Breather Soliton Research Articles

Propagation dynamics of Airy beams and nonlinear accelerating beams in biased photorefractive media with quadratic electro-optic effect

An investigation of propagation dynamics of Airy beams and nonlinear accelerating beams in biased photorefractive media with quadratic electro-optic effect is presented by means of numerical simulations. For linear Airy beams, it is found that single breathing soliton can be formed by adjusting the initial amplitude and the external bias electric field during the propagation of one Airy beam. The propagation direction of such soliton can be controlled by introducing a launch angle to the initial beam and it can propagate stably against small fluctuations. Interaction of two Airy beams shows that single breathing solitons and soliton pairs can be produced after collision and the external bias electric field has a big impact on their propagation. For nonlinear accelerating beams, because they have infinite energy, we truncate the oscillating tails to study the interaction of two in-phase and out-of-phase accelerating beams. The numerical results show that single solitons and soliton pairs can generate only within a certain range of external bias electric field.

Generation of breathing solitons in the propagation and interactions of Airy–Gaussian beams in a cubic–quintic nonlinear medium**Project supported by the National Natural Science Foundation of China (Grant No. 51602028), the Science and Technology Development Project of Jilin Province, China (Grant No. 20160520114JH), the Youth Science Fund of Changchun University of Science and Technology, China (Grant No. XQNJJ-2017-04), and the Natural

Using the split-step Fourier transform method, we numerically investigate the generation of breathing solitons in the propagation and interactions of Airy–Gaussian (AiG) beams in a cubic–quintic nonlinear medium in one transverse dimension. We show that the propagation of single AiG beams can generate stable breathing solitons that do not accelerate within a certain initial power range. The propagation direction of these breathing solitons can be controlled by introducing a launch angle to the incident AiG beams. When two AiG beams accelerated in opposite directions interact with each other, different breathing solitons and soliton pairs are observed by adjusting the phase shift, the beam interval, the amplitudes, and the light field distribution of the initial AiG beams.

Spatial multiplexing of soliton microcombs

Dual-comb interferometry utilizes two optical frequency combs to map the optical field's spectrum to a radio-frequency signal without using moving parts, allowing improved speed and accuracy. However, the method is compounded by the complexity and demanding stability associated with operating multiple laser frequency combs. To overcome these challenges, we demonstrate simultaneous generation of multiple frequency combs from a single optical microresonator and a single continuous-wave laser. Similar to space-division multiplexing, we generate several dissipative Kerr soliton states - circulating solitonic pulses driven by a continuous-wave laser - in different spatial (or polarization) modes of a $\mathrm{MgF_2}$ microresonator. Up to three distinct combs are produced simultaneously, featuring excellent mutual coherence and substantial repetition rate differences, useful for fast acquisition and efficient rejection of soliton intermodulation products. Dual-comb spectroscopy with amplitude and phase retrieval, as well as optical sampling of a breathing soliton, is realised with the free-running system. Compatibility with photonic-integrated resonators could enable the deployment of dual- and triple-comb-based methods to applications where they remained impractical with current technology.

Interaction of Airy beams in photorefractive media

We investigate numerical interactions between two Airy beams with different relative phases in photorefractive (PR) nonlinear media in one transverse dimension. It is shown that solitons and soliton pairs are generated that are not accelerating, although the incident beams exhibit acceleration property. When the relative phase is very small, the attraction between the two Airy beams is strengthened and their propagation behavior is similar to that in in-phase case, where the two Airy beams attract each other and the main lobes can fuse into one single breathing soliton, secondary lobes can generate soliton pairs with symmetric intensity distribution. While the relative phase is close to [Formula: see text], these beams present out-phase propagation with strong repulsive effect. Their interaction properties are sensitive to both the interval and relative phase between the two Airy beams.

