Variational formalism for the Klein-Gordon oscillon

The variational method employing the amplitude and width as collective coordinates of the Klein-Gordon oscillon leads to a dynamical system with unstable periodic orbits that blow up when perturbed. We propose a multiscale variational approach free from the blow-up singularities. An essential feature of the proposed trial function is the inclusion of the third collective variable: a correction for the nonuniform phase growth. In addition to determining the parameters of the oscillon, our approach detects the onset of its instability.

Most of the mathematical analysis of oscillons has been carried out using asymptotic [55][56][57] and numerical techniques [1,42,43,55,[58][59][60][61] while qualitative insights called on variational arguments.In Ref [1], the Φ 4 oscillon was approximated by a localised waveform where A(t) is an unknown oscillating amplitude and b is an arbitrarily chosen value of the width.(Ref [62] followed a similar strategy when dealing with the twodimensional sine-Gordon equation.)Once the ansatz (2) has been substituted in the lagrangian and the rdependence integrated away, the variation of action produces a second-order equation for A(t).
The variational method does not suggest any optimisation strategies for b.Making b(t) another collective coordinate -as it is done in the studies of the nonlinear Schrödinger solitons [63,64] -gives rise to an ill-posed dynamical system not amenable to numerical simulations.(See section II below.) With an obstacle encountered in (3+1) dimensions, one turns to a (1+1) dimensional version of the model for guidance.The analysis can be further simplified by considering oscillons approaching a symmetric vacuum as x → ±∞.A physically relevant model of this kind was considered by Kosevich and Kovalev [66]: Unlike its Φ 4 counterpart, the oscillon in the Kosevich-Kovalev model satisfies φ → 0 as x → ±∞ and oscillates, symmetrically, between positive and negative values.The asymptotic representation of this solution is where ω 2 = 4 − ǫ 2 and ǫ → 0 [66].Despite the difference in the vacuum symmetry, equations ( 1) and ( 3) belong to the same, Klein-Gordon, variety and share a number of analytical properties.
The purpose of the present study is to identify a set of collective coordinates and formulate a variational description of the Klein-Gordon oscillon.A consistent variational formulation would determine the stability range of the oscillon, uncover its instability mechanism and explain some of its properties such as the amplitudefrequency relationship.Using the (1+1)-dimensional Kosevich-Kovalev equation (3) as a prototype system, we transplant the idea of multiple time scales to the collective-coordinate Lagrangian method.With some modifications, our approach should remain applicable to oscillons in the (3+1)-dimensional Φ 4 theory and other Klein-Gordon models.
Before outlining the paper, three remarks are in order.First, equation (3) can be seen as a truncation of the sine-Gordon model.The fundamental difference between the Kosevich-Kovalev oscillon and the sine-Gordon breather is that the latter solution is exactly periodic while the amplitude of the former one decreases due to the third-harmonic radiation.(When the amplitude of the oscillations is small, the radiation is exponentially weak though; hence the decay is slow.) Second, it is appropriate to mention an alternative variational procedure [67] where one not only chooses the spatial part but also imposes the time dependence of the trial function.For instance, one may set For a fixed ω, the action becomes a function of two timeindependent parameters, A 0 and b.The shortcoming of this technique is that it does not allow one to examine the stability of the Klein-Gordon oscillon.Neither would it capture a slow modulation of the oscillation frequency -such as the one observed in numerical simulations of the Φ 4 model [42,58,60].
Our last remark concerns a closely related system, the nonlinear Schrödinger equation.The variational method has been highly successful in the studies of the Schrödinger solitons -scalar and vector ones, with a variety of nonlinearities, perturbations, and in various dimensions [63].Several sets of collective coordinates for the Schrödinger solitons have been identified.It is the remarkable simplicity and versatility of the variational method demonstrated in the nonlinear Schrödinger domain that motivate our search for its Klein-Gordon counterpart.
The outline of the paper is as follows.In the next section we show that choosing the collective coordinates similar to the way they are chosen for the nonlinear Schrödinger soliton leads to singular finite-dimensional dynamics.A consistent variational procedure involving fast and slow temporal scales is formulated in section III.We assess the approximation by comparing the variational solution to the "true" oscillon obtained numerically.Section IV adds remarks on the role of the third collective coordinate and the choice of the trial function, while an explicit construction of the oscillon with adiabatically changing parameters has been relegated to the Appendix A. Finally, section V summarises conclusions of this study.

