Generic quartic solitons in optical media

Our analysis suggests strongly that stationary pulses exist in nonlinear media with second-, third-, and fourth-order dispersion. A theory, based on the variational approach, is developed for finding approximate parameters of such solitons. It is obtained that the soliton velocity in the retarded reference frame can be different from the inverse of the group velocity of linear waves. It is shown that the interaction of the pulse spectrum with that of linear waves can affect the existence of stationary solitons. These theoretical results are supported by numerical simulations. Transformations between solitons of different systems are derived. A generalization for solitons in media with the highest even-order dispersion is suggested.


I. INTRODUCTION
A balance between second-order dispersion, or group velocity dispersion (GVD), and cubic (Kerr) nonlinearity results in a formation of optical solitons -stable pulses that propagate without change of parameters [1].Moreover, these pulses preserve their shapes and parameters after interaction with each other.Additional effects, such as higher-order dispersion, the Raman frequency shift, and self-steepening, change parameters of solitons, see e.g.Ref. [1].
In the present paper, we consider general quartic media described by the second-, third-(TOD), and fourthorder dispersion (FOD) terms.We demonstrate that such media also admits the propagation of stable localized pulses.Since all dispersion terms are involved, we call these pulses "generic quartic solitons" (GQS) to distinguish them from PQS. Parameters of stationary solitons are found approximately, using the variational approach.Regions of the GQS existence in the space of the system parameters are obtained.A relation between moving solitons in general quartic media and pure quartic media is established.A generalization of results to higher-order dispersion is discussed.* e.n.tsoy@gmail.com

II. MODEL AND STATIONARY SOLITONS
The dynamics of optical pulses in nonlinear dispersive media is described by the modified nonlinear Schrödinger (NLS) equation [1]: where ψ(τ, z) is the envelope of the electric field, τ is the time in the retarded frame, z is the propagation distance, β j is the parameter of dispersion of the j-th order, j = 2, 3, and 4, γ is the Kerr nonlinearity parameter.We consider dispersion terms of up to the fourth order only.We mention that at β 3 = β 4 = 0, the standard NLS equation is completely integrable [11], and has the soliton solution.At β 3 = 0, there is also an exact soliton solution [12].Soliton solutions of Eq. ( 1) for some sets of parameters β j are found in Ref. [3].These solutions have smooth, non-oscillating tails.The influence of higher-order dispersion on the dynamics of solitons was studied intensively, see e.g.Refs.[1][2][3][4][5][6][7][8][9][10][12][13][14][15][16].Usually, two extreme cases are investigated.Namely, either TOD and FOD are treated as perturbation to the GVD effect [13][14][15][16], or the FOD effect is considered as a dominant one [2][3][4][5][6][7][8][9][10]12].The former (latter) approach is valid far from (close to) zero dispersion points (ZDPs).In particular, the consideration of a system as a medium with pure-quartic dispersion is only valid near a specific ZDP, where both GVD and TOD are negligible.In contrast to previous works, we make no assumptions on values of GVD, TOD, and FOD effects.We show also that TOD does not result in the pulse asymmetry, if it acts together with FOD.Our results, based on the variational approach, indicate clearly that the joint action of GVD, TOD, and FOD can be balanced by Kerr nonlinearity, giving solitons with symmetric shapes.Numerical simulations of Eq. ( 1) support this conclusion.
Equation (1) has the following Lagrangian density: where the star means the complex conjugation.We use a trial function in the form of the Gaussian function: (4) Here, parameters A, a, τ c , b, c, and φ are the soliton amplitude, width, position of the center, linear phase parameter (the soliton frequency), chirp parameter, and phase parameter, respectively.All these parameters are assumed to be functions of z.The minus sign of a term proportional to b is taken for convenience.The actual form of solitons differs from Eq. (4).In particular, a soliton can have oscillating tails [5,10].However, numerical simulations show that trial function (4) captures well the overall soliton shape, so the parameter values predicted are close to the actual ones.
