ x v ) + gv x (cid:19) −
2i sin x∂ t v∂ x v (cid:21) , (2.11)which can, of course, equally well be expressed canonically through the momentum π v = ∂ t v conjugate to v . We start with setting g = 0 in (2.3) and considering a linear system with a quadratic Hamil-tonian H and a quadratic breathing mode B . Any linear system performing bounded motioncan be diagonalized and split into independent harmonic oscillators with normal frequencies ω n > α n ( t ) = e i ω n t α n (0) whose canonically conjugatemomenta are defined to be − i ¯ α n ( t ). In these variables, any quadratic Hamiltonian correspondingto bounded motion becomes simply H = (cid:88) n ω n ¯ α n α n . (3.1)We shall assume for the rest of our treatment that this diagonalization has been performed andour canonical variables are α n and − i ¯ α n .For this simple case, the structure of a general quadratic breathing mode B can be madeexplicit. Indeed, the most general possible expression is B = (cid:88) nm (cid:2) b nm ¯ α n α m + b + nm α n α m + b − nm ¯ α n ¯ α m (cid:3) , where b nm , b + nm and b − nm are numbers. We have to impose { H , B } = i (cid:88) k (cid:18) ∂H ∂ ¯ α k ∂B ∂α k − ∂H ∂α k ∂B ∂ ¯ α k (cid:19) = i B , which implies ω n b nm − ω m b nm = − b nm , ( ω n + ω m ) b ± nm = ± b ± nm . Since ω n > b − nm = 0. The rest exclusively depends, at least at the level of linearized theory, onthe spectrum of normal mode frequencies ω n . If there are two frequencies satisfying ω n + ω m = 1,the corresponding b + nm can have an arbitrary value. If there are two frequencies satisfying ω m = ω n + 1, the corresponding b nm can have an arbitrary value. (Evidently, b nn must be zero.)reathing Modes, Quartic Nonlinearities and Effective Resonant Systems 7While, within the linearized approximation, the above argument still leaves a huge amountof freedom in constructing breathing modes, provided that the spectrum ω n satisfies simpleconstraints, it is worth discussing upfront which of these breathing modes have a chance tosurvive inclusion of nonlinearities. In the linearized theory, all normal mode energies | α n | areindividually conserved. Generic nonlinearities induce energy transfer between the normal modes,typically in a way that essentially involves all modes. It is unrealistic to expect that a linearizedbreathing mode that only depends on a subset of α n will survive in a nonlinear theory, since it isoblivious of all the other α n , while all the degrees of freedom participate in a complex collectivedynamical process.Breathing modes based on b + nm are essentially eliminated by the above argument. Indeed, ω n + ω m = 1 can only be satisfied for ω m , ω n ≤
1. While it is possible to imagine artificiallyprepared sets of coupled oscillators with frequencies less than 1, where such breathing modesare relevant, in a realistic field theory, the normal frequencies grow without bound for short-wavelength modes. Therefore, ω n ≤ b + nm will depend only on a small subset of α n and has little chance to survive in an interacting theory.Breathing modes based on b mn may depend on α n if there exists m such that ω m = ω n − α n if there exists m such that ω m = ω n + 1. In order for B to depend onall α n and ¯ α n , as per the discussion above, one needs all ω n to fit in an evenly spaced ladder ω n = ω + n, (3.2)which we shall for simplicity assume nondegenerate. All the breathing modes mentioned in theprevious section are of this type. With this structure, the only nonvanishing b nm are b n,n +1 ≡ β n ,and hence we write B = (cid:88) n β n ¯ α n α n +1 . (3.3)For the rest of our treatment, we shall focus on including weak nonlinearities into systems definedby (3.1), (3.2) and (3.3). As explained in the previous section, a natural way for a breathing mode of the form (2.3) to besupported by the evolution is to have a theory with a linearized normal mode spectrum consistingof an infinite evenly spaced ladder of the form (3.2). In this case, there is a linearized breathingmode of the form (3.3) that one might hope to lift to the interacting theory to obtain (2.3).Assuming that has been accomplished (and our examples from Section 2 indeed demonstratethat it is possible in special cases), what are the properties of the corresponding interactingtheory in the weakly nonlinear regime g (cid:28) α n d t = i ω n α n + i g ∂H ∂ ¯ α n . It is convenient to switch to the ‘interaction picture’ by introducing a n so that α n = a n e i ω n t .Then,d a n d t = i g e − i ω n t ∂H ∂ ¯ α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n = a n e i ωnt = i g ∂∂ ¯ a n H (cid:0) a n e i ω n t , ¯ a n e − i ω n t (cid:1) . (4.1) O. EvninThe above equation is in what is known as the ‘periodic standard form’ in mathematical litera-ture [28], which facilitates its analysis at g (cid:28)