II. SINGULAR AMPLITUDE-WIDTH DYNAMICS A. Two-mode variational approximation
The variational approach to equation (3) makes use of its Lagrangian, Modelling on the nonlinear Schrödinger construction [63,64], we choose the amplitude and width of the oscillon as two collective variables: The amplitude A(t) is expected to oscillate between positive and negative values while the width ("breadth") b(t) should remain positive at all times.Substituting the Ansatz (6) in (5) gives the Lagrangian of a system with two degrees of freedom: In (7), the overdot stands for the derivative with respect to t.The equations of motion are where we have introduced a short-hand notation for a numerical factor

B. Asymptotic solution
The system (8) has a family of periodic solutions.For reasons that will become clear in what follows, these solutions are difficult to obtain by means of numerical simulations of equations (8).However the family can be constructed as a multiscale perturbation expansion -in the limit of small A and large b.
To this end, we let where A 1 , A 3 , ... and b 1 , b 3 , .... are functions of a sequence of temporal variables T 0 , T 2 , ..., with T 2n = ǫ 2n t and ǫ → 0. Writing d/dt = ∂/∂T 0 +ǫ 2 ∂T 2 +... and substituting the expansions ( 9) in (8a), we set coefficients of like powers of ǫ to zero.The order ǫ 1 gives a linear equation Without loss of generality we can take a solution in the form where ψ = ψ(T 2 , ....) = |ψ|e −2iθ is a complex-valued function of "slow" variables.The next order, ǫ 3 , gives Substituting for A 1 from (10) and imposing the nonsecularity condition we determine a solution of (11): Turning to equation (8b), the leading order is The general solution of this linear equation is given by where τ = T 0 − θ and C 1 is an arbitrary constant in front of a homogeneous solution.(The second homogeneous solution was absorbed in the term 1/ǫ in the expansion (9).)Letting C 1 = 0 and imposing the constraint selects a regular solution: Finally, the phase of the complex variable ψ is determined by equation (12).Substituting |ψ| from ( 16) we obtain Thus, the asymptotic solution of equations ( 8) has the form where ǫ → 0 and This solution describes a closed orbit in the phase space of the system (8).See Fig 1.

C. Singular dynamics
It is not difficult to realise that the asymptotic solution ( 18) is unstable.Indeed, the bounded solution (17) of equation ( 14) is selected by the initial condition ∂b 1 /∂T 0 = 0 at T 0 = θ.If we, instead, let ∂b 1 /∂T 0 = δ with a small δ, the tan 2τ component will be turned on in the expression (15) and b 1 will blow up at T 0 = θ + π/4.The numerical analysis of the system (8) indicates that periodic solutions with A(t) oscillating about zero are unstable for any value of the oscillation amplitude -and not only in the small-A asymptotic regime.The instability originates from the topology of the four-dimensional phase space of the system that features a singularity at A = 0.
Indeed, had the system not had a singularity and had the periodic orbit been stable, a small perturbation about it would have been oscillating, quasiperiodically, between positive and negative A. The corresponding trajectory would be winding on a torus in the four-dimensional phase space, with the points where the trajectory passes through A = 0 filling a finite interval on the ḃ-axis.In the presence of the singularity, however, such a torus cannot form because any trajectory crossing through A = 0 at time t * has to satisfy ḃ = 0 at the same time.
Trajectories that do not pass through the plane A = ḃ = 0 follow one of two scenarios.In the "spreading" scenario, the width b(t) escapes to infinity (Fig 2(a)).The corresponding A(t) approaches zero but remains on one side of it at all times.In the alternative scenario, the Due to the singularity of solutions emerging from generic initial conditions, the system (8) is not amenable to numerical simulations beyond a few oscillation cycles.What is even more important, the all-ω universal instability of periodic solutions of this four-dimensional system does not match up with the behaviour of the oscillon solutions of the full partial differential equation (3).Contrary to the predictions of the two-mode approximation, the simulations of equation ( 3) demonstrate that the nearly-periodic oscillons with frequencies in the range √ 2 ω < 2 are stable.The amplitude and frequency of such oscillons do change due to the third-harmonic radiation; however, these changes are slow and may only be noticeable over long temporal intervals.(See Fig 3(a)).
We note that an ill-posed system similar to (8) was encountered in the variational studies of the sine-Gordon breathers [65].
The spurious instability of periodic trajectories of the system (8) disqualifies the two-variable Ansatz (6) and prompts one to look for suitable alternatives.