The Lagrangian L = ∞ −∞ Ldτ is expressed in terms of the pulse parameters, using trial function (4).The Euler-Lagrange equations for L give the following equations: and b ′ = 0, where the prime denotes d/dz.Parameter and represents the initial energy of the pulse.Equations, similar to Eqs. ( 5)-( 8), have been obtained previously, see e.g.Refs.[15,16].However, these equations were used mainly to analyze the influence of higher order effects on the soliton of the unperturbed NLS equation.Pure quartic solitons have been studied by the same method in Ref. [9], but only for zero soliton frequency.Here, we are interested in the existence of stationary solitons in the presence of higher-order dispersion.Equations ( 5) and ( 6) constitute a closed set because their right-hand sides, f a (a, b, c) and f c (a, b, c, E 0 ), do not depend on τ c and φ.The right-hand sides of Eqs. ( 7) and ( 8) correspond to the soliton velocity 1/v in the retarded frame and the phase coefficient δ, respectively.Equations ( 5) and ( 6) have the invariant, which is the effective Hamiltonian of these equations: Using H(a, c) = H(a(0), c(0)), one can express variable c in terms of a, and substitute it into equation for a ′ , or a ′′ .In the latter case, the equation for the soliton width describes the motion of a particle with coordinate a in an effective potential.Stationary solutions are found from conditions f a (a, b, c) = 0 and f c (a, b, c, E 0 ) = 0. From Eqs. ( 5) and ( 6), it follows that stationary states exist only when c = 0.Then, the stationary soliton width a s > 0 is determined from the following equation: where s 1 = √ 2π η 2 (b)/(γE 0 ), and s 2 = √ 2π β 4 /(4γE 0 ).Applying the Sturm's theorem for the number of positive roots to Eq. ( 10), we obtain the following result: (i) If (s 1 > 0 and s 2 > 0), or (s 1 < 0 and s 2 > s 2,th ), then Eq. ( 10) does not have positive roots, where s 2,th = −4s 3  1 /27.(ii) For any s 1 , if s 2 < 0, then Eq. ( 10) has one positive root.
The first condition of Case (i) is reduced to (η 2 (b)γ > 0 and β 4 γ > 0), c.f. with the standard NLS equation.The second condition of Case (i) indicates that solitons do not exist also for negative η 2 (b)γ and corresponding β 4 .Solitons for parameters from Case (iii) are mostly nonstationary due to the interaction with linear waves, see the corresponding discussion below.Also, notice that solitons exist for any sign of γ.
Though Eq. ( 10) can be solved analytically, this gives a complicated dependence of a s on the system parameters.Therefore, it is useful to consider some limiting cases.Firstly, we consider the case of small β 4 , namely, if where p = 4γE 0 / √ 2π.The second case is for small β 2 , β 3 , and b, namely, if Therefore, for large |β 4 |, solitons exist when β 4 γ < 0. Having root a s of Eq. ( 10), the stationary amplitude is found as . Then (A s , a s , b) and c = 0, together with 1/v s and δ s , correspond to stationary parameters of a GQS.
The theory predicts that in absence of β 3 and β 4 , the stationary soliton velocity 1/v s coincide with the inverse of the group velocity η 1 (b) of linear waves.The inclusion of higher-order dispersion breaks this relation, see Eq. (7).In particular, even at the extremum of the dispersion relation, at b = 0, we have solitons, moving due to β 3 .This result is supported by numerical simulations of Eq. (1).A related observation is that a static soliton with 1/v s = 0 can have a phase dependence on time, b = 0.The difference of the soliton velocity 1/v s from η 1 (b) can be used for slow light and fast light applications of solitons.
Equations ( 5)-( 8) describe the adiabatic dynamics of a soliton.These equations do not take into account the interaction of the soliton with linear waves.However, this interaction can be accounted for qualitatively, using the following arguments.One can distinguish two different ways of generation of linear waves by solitons.When an initial pulse differs slightly from the stationary profile, the pulse adjusts its form to the stationary one, radiating the excess as linear waves.This adjustment is observed as damped oscillations of the soliton width and amplitude.Such a type of interaction is accounted for, to some extent, in Eqs. ( 5)-( 8) by the inclusion of the chirp parameter c.Namely, if the chirp is absent in trial function (4), c ≡ 0, width a is constant on z, even when a(0) is different from the stationary value.The variational approach with the chirp included treats the interaction with linear waves as a modulation of the soliton phase [17].In contrast to the actual dynamics, the method gives undamped oscillations of the soliton shape, but predicts reasonably well the frequency.
