Discrete Solitons Research Articles

Optical control of topological end states via soliton formation in a 1D lattice

Abstract Discrete spatial solitons are self-consistent solutions of the discrete nonlinear Schrödinger equation that maintain their shape during propagation. Here we show, using a pump-probe technique, that soliton formation can be used to optically induce and control a linear topological end state in the bulk of a Su–Schrieffer–Heeger lattice, using evanescently-coupled waveguide arrays. Specifically, we observe an abrupt nonlinearly-induced transition above a certain power threshold due to an inversion symmetry-breaking nonlinear bifurcation. Our results demonstrate all-optical active control of topological states.

Correction: Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Correction: Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Dynamics of discrete solitons in the fractional discrete nonlinear Schrödinger equationwith the quasi-Riesz derivative.

We elaborate a fractional discrete nonlinear Schrödinger (FDNLS) equationbased on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its Lévy index (LI). This FDNLS equationrepresents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wave numbers. Formation of single-site and two-site discrete solitons is explored, starting from the anticontinuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied.

Higher-order beyond-band discrete solitons in binary waveguide arrays

We study higher-order beyond-band discrete solitons (HOBBDSs) and quasi-HOBBDSs, which can be constructed by multiplying the solutions of fundamental single-peaked beyond-band discrete solitons by a soliton order parameter larger than unity. In the quasi-continuous regime when the HOBBDS peak amplitude is low (thus its width is large) and the soliton order parameter is a small integer number, HOBBDSs periodically evolve during propagation and their dynamics are similar to those of higher-order solitons governed by the nonlinear Schrödinger equation in an optical fiber, including the periodicity, pattern evolution, and independence of the period length on the soliton order parameter. If the soliton order parameter is still small but not an integer, then one can obtain the quasi-HOBBDSs whose profiles almost periodically evolve during propagation. The breathing length of quasi-HOBBDSs decreases if the soliton order parameter increases. Moreover, the breathing length of quasi-HOBBDSs is approximately inversely proportional to the square values of the peak amplitude of the fundamental beyond-band discrete solitons, just like what happens with the period length of the higher-order solitons governed by the nonlinear Schrödinger equation. If the fundamental beyond-band discrete solitons are intense enough and/or the soliton order parameter is large enough, then most of the energy of the beams is eventually trapped in a single waveguide.

INTERACTION BETWEEN A BEYOND-BAND DISCRETE SOLITON AND A DIRAC SOLITON IN BINARY WAVEGUIDE ARRAYS

We investigate the intricate interactions between a beyond-band discrete soliton (BBDS) and Dirac soliton in a binary waveguide array (BWA). Through comprehensive analysis, we demonstrate that the behavior of these soliton interactions remains largely unaffected by the presence of central peaks or dips at the center of each soliton, or the initial phase differences between them. Our research highlights the critical role of the BBDS intensity in determining the nature of its interaction with Dirac solitons. We reveal that low-intensity BBDS exhibit consistent attraction towards Dirac solitons across multiple encounters, while high-intensity BBDSs show an initial phase of attraction followed by a repulsion phase, causing a subsequent divergence. These findings not only enhance our understanding of soliton interactions on a fundamental level but also set the stage for the development of innovative optical communication and signal processing technologies leveraging the distinctive properties of BBDSs and Dirac solitons.

Radiation families emitted by a discrete soliton in parity-time-symmetric waveguide arrays

Radiation families emitted by a discrete soliton in parity-time-symmetric waveguide arrays

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Discrete solutions and interaction dynamics for coupled nonlocal discrete nonlinear Schrödinger equations in an inhomogeneous optical fiber

The nonlinear Schrödinger (NLS) equation has made great achievement, however the coupled nonlocal discrete nonlinear Schrödinger (CNDNLS) equations have less works. In this paper, we investigate the Darboux transformation (DT) method to obtain discrete soliton solutions of CNDNLS equations. We present two different kinds of solutions through choosing different seed solutions, and analysis the relations among them. Further many novel discrete 1-soliton and 2-soliton are derived with the zero and nonzero seed solutions. Meanwhile, the elastic interaction dynamic of two discrete solitons is displayed, it is shown that the amplitudes keep unchanged after the interactions in long time.

Matter-wave solitons in an array of spin-orbit-coupled Bose-Einstein condensates.

We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.

Discrete optical solitons: Perspectives and new trends

Fifty years ago, in 1973 temporal optical solitons were predicted by Hasegawa and Tappert, highlighting the possibilities of balancing dispersive and nonlinear effects in optical fibers. The next decade followed by similar advances on spatial and discrete solitons. This article brings perspectives and current research on topological photonics.

Beyond-band discrete soliton interaction in binary waveguide arrays.

We investigate different scenarios of interaction between two beyond-band discrete solitons (BBDSs), which are a new class of solitons in binary waveguide arrays and have been investigated just recently. In the quasi-continuous regime when solitons intensity is low and, thus, solitons are broad, two BBDSs with the same envelope in binary waveguide arrays interact with each other practically like two well-known fundamental solitons governed by the nonlinear Schrödinger equation in a single optical fiber. However, this similarity disappears if the discrete nature of the system is enhanced by increasing the intensity of BBDSs. In that case, two initially in-phase BBDSs with the same detuning cannot periodically collide during propagation. We also show that single-peaked BBDSs are more robust and less mobile than double-peaked BBDSs with the same detuning. This robustness stops two identical single-peaked BBDSs from interaction even at initial separations when double-peaked BBDSs can still strongly interact with each other or with single-peaked BBDSs.