Optimal control for soliton breathers of the Lakshmanan–Porsezian–Daniel, Hirota, and cmKdV models

We apply the method for systematic construction of the two-soliton (and in principal, N-soliton) breather solutions on the zero background to the Lakshmanan–Porsezian–Daniel model with varying dispersion and nonlinearity, gain or loss, and derive the specific conditions for the appearance of the so-called standing and moving soliton breathers. We show that both all parameters of constituent solitons forming the breather and the varying dispersion and nonlinearities should be appropriately chosen in accordance with the exact integrability conditions of the Lakshmanan–Porsezian–Daniel equation with vanishing boundary conditions. We compare the details of periodic breather dynamics for the complex modified KdV, Hirota, and Lakshmanan–Porsezian–Daniel models with the classical Satsuma–Yajima breather behavior. We derive simple formulas for the soliton breather periods and use them to understand the possibilities to control breather dynamics, and, in particular, to realize the so-called regimes of “artificial breathing”, when periods of soliton breathing in space and time are controlled by choosing appropriate complex spectral parameters defining the amplitudes and velocities of constituent solitons. Of fundamental importance is the remarkable theoretical fact that the higher-order nonlinear and dispersion effects included into (or removed from) the new higher-order equations of the AKNS hierarchy, as a rule, do not destroy soliton breathers discovered here and do not transform them into dispersive waves. On the contrary, we demonstrate that novel and novel soliton breathers can appear in the framework of the next, more and more higher orders of the AKNS and NLSE hierarchies.

Novel soliton breathers for the higher-order Ablowitz–Kaup–Newell–Segur hierarchy

In the framework of the nonautonomous exactly integrable higher-order equations belonging to the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy, the method for systematic construction of the bound soliton states (localized breathers on the zero background) is presented, and general and specific conditions for their appearance are formulated. We show that both all parameters of solitons forming the breather, and the varying dispersion and nonlinearities should be appropriately chosen in accordance with the exact integrability conditions of the models. Novel breather solutions of the higher-order nonlinear evolution equations with vanishing boundary conditions generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. We demonstrate that novel soliton breather solutions substantially extend the breather concept known so far. The revealed specific conditions for the breather formation can be generalized and applied to the evolution equations of the next, more and more higher orders of the AKNS hierarchy.

Do N-soliton breathers exist for the Hirota equation models?

We reveal the method for systematic construction of unexpected bound soliton states (localized breathers on the zero background) and formulate specific conditions for their appearance both in the framework of the canonical and the generalized nonautonomous Hirota models. These conditions show that both all parameters of solitons forming the breather and the dispersion and nonlinearity variations cannot be chosen independently, they are related by the exact integrability of the model. Novel N-soliton breathers derived for the nonautonomous Hirota equation with vanishing boundary conditions differ in that they generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain, or losses. We apply the specific conditions for the breather formation to derive novel soliton breather solutions of the nonlinear Schrödinger and the complex modified Korteweg–de Vries equations. Consequently, we demonstrate that the way is opened to generalize and apply the presented method for other evolution equations of the Ablowitz–Kaup–Newell–Segur hierarchy.

Novel conditions for soliton breathers of the complex modified Korteweg–de Vries equation with variable coefficients

We reveal novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries equation with variable coefficients and the loss (or gain) term (vc cmKdV). Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. This finding allows us to systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.

Incoherent interactions of Airy beams in nonlocal nonlinear media

We investigate, for the first time to our best knowledge, the incoherent interactions of incoherently coupled vector Airy beams in both local and nonlocal nonlinear media. In local media, an anomalous phenomena happens with repulsion appears in the incoherent interactions of Airy beams for some beam intervals. Otherwise, bound states of soliton pairs can form. In the regime of strong light (large amplitude or power), repulsion appears for all the beam intervals. In particular, repulsion and attraction of breathing solitons can coexist. Nonlocality provides a long range attractive force between Airy beams, leading to the formation of bound states of breathing Airy solitons. We also demonstrate the incoherent interactions of Airy beams with different amplitudes in detail.

Akhmediev breathers, Peregrine solitons and Kuznetsov-Ma solitons in optical fibers and PCF by Laplace-Adomian decomposition method

In this paper, we apply the Adomian decomposition method combined with the Laplace transform (LADM) in order to solve the cubic nonlinear Schrödinger equation for special cases Akhmediev breather (AB), Peregrine (PS) and Kuznetsov-Ma (KM) solitons. The proposed iterative scheme finds the solution without the need for a linearization or discretization process. Finally, five numerical examples are given to verify the high accuracy and reliability of the proposed analytical approach.