The second type is due to the resonance interaction of a soliton with linear waves [13].It occurs at frequencies where the soliton dispersion relation intersects with the dispersion relation of linear waves.The soliton dispersion relation is usually a straight line obtained from the following procedure.Let the stationary soliton has the form see Eq. ( 4), where real f (τ ) describes the soliton profile.Then, the soliton spectrum Ψ s (ω, z), obtained from the Fourier transform, is written as z}, where F (ω) is the Fourier transform of f (τ ).This expression indicates that the soliton dispersion relation, or the dependence of the soliton propagation constant η sol (ω) on frequency, is determined as the following The linear dependence (13) means that a soliton propagates without dispersion, d 2 η sol (ω)/dω 2 = 0, since it is balanced by nonlinearity.At frequencies ω r , defined by resonance linear waves are generated due to the phase matching condition, see Refs.[13][14][15][16].The rate at which the soliton energy goes to linear waves depends on values of the soliton spectrum at these resonant frequencies.The arguments presented above are valid for media with an arbitrary order of dispersion.Resonance condition ( 14) can be obtained rigorously from the analysis of small modulations of the soliton, see e.g.Refs.[13,15].In absence of higher-order dispersion (β 3 = β 4 = 0), parameter 1/v s = η 1 (b) = dη(b)/db, see Eq. ( 7).Therefore, for media with quadratic dependence only, η sol (ω) is a straight line that is parallel to the tangent to the dispersion relation of linear waves at frequency b, and shifted up by the amount depending on peak power A 2 s [13,15,16].In presence of higher-order terms (β 3 = 0, β 4 = 0), the soliton velocity 1/v s differs from η 1 (b), therefore η sol (ω) is not parallel to the tangent, see Eq. ( 7).
In media with quadratic and cubic dispersion terms (β 4 = 0), resonant linear waves are always generated because η sol (ω) intersects η(ω).Though the theory developed predicts the presence of solitons in media with cubic dispersion, these solitons are not stationary due to the continuous transfer of energy from solitons to linear waves.The lifetime of such solitons can be large if the resonant frequency is far from center b of the soliton spectrum.In contrast, when the quartic term is included, one can find a range of frequencies b, for which η sol (ω) does not intersect with η(ω).For example, such frequencies can be found near the extrema of η(ω).
The discussion above can be generalized with the following statement.Localized pulses are possible in nonlinear media with any order of dispersion.If the highestorder dispersion term is odd, then these pulses are nonstationary (quasi-stationary) due to the continuous radiation of linear waves.If the highest-order dispersion term is even, stationary stable pulses may exist.A necessary condition in the latter case is that the soliton dispersion relation does not intersect with the dispersion relation of linear waves.
We mention also about embedded solitons.The spectrum of these localized waves is located within the spectrum of linear waves, see e.g.Refs.[18,19].Embedded solitons appear mainly in multi-component systems [19], though they also exist in scalar systems with cubicquintic nonlinearity [18].However, these solitons exist for particular relations of the systems parameters.To the best of our knowledge, embedded solitons are not found for a system with cubic nonlinearity only.Extensive numerical simulations of Eq. (1) show that when the intersection of spectra occurs (with the soliton parameters found from the variational approach), no stationary solitons exist for β 4 γ > 0.

III. NUMERICAL SIMULATIONS AND TRANSFORMATIONS
In order to check theoretical predictions, we perform numerical simulations of Eq. (1).For this purpose, we take all variables as dimensionless.We consider three cases: i) small |β 4 |, ii) small |β 2 |, and iii) β 2 > 0. We take such values of parameters that there is no intersection of η sol (ω) and η(ω).The split-step Fourier method [1] is used.The size of the computational window is T num = 30-50, and the number of discretization points is 512-1024.Absorbing boundary conditions are used to prevent the reflection of linear waves from edges.Initial conditions are in the form of Eq. ( 4).A relatively small value of β 3 = 0.2 is taken for convenience to restrict the size of the computational window because 1/v grows with an increase of β 3 , see Eq. (7).Theoretical predictions have the similar accuracy for larger β 3 as well.