Discrete solitons in competitive zigzag waveguide arrays with cubic-quintic nonlinearity

In this paper, we study one-dimensional discrete solitons in zigzag waveguide arrays with competitive cubic-quintic nonlinearity and competitive linear mixing between the nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. The competitive nonlinearity features a cubic self-focusing associated with a quintic self-defocusing nonlinearities. The competitive linear mixing between the NN and NNN couplings is induced by making the two coefficients opposite of each other, which is assumed to be induced by the embedding synthetic gauge phase within the coupling constants. The combination of these two types of competition, linear mixing and nonlinearity can create four types of soliton: multipeak bell-shaped solitons, multipeak flat-top solitons, staggered bell-shaped solitons, and staggered flat-top solitons. The stability and dynamics of these types of solitons are verified systematically through the paper. The total power and the types of competition between the linear mixing play important roles in tuning these solitons.

The existence of discrete solitons for the discrete coupled nonlinear Schrödinger system

In this paper, we investigate the nonlinear coupled discrete Schrödinger equations with unbounded potentials. We find simple sufficient conditions for the existence of discrete soliton solution by using the Nehari manifold approach and the compact embedding theorem. Furthermore, by comparing the value of the action functional at the discrete soliton solution with those at nonzero solutions of one component zero, we demonstrate that both components of the discrete soliton solution are nontrivial.

Collision dynamics of a discrete soliton in uniform waveguide arrays

We investigate the collision dynamics of a discrete soliton (DS) pair in a realistic semi-infinite nonlinear waveguide array (WA) in the context of diffractive resonant radiation (DifRR). Depending on the initial amplitude ( A 0 ) and wavenumber ( k 0 ), a co-moving pair of identical DSs either collide elastically or merge to form a discrete breather. For a large amplitude and small wavenumber, DSs do not interact to form a bound state. We map the domain of interaction by iterative simulation and present a phase plot in ( A 0 − k 0 ) parameter space. To grasp the collision dynamics, we develop an analytical treatment based on a variational method that leads us to a set of ordinary differential equations describing the collision mechanism. Our analytical results corroborate our numerical data. We further investigate the role of initial phase detuning between two identical DSs and observe a periodic energy exchange during their interaction. A varied degree of energy exchange occurs when two DSs with different wavenumbers collide, which results in three distinct output states accompanied by DifRR generation. Extending our investigation to a more generalized condition by taking a nonidentical DS pair, we find unique secondary radiation in -space that originates due to the interaction of DSs during collision. The nature of this collision mediated secondary radiation is found to be different from usual DifRR. Our results shed light on the interesting aspects of the collision dynamics of a DS pair in a nonlinear WA and are useful in understanding the complex mechanism.

Comment on "A new class of out-gap discrete solitons in binary waveguide arrays" [Chaos 32, 073113 (2022)

Recent results [M. C. Tran and T. X. Tran, Chaos 32, 073113 (2022)] are put into the context of earlier work [A. V. Gorbach and M. Johansson, Eur. Phys. J. D 29, 77-93 (2004); M. Johansson and A. V. Gorbach, Phys. Rev. E 70, 057604 (2004)]. The two newly found families of "beyond-band discrete solitons" [M. C. Tran and T. X. Tran, Chaos 32, 073113 (2022)] are found to be smooth continuations of "on-top breathers" briefly mentioned by Gorbach and Johansson [Eur. Phys. J. D 29, 77-93 (2004)] and Johansson and Gorbach [Phys. Rev. E 70, 057604 (2004)] and the relevant bifurcation scenarios are described. One family is shown to be linearly unstable with respect to symmetry-breaking oscillations.

Response to "Comment on 'A new class of out-gap discrete solitons in binary waveguide arrays'" [Chaos 32, 073113 (2022)

Response to "Comment on 'A new class of out-gap discrete solitons in binary waveguide arrays'" [Chaos 32, 073113 (2022)

Variational quantum algorithm for Gaussian discrete solitons and their boson sampling

In the context of quantum information, highly nonlinear regimes, such as those supporting solitons, are marginally investigated. We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine-slearning model, we find different soliton solutions, varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves.

A new class of out-gap discrete solitons in binary waveguide arrays.

We analytically and numerically investigate beyond-band discrete solitons, which present a completely new class of stable localized out-gap solitons with detunings being located beyond the two bands of the linear plane waves in a periodic binary waveguide array. Each of the even and odd components of these discrete solitons does not change its sign across the transverse direction of the binary waveguide array. The even and odd components of these newly found discrete solitons can be approximately presented by two hyperbolic secant functions with the only difference in their peaks. This approximation is especially good in the low-intensity regime in which the detuning of these solitons can asymptotically reach the two limits of a linear spectrum. These distinguishing features altogether make the newly found discrete solitons different from all other classes of discrete solitons investigated earlier in binary waveguide arrays. Two transformation rules for constructing even and odd components of these discrete solitons are also found for various combinations of signs of the propagation mismatch σ and nonlinear coefficient γ.

Infinite Homoclinic Solutions of the Discrete Partial Mean Curvature Problem with Unbounded Potential

The mean curvature problem is an important class of problems in mathematics and physics. We consider the existence of homoclinic solutions to a discrete partial mean curvature problem, which is tied to the existence of discrete solitons. Under the assumptions that the potential function is unbounded and that the nonlinear term is superlinear at infinity, we obtain the existence of infinitely many homoclinic solutions to this problem by means of the fountain theorem in the critical point theory. In the end, an example is given to illustrate the applicability of our results.