Semi-discrete breather in a helicoidal DNA double chain-model

The nonlinear dynamics of DNA molecular chain is studied for longitudinal and transversal motions through a new discrete helicoidal zigzag model with four degrees of freedom. We take into account the Stokes and hydrodynamical viscous forces. In the semi-discrete approximation, we show that the coupled nonlinear partial differential equations for the longitudinal and transversal out-of-phase motions can be reduced to the nonlinear Schrödinger equation with complex coefficients, allowing analytical breather soliton solution. We found analytically as well as numerically that increasing the damping constant reduces the amplitude and increases the width of the soliton. When the zigzag angle decreases, the height of the soliton increases, but its width remains constant. The linear stability analysis of the system is performed. The growth rate of the instability and the instability regions are discussed as the functions of damping constant, zigzag angle and system parameters.

Generation of single or double parallel breathing soliton pairs, bound breathing solitons, moving breathing solitons, and diverse composite breathing solitons in optical fibers.

Interactions of two truncated Airy pulses with arbitrarily initial relative phases, initial pulse intervals, and different soliton order are numerically investigated in optical fibers. When the soliton order is 1, depending on different initial pulse intervals, the initial in-phase Airy pulses may evolve to single breathing solitons, bound breathing solitons, and single parallel breathing solitons. While the out-of-phase Airy pulses may evolve to parallel or repulsive soliton pairs with breathing or weak breathing, after radiating away some dispersive waves. When the initial relative phases take arbitrary values except 0 and π, moving single breathing solitons and repulsive or parallel soliton pairs will form. Moreover, the whole temporal profiles may become asymmetric. The repulsive soliton pairs consist of two moving breathing solitons with different intensities, moving velocities, and breathing periods. The most interestingly is that, when the soliton order is larger than one, we observe double bound breathing solitons, double parallel breathing soliton pairs, and diverse composite breathing solitons which consist of two or more different breathing solitons. one can effectively manipulate and select the soliton expected and its evolution dynamics by adjusting the soliton order, initial pulse intervals, and initial relative phases.

Knot soliton solutions for the one-dimensional non-linear Schrödinger equation

We identify that a breather soliton solution for the one-dimensional non-linear Schrödinger equation, presented here, is characteristically distinct when one studies the associated space curve, specifically that this space curve is knotted. The significance of these solutions with such a non-trivial geometrical element is pre-eminent on two counts: it is a one-dimensional model wherein structures with such non-trivial geometry are unexpected, and that the nonlinear Schrödinger equation is well known to model a plethora of physical systems.

Localized nonlinear waves and dynamical stability in spinor Bose–Einstein condensates with time–space modulation

Nonlinearity is one of the most remarkable characteristics of Bose–Einstein condensates (BECs). Much work has been done on one- and two-component BECs with time- or space-modulated nonlinearities, while there is little work on spinor BECs with space–time-modulated nonlinearities. In the present paper we investigate localized nonlinear waves and dynamical stability in spinor Bose–Einstein condensates with nonlinearities dependent on time and space. We solve the three coupled Gross–Pitaevskii equations by similarity transformation and obtain two families of exact matter wave solutions in terms of Jacobi elliptic functions and the Mathieu equation. The localized states of the spinor matter wave describe the dynamics of vector breathing solitons, moving breathing solitons, quasi-breathing solitons and resonant solitons. The results show that one-order vector breathing solitons, quasi-breathing solitons, resonant solitons and the moving breathing solitons ψ±1 are all stable, but the moving breathing soliton ψ0 is unstable. We also present the experimental parameters to realize these phenomena in future experiments.

Darboux transformation of a new generalized nonlinear Schrödinger equation: soliton solutions, breather solutions, and rogue wave solutions

In this paper, a new generalized nonlinear Schrodinger (GNLS) equation is investigated by Darboux matrix method. Firstly, the n-fold Darboux transformation (DT) of the GNLS equation is constructed. Then, the soliton solutions, breather solutions, and rogue wave solutions of the GNLS equation are studied based on the DT by choosing different seed solutions. Furthermore, the dynamic features of these solutions are explicitly delineated through some figures with the help of Maple software.