A. Amplitude, width and phase correction
To rectify the flaws of the "naive" variational algorithm, we consider φ to be a function of two time variables, T 0 = t and T 1 = ǫt.The rate of change is assumed to be O(1) on either scale: ∂φ/∂T 0 , ∂φ/∂T 1 ∼ 1.We require φ to be periodic in T 0 , with a period of T : As ǫ → 0, the variables T 0 and T 1 become independent and the Lagrangian (5) transforms to The action Ldt is replaced with We choose the trial function in the form where A, b and θ are functions of the "slow" time variable T 1 while ω = 2π/T .(Note that φ does not have to be assumed small.)The interpretation of the width b is the same as in the Ansatz (6) while A represents the maximum of the oscillon's amplitude rather than the amplitude itself.Unlike the previous trial function ( 6), the variable A in ( 21) is assumed to remain positive at all times.The phase correction θ is a new addition to the set of collective coordinates; its significance will be elucidated later (section IV A).The choice of the spatial part of the Ansatz will also be discussed below (section IV B).
Once the explicit dependence on x and T 0 has been integrated away, equations (19) and (20) give an effective action and D = ǫ ∂ ∂T1 .Two Euler-Lagrange equations are and The last equation can be integrated to give where ℓ is a constant of integration.Eliminating the cyclic variable θ between ( 23) and ( 24) we arrive at The third Euler-Lagrange equation for the Lagrangian (22) does not involve θ: Equations ( 25) constitute a four-dimensional conservative system with a single control parameter ℓ 2 .

B. Slow dynamics and stationary points
The oscillon corresponds to a fixed-point solution of the system (25).There are two coexisting fixed points for each ℓ 2 in the interval (0, 64  9 ).We denote their components by (A + , b + ) and (A − , b − ), respectively.Here Turning to the stability of these, we note that all derivatives in equations ( 25) carry a small factor ǫ. Accordingly, most of the time-dependent solutions of that system evolve on a short scale T 1 ∼ ǫ.This is inconsistent with our original assumption that ∂φ/∂T 1 = O(1).There is, however, a particular ℓ-regime where solutions change slowly and the system (25) is consistent.Specifically, slowly evolving nonstationary solutions can be explicitly constructed in the vicinity of the value ℓ 2 c = 64 9 ; see Appendix A. This value proves to be a saddle-centre bifurcation point separating a branch of stable equilibria, namely (A − , b − ), from an unstable branch, (A + , b + ).
Since the asymptotic construction presented in the Appendix is limited to the neighbourhood of the bifurcation value ℓ c , we do not have access to the oscillon perturbations outside that parameter region.Nevertheless, it is not difficult to realise that the two fixed points maintain their stability properties over their entire domain of existence, 0 ≤ ℓ 2 < ℓ 2 c .Indeed, the stability may only change as ℓ passes through the value ℓ 0 given by a root of det M = 0, where M is the linearisation matrix.(The evolution is slow and the system (25) is consistent in the vicinity of that point.)There happens to be only one such root and it is given exacty by ℓ c ; see Appendix A.
In order to compare the variational results to conclusions of the direct numerical simulations of equation ( 3), we return to the oscillon Ansatz (21).Switching from the parametrisation by ℓ to the frequency parameter ω, two branches of fixed points (26) can be characterised in a uniform way: (The relations (27) result by letting ℓ = ωbA 2 in (26).) The frequencies ω c ≤ ω < 2 correspond to stable oscillons and those in the interval 0 ≤ ω < ω c to unstable ones. Here The third collective coordinate in ( 21) -the phase correction θ -can be assigned an arbitrary constant value.Note that the expressions ( 27) agree with the asymptotic result (4) in the A, b −1 → 0 limit.