1. Qualitatively, a n ( t ) evolve very slowly, varyingappreciably on time scales of order 1 /g , while being essentially constant on time scales of order 1.By contrast, the right-hand side contains explicit oscillatory factors e i ω n t varying on time scalesof order 1. By the standard lore of time-averaging [28], (4.1) can be approximated arbitrarilywell for sufficiently small g on long time scales of order 1 /g by the corresponding averaged (or resonant ) system of the formd a n d t = i g ∂∂ ¯ a n (cid:18) π (cid:90) π d t H (cid:0) a n e i ω n t , ¯ a n e − i ω n t (cid:1)(cid:19) . Note that the t -integral only applies to the explicit dependence on t through the oscillatoryfactors e ± i ω n t , and not to the implicit dependence on t in a n and ¯ a n , which are treated asconstants for the purposes of the t -integration. The result of the integration is an explicitfunction of a n and ¯ a n , while all the explicit dependence on t disappears and g can be absorbedby defining the slow time τ = gt . The resulting equations for a n are again in a Hamiltonianform, but with a new ‘resonant’ Hamiltonian H res ,d α n d τ = i ∂H res ∂ ¯ α n , H res = 12 π π (cid:90) d t H (cid:0) a n e i ω n t , ¯ a n e − i ω n t (cid:1) . (4.2)Detailed justification of the time-averaging method and the resulting resonant approximationcan be found in [26, 28]. One can more formally (and more generally) write H res through theevolution operator of H denoted as ˆ S t , whose action on any phase space function F is definedby d (cid:0) ˆ S t F (cid:1) / d t = (cid:8) H , ˆ S t F (cid:9) . This gives simply H res = 12 π π (cid:90) d t ˆ S t H , and we shall make use of this representation in our subsequent treatment.We can now ask whether the breathing mode B has any implications for the resonant sys-tem (4.2). To this end, consider the Poisson brackets { H res , B } : { H res , B } = 12 π π (cid:90) d t (cid:8) ˆ S t H , B (cid:9) = 12 π π (cid:90) d t ˆ S t (cid:8) H , ˆ S − t B (cid:9) = 12 π π (cid:90) d t e − i t ˆ S t { H , B } = 12 π π (cid:90) d t e − i t ˆ S t (i B − { H , B } ) = − π π (cid:90) d t dd t (cid:0) e − i t ˆ S t B (cid:1) = B − ˆ S π B π , where we have used the evident properties of the evolution operator ˆ S t ˆ S − t = 1 and ˆ S t { F, G } = (cid:8) ˆ S t F, ˆ S t G (cid:9) , which is easily proved by differentiating with respect to t and using Jacobi identitiesfor the Poisson brackets; in going from the first line to the second line, we used (2.4). One canalso write equivalently and more explicitly { H res , B } = B ( a n , ¯ a n ) − B (cid:0) a n e π i ω n , ¯ a n e − π i ω n (cid:1) π . (4.3)The simple expression on the right-hand side easily vanishes in special cases. For example, itwould vanish if B = 0, as is the case in (2.6), or if ω in (3.2) is integer, or if ω is half-integerreathing Modes, Quartic Nonlinearities and Effective Resonant Systems 9and B is quartic in a n and ¯ a n . If the right-hand side of (4.3) is zero, B is conserved by theHamiltonian evolution defined by H res .The bottom line, and a key message of our treatment is then that B , the quadratic part ofthe breathing mode B defined by (2.3)–(2.4), becomes an ordinary conserved quantity withinthe resonant approximation at g (cid:28)
1, provided that B ( a n , ¯ a n ) = B (cid:0) a n e π i ω , ¯ a n e − π i ω (cid:1) . (4.4)This statement is independent of the form of B and H , as long as they satisfy the defini-tion (2.4), and the linearized breathing mode B is realized as (3.2)–(3.3). The specific examplesof breathing modes given in section 2 follow this pattern (at least after the evolution has beentruncated to appropriate subsets of modes labelled by a single integer n ).Assume now that (4.4) is satisfied so that B of the form (3.3) is a conserved quantity of H res .If H , and hence H res , is quartic in a n and ¯ a n , further implications of the breathing mode canbe exposed. The most general quartic H res one could write is H