Embedded Solitons Research Articles

Embedded solitons in the double sine-Gordon lattice with next-neighbor interactions.

Topological solitons can propagate without radiation in discrete media. These solutions are known as embedded solitons (ES). They come as isolated solutions and exist despite their resonance with the linear spectrum of the respective lattices. In this paper the properties of embedded solitons in the discrete double sine-Gordon equation with next-neighbor and second-neighbor interactions are investigated. Depending on the sign of these interactions they can be either destructive or favorable for the ES creation. The ES existence area depends on the width of the linear spectrum: narrowing of the spectrum widens the ES existence range and vice versa. The application to the Josephson junction arrays is discussed.

Embedded Soliton Solutions: A Variational Study

Embedded solitons(ES) represent solitary waves which reside inside the continuous spectrum of a nonlinear-wave system. This type of solitons was first reported by Yang et al. [1] in optical models characterized by opposing dispersion and competing nonlinearities at fundamental and second harmonics. More specifically, optical media with quadratic χ and cubic χ nonlinear susceptibilities can support ES solutions. The evolution of ES is governed by the coupled nonlinear partial differential equations [2]:

Detection of symmetric homoclinic orbits to saddle-centres in reversible systems

We present a perturbation technique for the detection of symmetric homoclinic orbits to saddle-centre equilibria in reversible systems of ordinary differential equations. We assume that the unperturbed system has primary, symmetric homoclinic orbits, which may be either isolated or appear in a family, and use an idea similar to that of Melnikov’s method to detect homoclinic orbits in their neighbourhood. This technique also allows us to identify bifurcations of unperturbed or perturbed, symmetric homoclinic orbits. Our technique is of importance in applications such as nonlinear optics and water waves since homoclinic orbits to saddle-centre equilibria describe embedded solitons (ESs) in systems of partial differential equations representing physical models, and except for special cases their existence has been previously studied only numerically using shooting methods and continuation techniques. We apply the general theory to two examples, a four-dimensional system describing ESs in nonlinear optical media and a six-dimensional system which can possess a one-parameter family of symmetric homoclinic orbits in the unperturbed case. For these examples, the analysis is compared with numerical computations and an excellent agreement between both results is found.

Accumulation of embedded solitons in systems with quadratic nonlinearity

Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multiwave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model. An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter epsilon, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of Wentzel-Kramers-Brillouin type yields an asymptotic formula for the distribution of discrete values of epsilon at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, -65, and a fairly good approximation for the numerical factor in front of the scaling term.

Discrete embedded solitons

We address the existence and properties of discrete embedded solitons (ESs), that is, localized waves existing inside the phonon band in a nonlinear dynamical-lattice model. The model describes a one-dimensional array of optical waveguides with both χ(2) (second-harmonic generation) and χ(3) (Kerr) nonlinearities, for which a rich family of ESs are known to occur in the continuum limit. First, a simple motivating problem is considered, in which the χ(3) nonlinearity acts in a single waveguide. An explicit solution is constructed asymptotically in the large wavenumber limit. The general problem is then shown to be equivalent to the existence of a homoclinic orbit in a four-dimensional reversible map. From properties of such maps, it is shown that (unlike ordinary gap solitons) discrete ESs have the same codimension as their continuum counterparts. A specific numerical method is developed to compute homoclinic solutions of the map, that are symmetric under a specific reversing transformation. Existence is then studied in the full parameter space of the problem. Numerical results agree with the asymptotic results in the appropriate limit and suggest that the discrete ESs may be semi-stable as in the continuous case.