Propagation and interaction of finite-energy Airy–Hermite–Gaussian beams in photorefractive media

We investigate numerically the propagation and interaction characteristics of finite-energy Airy–Hermite–Gaussian beams in biased photorefractive media. For the case of first-order Hermite polynomial, two main lobes of the initial input beam can form breathing solitons with two components. The interval between two soliton components in the \(y{\text{-direction}}\) increases gradually with the propagation distance, while the central position of two soliton components in the \(x{\text{-direction}}\) is almost unchanged during propagation. Moreover, in interaction situations, four main lobes of the two Airy–Hermite–Gaussian beams can also form breathing solitons with four components under the in-phase and out-of-phase conditions, respectively.

Vector breather-to-soliton transitions and nonlinear wave interactions induced by higher-order effects in an erbium-doped fiber

Vector breather-to-soliton transitions for the higher-order nonlinear Schrödinger–Maxwell–Bloch (NLS–MB) system with sextic terms are investigated. The Lax pair and Darboux transformation (DT) of such system are constructed. With the DT, analytic vector breather solutions up to the second order are obtained. With appropriate choices of the spectra parameters, vector breather-to-soliton transitions happen. Interaction mechanisms of vector nonlinear waves (breather–soliton or soliton–soliton interactions) are displayed.

Experimental observation of breathing solitons and a third harmonic in a tapered photonic crystal fiber

We report the experimental observation of breathing solitons and a third harmonic in a tapered fluorotellurite photonic crystal fiber (PCF) pumped by a 1560 nm femtosecond fiber laser. The PCF has a core diameter that varies continuously along its length, resulting in a zero-dispersion wavelength that moves from 1325 nm to 906 nm over the transition region. By finely controlling the dispersion map of the tapered PCF and increasing the order of the optical solitons, their breathing behavior is observed in the frequency domain and the number of breaths goes up to 9. Furthermore, the breathing behavior of the optical soliton is transferred to the third harmonic through inter-modal phase-matched processes in the tapered PCF, and the third harmonic also breathes with an increase in the pump power.

Preventable interaction of attosecond soliton in an inhomogeneous lossy fiber: Application to dispersion and nonlinearity management

In this paper, inhomogeneous sixth order nonlinear Schrödinger equation with variable dispersion, nonlinearity and loss/gain is considered which describes the propagation of attosecond soliton in an inhomogeneous nonlinear fiber medium. Multi-soliton solutions are obtained by the Darboux transformation technique based on the newly constructed Lax pair. In order to study the impact of variable coefficients on soliton dynamics in an inhomogeneous fiber, various forms of inhomogeneous profiles for variable coefficients are considered. By symbolic computation, characteristics of these coefficients are discussed including the soliton amplification, soliton compression, boomerang soliton and breathing soliton. We further discuss the preventable interaction between the attosecond solitons which are attained through manipulating the control parameters in various inhomogeneous functions. Our results may be useful for the experimental control and manipulation of the attosecond soliton in inhomogeneous optical fiber system.

Intermode Breather Solitons in Optical Microresonators

The observation of temporal dissipative Kerr solitons in optical microresonators provides, on the applied side, compact sources of coherent optical frequency combs that have already been applied in coherent communications, dual comb spectroscopy and metrology. On a fundamental level, it enables the study of soliton physics in driven nonlinear cavities. Microresonators are commonly multimode and, as a result, inter-mode interactions inherently occur among mode families - a condition referred to as "avoided mode crossings". Avoided mode crossings can cause soliton decay, but can also modify the soliton spectrum, leading to e.g. the formation of dispersive wave and inducing a spectral recoil. Yet, to date, the entailing temporal soliton dynamics from inter-mode interactions has rarely been studied, but is critical to understand regimes of soliton-stability. Here we report the discovery of an inter-mode breather soliton. Such breathing dynamics occurs within a laser detuning range where conventionally stationary dissipative solitons are expected. We demonstrate experimentally the phenomenon in two microresonator platforms (crystalline magnesium fluoride and photonic chip-based silicon nitride microresonators), and theoretically describe the dynamics based on a pair of coupled Lugiato-Lefever equations. We demonstrate experimentally that the breathing is associated with a periodic energy exchange between the soliton and another optical mode family. We further show that inter-mode interactions can be modeled by a response function acting on dissipative solitons. The observation of breathing dynamics in the conventionally stable soliton regime is critical to applications, ranging from low-noise microwave generation, frequency synthesis to spectroscopy. On a fundamental level, our results provide new understandings of the rich dissipative soliton dynamics in multimode nonlinear cavities.