res = (cid:88) ω n + ω m = ω k + ω l C nmkl ¯ a n ¯ a m a k a l + (cid:88) ω n = ω m + ω k + ω l S nmkl ¯ a n a m a k a l + (cid:88) ω n = ω m + ω k + ω l ¯ S nmkl a n ¯ a m ¯ a k ¯ a l , (4.5)where C and S are numerical coefficients. Terms with four a ’s or four ¯ a ’s could not possiblysurvive the time averaging in the definition (4.2). Assuming that (4.4) is satisfied and hence B ofthe form (3.3) is a conserved quantity of H res , the following symmetry transformation associatedto B must be respected by H res : a n → a n + i ηβ n a n +1 + i¯ η ¯ β n − a n − , (4.6)which imposes relations between the coefficients C nmkl and S nmkl in H res . (In all of our formulasit should be understood that if a mode number index is outside the standard range [0 , ∞ ), thecorresponding expression is 0.) The actual form of the constraints on C and S from the abovesymmetry transformations is β n C n +1 ,m,k,l + β m C n,m +1 ,k,l = β k − C n,m,k − ,l + β l − C n,m,k,l − , (4.7) β n S n +1 ,m,k,l = β m − S n,m − ,k,l + β k − S n,m,k − ,l + β l − S n,m,k,l − . (4.8)The equation for S , in fact, guarantees that S nmkl = 0. Indeed, setting m = k = l = 0 in (4.8),we obtain S n = 0, and then one proceeds recursively increasing m , k and l in steps of 1 toprove that S nmkl = 0. This is closely related to the selection rules for AdS mode couplingsdiscussed in [21]. With only C in place, the resonant Hamiltonian takes the simple form H res = (cid:88) n + m = k + l C nmkl ¯ a n ¯ a m a k a l , (4.9)which is familiar from [3, 4, 5, 6, 8, 9, 17]. Note that, with S having dropped out, the resonantHamiltonian enjoys two conservation laws N = (cid:88) n | a n | , E = (cid:88) n n | a n | , (4.10)irrespectively of the values of C .We have just seen that, for a system with quartic nonlinearities, if (4.4) is satisfied and B becomes a conserved quantity of H res , a number of possible terms in H res drop out, leaving the0 O. Evninsimple expression (4.9). The converse is also true: if it happens that the S -couplings in (4.5) van-ish for a specific quartic system, and the resonant Hamiltonian is of the form (4.9), then (4.4) issatisfied, and hence B becomes a conserved quantity of H res . Indeed, H res of (4.9) is bilinear in a and bilinear in ¯ a , while B is linear in a and linear in ¯ a . Therefore, { H res , B } is likewise bilinearin a and bilinear in ¯ a , and should arise, by (4.3), from terms in B bilinear in a and bilinear in ¯ a .But any such terms would give a vanishing contribution to B ( a n , ¯ a n ) − B (cid:0) a n e π i ω , ¯ a n e − π i ω (cid:1) ,leaving nothing on the right-hand side of (4.3), and yielding { H res , B } = 0.We now reexamine the breathing mode (3.3) that has become a conserved quantity of (4.9).If B is a conserved quantity of H res , so are ¯ B and (cid:8) ¯ B , B (cid:9) , which is explicitly given by (cid:8) ¯ B , B (cid:9) = i ∞ (cid:88) n =0 (cid:0) | β n | − | β n − | (cid:1) | a n | . This conserved quantity is itself of a form similar to (4.10), being a weighted sum of the individuallinearized mode energies | a n | . Each such conserved quantity constrains the way nonlinearitiesmay dynamically redistribute the energy among the normal modes. It may be reasonable todemand that no further constraints of this sort, beyond the generic conservation of N and E ,are present. In this case, (cid:8) ¯ B , B (cid:9) must be a linear combination of N and E , which we canwrite as (cid:8) ¯ B , B (cid:9) = i (cid:18) N + 2 EG (cid:19) . Here, G is an arbitrary number, while the numerical coefficient in front of N has been set to 1 asa matter of fixing the normalization of B , which has been until now kept undetermined. Onethen has | β n | − | β n − | = 1 + 2 nG , which is solved by | β n | = (1 + n )(1 + n/G ) . As the phases of β n can be arbitrarily shifted by adjusting the phases of a n , one can simplydefine β n to be the square root of the right-hand side, β n = (cid:112) (1 + n )(1 + n/G ) , reducing B to B = (cid:88) n (cid:112) (1 + n )(1 + n/G )¯ a n a n +1 . (4.11)Thus, with a series of simple and generic assumptions on how the breathing mode is realizedin the linearized theory, how simple conditions are met to promote the breathing mode toa conserved quantity of the resonant approximation to the weakly nonlinear theory, and howtaking Poisson brackets of the breathing mode with its own complex conjugate does not