Four-wave solitons in Bragg cross-gratings

A temporal-domain model for a two-dimensional (2D) waveguide with two transversal Bragg gratings (BGs) is introduced. In this medium, four waves are coupled linearly by the Bragg reflections, and nonlinearly by XPM and four-wave mixing. The same model in the spatial domain applies to a 3D photonic crystal. The subject of the work is (effectively) 1D solitons in this system. Equations which govern their dynamics also describe the evolution of two circular polarizations in a fibre BG. Depending on the relative strength of the two gratings, the system may or may not have a gap in its spectrum, and it always has 'semi-gaps', where embedded solitons (ES) may be found. In the case when the propagation constant k (frustration parameter) is zero, the system admits two reductions, symmetric (S) and anti-symmetric (anti-S), to two-wave systems, which are equivalent to the standard fibre-BG model. These systems give rise to exact solutions, which, in terms of the full system, may be gap solitons (GS) or ES. Asymmetric (aS) solitons for k = 0, and solutions generated by the continuation of the S, anti-S and aS solitons to , are found by means of the variational approximation and in a numerical form. In the semi-gaps, these solutions may be both ES and quasi-solitons (QS) with nonvanishing tails. Stability of the solitons is first investigated by means of the Vakhitov–Kolokolov (VK) criterion, which predicts no instability for the extended solution families of the S and anti-S type. For the aS solitons, the VK criterion predicts a nontrivial stability border inside the bandgap, the location of which is controlled by k: at k = 0 the entire bandgap is VK-stable, while for large k the area of the VK stability shrinks to nothing. This intra-gap stability border is a new feature of GS models. Direct simulations demonstrate that, at k = 0, the S and anti-S reductions are always stable against perturbations violating the reduction, hence the stability of the exact S and anti-S solitons can be predicted on the basis of known results for the standard fibre-BG model. Continuing these solutions in k and testing their stability by simulations, we conclude that the solution branches stemming from the stable and unstable exact solitons at k = 0 always remain, respectively, stable or unstable. As a result, the extended soliton family of the S type is chiefly (but not exactly) stable inside the bandgap and unstable in the semi-gaps, while the opposite is true for the extended family of the anti-S type solutions. The aS solitons which are predicted to be VK-unstable quickly decay indeed, while their VK-stable counterparts survive for a long time, but are finally destroyed by oscillatory perturbations, which the VK criterion does not take into consideration. Moving solitons are constructed too, and a limit velocity is found for them.

Embedded solitons in Lagrangian and semi-Lagrangian systems

We develop the technique of the variational approximation (VA) for solitons in two directions. First, one may have a physical model which does not admit the usual Lagrangian representation, as some terms were discarded for various reasons. For instance, the second-harmonic-generation (SHG) model considered here, which includes the Kerr nonlinearity, lacks the usual Lagrangian representation if one ignores the Kerr nonlinearity of the second-harmonic, as compared to that of the fundamental. However, we show that, with a natural modification, one may still apply the VA to those seemingly flawed systems as efficiently as it applies to their fully Lagrangian counterparts. We call such models, that do not admit the usual Lagrangian representation, semi-Lagrangian systems. Second, we show that, upon adding an infinitesimal tail that does not vanish at infinity, to a usual soliton ansatz, one can obtain an analytical criterion which (within the framework of VA) gives a condition for finding embedded solitons (ESs), i.e., isolated truly localized solutions existing inside the continuous spectrum of the radiation modes. The criterion takes a form of orthogonality of the radiation mode in the infinite tail to the soliton core. To test the criterion, we have applied it to both the semi-Lagrangian truncated version of the SHG model and to the same model in its full form. In the former case, the criterion (combined with VA for the soliton proper) yields an exact solution for the ES. In the latter case, the criterion selects the ES with a relative error ≈1%.

Embedded solitons: solitary waves in resonance with the linear spectrum

It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton’s frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of codimension-one solitons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave’s tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive properties of semi-stability of ESs, and moving ESs. Results are presented for four different physical models, including an extended fifth-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear-optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. In particular, a new result is that when both harmonics have anomalous dispersion, an ES can exist which is, virtually, completely stable.

Moving embedded solitons

The first theoretical results are reported predicting moving solitons residing inside (embedded into) the continuous spectrum of radiation modes. The model taken is a Bragg-grating medium with Kerr nonlinearity and additional second-derivative (wave) terms. The moving embedded solitons (ESs) are doubly isolated (of codimension two), but, nevertheless, structurally stable. Like quiescent ESs, moving ESs are argued to be stable to linear approximation, and semi-stable nonlinearly. Estimates show that moving ESs may be experimentally observed as ~10 fs pulses with velocity 1/10th that of light.