1(a)-(c) shows the dynamics of solitons for the three sets of parameters, and E 0 = 2. Since initial profiles are approximate, solitons adjust their shapes, emitting linear waves.In Fig. 1(c), the field at large z has oscillating tails.Though, trial function ( 4) is different from this form, the theory gives acceptable values for stationary parameters with a deviation of 10-20%, even for larger values of β 2 (≥ 0.5).In Fig. 1(d), variations of the soliton amplitudes for the dynamics in Fig. 1(a)-(c) are presented.The amplitudes tend to stationary values via damped oscillations.Also, Fig. 1 demonstrates the stability of solitons to small modulations.Dependencies of A s and 1/v s on E 0 and b are presented in Fig. 2. Soliton velocity 1/v s is found as the average velocity over range z ∼ 20-50 after the adjustment process.There are small deviations of the predicted values from those found from numerical simulations.However, the theory gives correctly the overall trend of all dependencies in Fig. 2. The soliton amplitude increases on E 0 , and correspondingly, the soliton width a s decreases on E 0 .Contributions of the dispersion terms can be compared using the characteristic lengths [1] The smaller the length is, the more important is the contribution of the corresponding effect.Since a s varies on E 0 and b, relative contributions of the dispersion terms are changed as well.For example, for (β 2 , β 3 , β 4 ) = (−1, 0.2, −0.2), the corresponding lengths are L GVD = 1.67,L TOD = 10.8, and L FOD = 13.9 at E 0 = 2, while at E 0 = 10, L GVD = 0.124, L TOD = 0.218, and L FOD = 0.0767.Therefore, as E 0 increases, the influence of TOD and FOD increases as well.Points corresponds to parameters found from numerical simulations of Eq. ( 1).
Ratios of |β 2 | and |β 4 |, as in Figs. 1 and 2, can be obtained in optical media at wavelengthes close to zero dispersion points.We consider, as an example, the structure in Ref. [2].To find values of β and β 4 , we fit the dependence β 2 (λ) in Fig. 1(d The standard NLS equation, i.e.Eq. ( 1) with β 3 = β 4 = 0, is invariant under the Galilean transformation.It means that a moving solution of the NLS equation can be obtained from a static solution by a corresponding change of variables.In contrast, the full Eq.( 1) is not Galilean invariant.The shape of the soliton can be altered as the velocity changes.This property is ignored in trial function (4).Nevertheless, this function gives a reasonable approximation for solitons.

IV. CONCLUSIONS
In conclusion, we have demonstrated that stationary pulses, generic quartic solitons, can propagate in media with GVD, TOD, and FOD.Numerical simulations of Eq. (1) show that these pulses are stable for sufficiently long distances.Conditions in terms of the system parameters have been identified for the existence of GQS.In particular, these solitons exist both for the positive GVD and negative GVD parameters.Parameters of stationary solitons for different energies and soliton frequencies have been found approximately.Values of these parameters are close to those found numerically.It has been demonstrated that the soliton velocity in general quartic media differs, in principle, from the inverse of the group velocity of linear waves.It has been shown that the resonance interaction of a pulse with linear waves can prevent the existence of stationary solitons.Transformations, that connect solutions of Eq. ( 1) with those of Eq. ( 15), have been obtained.Our analysis provides strong support for a conjecture that stable solitons can exist in media with a general form of dispersion if the highest-order dispersion term is even.
Our results suggest an alternative view on the dynamics of pulses in dispersive nonlinear media, in particular, during supercontinuum generation.Usually, the dynamics is considered as a perturbation of solitons of the standard (with GVD only) NLS model.However, the dynamics can also be treated as an adjustment of pulses to stationary solitons associated with higher-order dispersion.