generatenew conserved quantities, we have arrived at the class of ‘solvable’ resonant systems developedin [5, 6]. Indeed, the resonant Hamiltonian (4.9) explicitly matches the constructions of [5, 6],while the conserved quantity B of (4.11) corresponds, in the notation of [5, 6] to ¯ Z/ √ G . Weshall therefore conclude by simply restating the consequences of (4.9) and (4.11) already exploredin [5, 6].reathing Modes, Quartic Nonlinearities and Effective Resonant Systems 11With β n = (cid:112) (1 + n )(1 + n/G ), (4.7) imposes constraints on the coefficients of the resonantsystem (4.9), which are identical to the ones used in [5, 6] to define the ‘solvable’ class of resonantsystem. ‘Solvability’ is understood here in a very restricted sense, namely, as having an explicitfamily of nontrivial solutions. This family is defined by the ansatz a n ( t ) = (cid:114) G ( G + 1)( G + 2) · · · ( G + n − G n n ! ( b ( t ) + a ( t ) n )( p ( t )) n , (4.12)where b ( t ), a ( t ) and p ( t ) are complex-valued functions of time (and the conventions in the aboveformula differ slightly and inessentially from [5, 6]). The Hamiltonian equations of motionof (4.9) are da n dt = i ∞ (cid:88) m =0 n + m (cid:88) k =0 C nmk,n + m − k ¯ a m a k a n + m − k . (4.13)It is a nontrivial fact that the ansatz (4.12) is consistent with these equation of motion, and yetit is true by virtue of the conservation of B and the identity (4.7) it implies, as demonstratedin [5, 6]. A key point of the proof is that finite-difference identities (4.7) imply summationidentities for C nmkl adapted to the summation structure in (4.13). As a result, one obtainsa closed system of three ODEs for b ( t ), a ( t ) and p ( t ), which is furthermore superintegrablebecause of the conservation of H , N , E and B . The ODEs can be integrated to show that | p ( t ) | is always a strictly periodic function for all solutions, and the same is true for the spectrum | a n | .An explicit bound can be given on the turbulent transfer of energy toward large n modes for thesolutions in the ansatz (4.12). The reader is referred to [5, 6] for detailed derivations of theseresults.It is worth noting that the above properties were developed in [5, 6] completely in the lan-guage of resonant systems of the form (4.9), without any specific attention to how such featurescould emerge in resonant systems arising as weakly nonlinear approximations to realistic PDEs.Our present treatment closes this gap. We also remark that it is outside the normal range ofimplications of symmetries that explicit families of solutions, as given by (4.12), are generated.Symmetries produce new solutions out of known solutions, but (4.12) does not follow by appli-cation of transformations (4.6) to any other, more obvious solutions of (4.13). Rather, the logichere is that the identities (4.7) imposed on the mode couplings by the symmetries have furtherimplications and allow for the closure of the ansatz (4.12). This feature is specific to quarticnonlinearities, and does not immediately generalize to other cases. We have revisited the topic of breathing modes in the dynamics of nonlinear PDEs, and in partic-ular, the implications of the breathing modes for the weakly nonlinear regime. We have assumedthat both the Hamiltonian and the breathing mode are linear functions of a coupling parame-ter, and that setting the coupling parameter to zero results in a linear dynamical system, witha quadratic Hamiltonian, wherein the breathing mode also becomes quadratic in the canonicalvariables, which corresponds to a linear realization of the corresponding kinematic symmetry.Such setup is very generic from a physical perspective, commonly occurring in classical fieldtheories. We have presented a collection of explicit breathing modes related to the dynamicsof Bose–Einstein condensates and anti-de Sitter spacetimes. While the breathing modes (2.6)and (2.7) are standard in the Bose–Einstein literature, the corresponding relativistic breathingmodes (2.10) and (2.11) are in principle known from the symmetry properties of AdS spacetimes,but we believe our explicit expressions are compact and convenient.2 O. EvninWe have discussed how breathing modes of our type may be realized in a linear theory.The most natural realization is for systems whose normal mode frequencies form evenly spacedladders, as in (3.2). Such an evenly-spaced