Standard and embedded solitons in nematic optical fibers

A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier-Lebesgue Spaces

We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2021), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside H 1 2 ( T ) . Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061-1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier-Lebesgue spaces F L s , p ( T ) for s ≥ 1 2 and 1 ≤ p < ∞ . As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier-Lebesgue spaces with infinite momentum.

Discrete rational and breather solution in the spatial discrete complex modified Korteweg-de Vries equation and continuous counterparts

In this paper, a spatial discrete complex modified Korteweg-de Vries equation is investigated. The Lax pair, conservation laws, Darboux transformations, and breather and rational wave solutions to the semi-discrete system are presented. The distinguished feature of the model is that the discrete rational solution can possess new W-shape rational periodic-solitary waves that were not reported before. In addition, the first-order rogue waves reach peak amplitudes which are at least three times of the background amplitude, whereas their continuous counterparts are exactly three times the constant background. Finally, the integrability of the discrete system, including Lax pair, conservation laws, Darboux transformations, and explicit solutions, yields the counterparts of the continuous system in the continuum limit.

2D in-band solitons in PT-symmetric waveguide arrays

We address two types of two-dimensional (2D) localized solitons in Kerr media with an imprinted quasi-one-dimensional lattice featuring a parity-time (PT) symmetry. Solitary waves originating from the edges of the Bloch bands are stable in their entire existence domains. Purely nonlinear multipeaked states propagate stably in wide parameter windows. Both types of nonlinear waves exist in the finite bandgaps of the corresponding linear system and, cross-continuously, the Bloch band (continuous spectrum) sandwiched between (or neighboring) them. To the best of our knowledge, our findings thus provide the first example of "embedded solitons" in 2D PT periodic systems.

Rogue-wave bullets in a composite (2+1)D nonlinear medium

We show that nonlinear wave packets localized in two dimensions with characteristic rogue wave profiles can propagate in a third dimension with significant stability. This unique behavior makes these waves analogous to light bullets, with the additional feature that they propagate on a finite background. Bulletlike rogue-wave singlet and triplet are derived analytically from a composite (2+1)D nonlinear wave equation. The latter can be interpreted as the combination of two integrable (1+1)D models expressed in different dimensions, namely, the Hirota equation and the complex modified Korteweg-de Vries equation. Numerical simulations confirm that the generation of rogue-wave bullets can be observed in the presence of spontaneous modulation instability activated by quantum noise.

Solitary waves in diatomic chains

We consider the mechanism of formation of isolated localized wave structures in the diatomic Fermi-Pasta-Ulam (FPU) model. Using a singular multiscale asymptotic analysis in the limit of high mass mismatch between the alternating elements, we obtain the typical slow-fast time scale separation and formulate the Fredholm orthogonality condition approximating a sequence of mass ratios supporting the formation of solitary waves in the general type of diatomic FPU models. This condition is made explicit in the case of a diatomic Toda lattice. Results of numerical integration of the full diatomic Toda lattice equations confirm the formation of these genuinely localized wave structures at special values of the mass ratio that are close to the analytical predictions when the ratio is sufficiently small.

Diversity of solitons in a generalized nonlinear Schrödinger equation with self-steepening and higher-order dispersive and nonlinear terms

In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.

Chaotic solitons in the quadratic-cubic nonlinear Schrödinger equation under nonlinearity management

We analyze the response of rational and regular (hyperbolic-secant) soliton solutions of an extended nonlinear Schrödinger equation (NLSE) which includes an additional self-defocusing quadratic term, to periodic modulations of the coefficient in front of this term. Using the variational approximation (VA) with rational and hyperbolic trial functions, we transform this NLSE into Hamiltonian dynamical systems which give rise to chaotic solutions. The presence of chaos in the variational solutions is corroborated by calculating their power spectra and the correlation dimension of the Poincaré maps. This chaotic behavior (predicted by the VA) is not observed in the direct numerical solutions of the NLSE when rational initial conditions are used. The solitary-wave solutions generated by these initial conditions gradually decay under the action of the nonlinearity management. On the contrary, the solutions of the NLSE with exponentially localized initial conditions are robust solitary-waves with oscillations consistent with a chaotic or a complex quasiperiodic behavior.

Moving embedded lattice solitons

It was recently proved that solitons embedded in the spectrum of linear waves may exist in discrete systems, and explicit solutions for isolated unstable embedded lattice solitons (ELS) of a differential-difference version of a higher-order nonlinear Schrodinger equation were found [Gonzalez-Perez-Sandi, Fujioka, and Malomed, Physica D 197, 86 (2004)]. The discovery of these ELS gives rise to relevant questions such as the following: (1) Are there continuous families of ELS? (2) Can ELS be stable? (3) Is it possible for ELS to move along the lattice? (4) How do ELS interact? The present work addresses these questions by showing that a novel equation (a discrete version of a complex modified Korteweg-de Vries equation that includes next-nearest-neighbor couplings) has a two-parameter continuous family of exact ELS. These solitons can move with arbitrary velocities across the lattice, and the numerical simulations demonstrate that these ELS are completely stable. Moreover, the numerical tests show that these ELS are robust enough to withstand collisions, and the result of a collision is only a shift in the positions of the solitons. The model may apply to the description of a Bose-Einstein condensate with dipole-dipole interactions between the atoms, trapped in a deep optical-lattice potential.

Spatial solitons in chiral media

We study theoretically the nonlinear propagation of a narrow optical wave packet through a cholesteric liquid crystal. We derive the equations governing the weakly nonlinear dynamics of an optical field by taking into account the coupling with the liquid crystal. We constructed the solution as the superposition of four narrow wave packets centered around the linear eigenmodes of the helical structure whose corresponding envelopes A are slowly varying functions of their arguments. We found a system of four coupled equations to describe the resulting vector wave packet which has some integration constants and that under special conditions reduces to the nonlinear Schrödinger equation with space-dependent coefficients. We solved this equation both, using a variational approach and performing numerical calculations. We calculated analytically the soliton spatial scales, the transported power, the nonlinear refraction index, and its wavelength dependence, showing that this has its maxima at the edges of the reflection band. We also exhibit the existence of some other exact but non-self-focused solutions.

Symmetric and asymmetric solitons in linearly coupled Bragg gratings

We demonstrate that a symmetric system of two linearly coupled waveguides, with Kerr nonlinearity and resonant grating in both of them, gives rise to a family of symmetric and antisymmetric solitons in an exact analytical form, a part of which exists outside of the bandgap in the system's spectrum, i.e., they may be regarded as embedded solitons (ES's, i.e., the ones partly overlapping with the continuous spectrum). Parameters of the family are the soliton's amplitude and velocity. Asymmetric ES's, unlike the regular (nonembedded) gap solitons (GS's), do not exist in the system. Moreover, ES's exist even in the case when the system's spectrum contains no bandgap. The main issue is the stability of the solitons. We demonstrate that some symmetric ES's are stable, while all the antisymmetric solitons are unstable; an explanation is given to the latter property, based on the consideration of the system's Hamiltonian. We produce a full stability diagram, which comprises both embedded and regular solitons, quiescent and moving. A stability region for ES's is found around the point where the constant of the linear coupling between the two cores is equal to the Bragg-reflectivity coefficient accounting for the linear conversion between the right- and left-traveling waves in each core, i.e., the ES's are the "most endemic" solitary solitons in this system. The stability region quickly shrinks with the increase of the soliton's velocity c, and completely disappears when c exceeds half the maximum velocity. Collisions between stable moving solitons of various types are also considered, with a conclusion that the collisions are always quasielastic.