C. Numerical verification
We simulated the partial differential equation ( 3) using a pseudospectral numerical scheme with 2 13 Fourier modes.The scheme imposes periodic boundary conditions φ(L) = φ(−L) and φ x (L) = φ x (−L), where the interval should be chosen long enough to prevent any radiation re-entry.(Our L was pegged to the estimated width of the oscillon, varying between L = 20 and L = 100.) Using the initial data in the form and varied A 0 , we were able to create stable oscillons with frequencies ranging from ω = 1.03 ω c to ω = 2. (Here ω = 2π/T , where T is the observed period of the localised periodic solution.)This "experimental" stability domain is in good agreement with the variational result ω c ≤ ω < 2.
The 3% discrepancy between two lower threshold values can be attributed to the emission of radiation and the oscillon's core deformation due to the third harmonic excitation.(The presence of the third harmonic in the oscillon's core is manifest already in the asymptotic solution (4).)The radiation intensifies and deformation becomes more significant as the oscillon's amplitude grows (Fig 3(a)); yet the variational approximation disregards both effects (see Fig 3(b)).
Once the evolution has settled to an oscillon with a period T , we would measure its amplitude and evaluate its width which we define by In ( 29)-( 30), the maximum is evaluated over the time interval t 0 ≤ t < t 0 + T , where t 0 was typically chosen as the position of the third peak of φ(0, t). Figure 4 compares the amplitude and width of the numerically generated oscillon with their variational approximations (27).The difference between the numerical and variational results grows as ω approaches 1.03 ω cyet the relative error in the amplitude remains below 8% and the error in the width does not exceed 12.5%.

FIG. 3.
Top panel: the Kosevich-Kovalev oscillon with ω = 1.06 ωc (where ωc = √ 2).The oscillon is stable: despite the energy loss to radiation waves, any changes in its period and amplitude are hardly visible.This figure is obtained by the numerical simulation of equation (3).Bottom panel: the variational approximation (21) with the matching ω.Here A and b are as in (27) with ω = 1.06 ωc, and θ = 0. Except for the absence of the radiation waves, the variational pattern is seen to be a good fit for the true oscillon.

IV. TWO REMARKS ON THE METHOD A. Modulation, instability and significance of θ
The inclusion of the cyclic coordinate θ(T 1 ) is crucial for our variational approach.To show that, we compare the system (25) incorporating, implicitly, three degrees of freedom with its two-degree (A and b) counterpart.
Linearising equations (25) about the fixed point ( 27) and considering small perturbations with the time dependence e (λ/ǫ)T1 , we obtain a characteristic equation When A 2 is away from 0 or 8/3, all eigenvalues λ are of order 1.This means that contrary to the assumption under which the system (25) was derived, small perturbations evolve on a short scale T 1 ∼ ǫ rather than T 1 ∼ 1.
The variational method cannot provide trustworthy information on the stability or modulation frequency of the oscillons with those A.
There are two regions where a pair of O(ǫ)-eigenvalues occurs and, consequently, our approach is consistent.One region consists of small A ∼ ǫ; this range accounts for the asymptotic regime (4).The second region is defined by |A 2 − 8/3| = O(ǫ 2 ) or, equivalently, by |ω − ω c | ∼ ǫ 2 .As ω is reduced through ω c , a pair of opposite imaginary eigenvalues converges at the origin and moves onto the positive and negative real axis: At this point, a slow modulation of the principal harmonic cos(ω c t) with the modulation frequency ∼ (ω − ω c ) 1/2 gives way to an exponential growth of the perturbation.(For an explicit construction of the timedependent solutions of the system (25), see Appendix A.) Had we not included θ(T 1 ) in our trial function -that is, had we set θ = 0 in equation ( 21) -we would have ended up with the same fixed point (27) but a different characteristic equation: Equation ( 32) does not have roots of order ǫ outside the asymptotic domain A ∼ ǫ.Therefore, the multiscale variational Ansatz excluding the cyclic variable θ(T 1 ) is inconsistent with the slow evolution of the collective coordinates A(T 1 ) and b(T 1 ).