spectrum is highly resonant and, by the standard loreof weakly nonlinear dynamics, creates a possibility for arbitrarily small nonlinearities of order g to produce arbitrarily large effects on time scales of order 1 /g . On these specific time-scales, theoriginal dynamics may be accurately approximated by the time-averaged dynamics, describedby the resonant system (4.2). A simple condition (4.4), which is easily satisfied in special casesof interest, ensures that the quadratic part of the original breathing mode becomes a conservedquantity of the effective resonant dynamics (4.2).If the nonlinearities are quartic, as is common in field theories, further consequences resultfrom the conservation law in the resonant system inherited from the breathing mode of theoriginal system. First, only one of the possible quartic terms may remain in the resonant Hamil-tonian, leaving a simple expression (4.9). Two conservation laws (4.10) are then obeyed bythe resonant system. Assuming that the algebra of conserved quantities closes on the resonantHamiltonian, the breathing mode and these two extra quantities fixes the functional form ofthe breathing mode in terms of one free parameter (4.11). This recovers, starting from physi-cally motivated PDE problems, resonant systems of the solvable class considered in [5, 6]. Asa consequence, one obtains explicit solutions of the form (4.12) at the level of the resonantapproximation, which can be thoroughly analyzed as in [5, 6].Our treatment explains in a uniform fashion the emergence of solvable features within theresonant approximation in a number of physically motivated PDEs in the recent literature [3, 4,8, 9, 17, 24]. In particular, the progenitors of these solvable features in the resonant systems areidentified as breathing modes in the PDEs whose dynamics the resonant systems approximate.With respect to the solvable resonant systems of [5, 6], our treatment provides a mechanism bywhich they can emerge as approximations to specific PDEs of mathematical physics. Systemswith breathing modes may be engineered starting with linear systems whose normal frequenciesform evenly spaced ladders (3.2), which creates a lot of room for concrete applications of ouranalysis. In relation to the concrete physical problems that have motivated our considerations,beyond what has been explicitly treated in the literature, one is led to expect solvable featuresin the resonant systems corresponding to (1) one-dimensional nonlinear Schr¨odinger equation ina harmonic trap with arbitrary 2-body interactions, (2) Landau-level truncations, in the styleof [3, 4], of nonlinear Schr¨odinger equations in isotropic harmonic traps with arbitrary 2-bodyinteractions in any number of dimensions, (3) maximally rotating truncations of the resonantdynamics in AdS, in the style of [17], with arbitrary quartic local interactions. The last topicconnects to extensive studies of nonlinear dynamics in AdS [1, 12, 13, 14, 16, 18, 19], in particular,outside spherical symmetry [15, 20, 22, 27, 33]. Some of the results presented here, in particularexplicit analytic solutions within the resonant approximation, are specific to the case of quarticnonlinearities. It would be interesting to investigate whether generalizations of these results(which are expected to be non-straightforward) exist for more general nonlinearities. Note added:
An anonymous referee has aptly observed that the last condition listed in (2.4),namely { H , B } = 0, has never been used in our derivations. This means that, technically, it issufficient for the breathing mode definition (2.1) to be satisfied up to linear order in g to ensurethat the formalism developed here is applicable. Acknowledgments
I have benefitted from discussions with Anxo Biasi, Piotr Bizo´n, Ben Craps and Andrzej Ros-tworowski. This research is supported by CUniverse research promotion project at Chula-longkorn University (grant CUAASC) and by FWO-Vlaanderen through project G006918N.Part of this work was developed during a visit to the physics department of the Jagiellonianreathing Modes, Quartic Nonlinearities and Effective Resonant Systems 13University (Krakow, Poland). Support of the Polish National Science Centre through grantnumber 2017/26/A/ST2/00530 and personal hospitality of Piotr and Magda Bizo´n are grate-fully acknowledged.
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