B. Insensitivity to spatial shape variations
The x-part of the trial function (21) was chosen so as to reproduce the asymptotic representation (4) and match the amplitude-frequency relationship as ω → 2. As for the global behaviour of the A(ω) curve, the variation of the spatial profile of the trial function has little effect on it -as long as the function remains localised.
To exemplify this insensitivity to the Ansatz variations, we replace the exponentially localised trial function (21) with a gaussian: As in (21), the amplitude A, width b and phase shift θ are assumed to be functions of the slow time variable T 1 = ǫt.Substituting in (20) gives an effective action with the Lagrangian (Here, as before, D = ǫ∂/∂T 1 ).Equation ( 34) has the same form as (22) with the only difference residing in the value of some of the coefficients.The Euler-Lagrange equations resulting from (34) have a fixed-point solution Note that the gaussian amplitude and width are related to ω by exactly same laws as the amplitude and width of the secant-shaped approximation (equations ( 27)).If A g stands for the amplitude (35) and A s for the secantbased result (27), the ratio A g (ω)/A s (ω) is given by 4 8/9 ≈ 0.971.Thus the gaussian-based amplitudefrequency curve reproduces the qualitative behaviour of the curve (27), with the gaussian amplitude being only 3%-different from the amplitude of the secant-shaped variational oscillon.Linearising the Euler-Lagrange equations about the fixed point (35) we obtain a gaussian analog of the characteristic equation (31): (36) The critical value of A 2 above which a pair of opposite eigenvalues moves onto the real axis is 32/9 √ 2. Remarkably, the corresponding threshold frequency ω c = √ 2 coincides with the value (28) afforded by the secant Ansatz.

V. CONCLUSIONS
This study was motivated by the numerous links and similarities between the Klein-Gordon oscillons and solitons of the nonlinear Schrödinger equations.A simple yet powerful approach to the Schrödinger solitons exploits the variation of action.By contrast, the variational analysis of the Klein-Gordon oscillons has not been nearly as successful.
One obstacle to the straightforward ("naive") variational treatment of the oscillon is that its width proves to be unsuitable as a collective coordinate in that approach.The soliton's amplitude and width comprise a standard choice of variables in the Schrödinger domain, but making a similar choice in the Klein-Gordon Lagrangian results in a singular four-dimensional system.
This paper presents a variational method free from singularities.The method aims at determining the oscillon's parameters, domain of existence and stability-instability transition points.The proposed formulation is based on a fast harmonic Ansatz supplemented by the adiabatic evolution of the oscillon's collective coordinates.An essential component of the set of collective coordinates is the "lazy phase": a cyclic variable accounting for nonuniform phase acquisitions.
We employed the Kosevich-Kovalev model as a prototype equation exhibiting oscillon solutions.Our variational method establishes the oscillon's domain of existence (0 < ω < 2) and identifies the frequency ω c at which the oscillon loses its stability (ω c = √ 2).The stability domain is in good agreement with numerical simulations of the partial differential equation (3) which yield stable oscillons with frequencies 1.03 ω c ≤ ω < 2. The variational amplitude-frequency and width-frequency curves are consistent with the characteristics of the numerical solutions.
Fig 1 illustrates the evolution of a small perturbation of the periodic orbit.

FIG. 2 .
FIG. 2. Two types of unstable evolution in equations (8).(a)A(t) approaches zero while b(t) grows exponentially.(b) A(t) grows to infinity (negative infinity in this simulation) while b(t) shrinks to zero.

FIG. 4 .
FIG.4.The amplitude and width of the oscillon as functions of its frequency.The solid curves depict results of the numerical simulations of the partial differential equation (3).The blue curve traces the amplitude-frequency and the brown one gives the width-frequency dependence.The nearby dashed lines describe the corresponding variational approximations(27).