Bright-bright Solitons Research Articles

Interaction of lump, periodic, bright and kink soliton solutions of the (1+1)-dimensional Boussinesq equation using Hirota-bilinear approach

In this paper, we explore the characteristics of lump and interaction solutions for a (1+1) dimensional Boussinesq equation. By employing the Hirota bilinear method, we derive and analyze the exact solutions of this equation. Specifically, we achieve the lump with bright-bright soliton solution, 1-lump,2-lumps and 3-lumps with single bright soliton solution, lump with periodic, kink, and anti-kink soliton solutions. Alongside deriving these solutions, we also illustrate their dynamic properties through graphical simulations. The Boussinesq equation holds significant importance due to its applications in various domains, such as water wave modeling, coastal engineering, and the numerical simulation of water wave dynamics in harbors and shallow seas. Our research shows that the employed method is straightforward, easy to understand, and highly efficient, providing valuable insights into the equation’s nature and its practical applications.

Soliton and breather solutions of a reverse time nonlocal coupled nonlinear Schrödinger equation with four-wave mixing effect

Under investigation is a reverse time nonlocal coupled nonlinear Schrödinger equation with four-wave mixing effect. The N-fold Darboux transformation is constructed in a compact determinant representation. As an application, the multi-bright–bright soliton, multi-dark–bright soliton and multi-breather solutions are presented. Dynamical behaviors of the degenerate bright–bright solitons, the regular and periodically oscillating dark–bright solitons and the multiple breather interactions are graphically shown.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones

In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

Vector valley Hall edge solitons in distorted type-II Dirac photonic lattices.

Topological edge states have recently garnered a lot of attention across various fields of physics. The topological edge soliton is a hybrid edge state that is both topologically protected and immune to defects or disorders, and a localized bound state that is diffraction-free, owing to the self-balance of diffraction by nonlinearity. Topological edge solitons hold great potential for on-chip optical functional device fabrication. In this report, we present the discovery of vector valley Hall edge (VHE) solitons in type-II Dirac photonic lattices, formed by breaking lattice inversion symmetry with distortion operations. The distorted lattice features a two-layer domain wall that supports both in-phase and out-of-phase VHE states, appearing in two different band gaps. Superposing soliton envelopes onto VHE states generates bright-bright and bright-dipole vector VHE solitons. The propagation dynamics of such vector solitons reveal a periodic change in their profiles, accompanied by the energy periodically transferring between the layers of the domain wall. The reported vector VHE solitons are found to be metastable.

Two-colour dissipative solitons and breathers in microresonator second-harmonic generation

Frequency conversion of dissipative solitons associated with the generation of broadband optical frequency combs having a tooth spacing of hundreds of giga-hertz is a topical challenge holding the key to practical applications in precision spectroscopy and data processing. The work in this direction is underpinned by fundamental problems in nonlinear and quantum optics. Here, we present the dissipative two-colour bright-bright and dark-dark solitons in a quasi-phase-matched microresonator pumped for the second-harmonic generation in the near-infrared spectral range. We also found the breather states associated with the pulse front motion and collisions. The soliton regime is found to be typical in slightly phase-mismatched resonators, while the phase-matched ones reveal broader but incoherent spectra and higher-order harmonic generation. Soliton and breather effects reported here exist for the negative tilt of the resonance line, which is possible only via the dominant contribution of second-order nonlinearity.

Soliton propagation for a coupled Schrödinger equation describing Rossby waves

We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method. For the coupled Schrödinger equation, we obtain the Manakov model of all-focusing, all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions. Especially, we find that the all-defocusing type Manakov model admits bright-bright soliton solutions. Furthermore, we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions. Therefrom, we go over how the free parameters affect the Manakov model’s all-focusing type’s two-soliton and three-soliton solutions’ collision locations, propagation directions, and wave amplitudes. These findings are useful for setting a simulation scene in Rossby waves research. The answers we have found are helpful for studying physical properties of the equation in Rossby waves.

Data-driven solutions and parameter discovery of the nonlocal mKdV equation via deep learning method

In this paper, we systematically study the integrability and data-driven solutions of the nonlocal mKdV equation. The infinite conservation laws of the nonlocal mKdV equation and the corresponding infinite conservation quantities are given through Riccti equation. The data-driven solutions of the zero boundary for the nonlocal mKdV equation are studied by using the multilayer physical information neural network algorithm, which include kink soliton, complex soliton, bright-bright soliton and the interaction between soliton and kink-type. For the data-driven solutions with nonzero boundary, we study kink, dark, anti-dark and rational solution. By means of image simulation, the relevant dynamic behavior and error analysis of these solutions are given. In addition, we discuss the inverse problem of the integrable nonlocal mKdV equation by applying the physics-informed neural network algorithm to discover the parameters of the nonlinear terms of the equation.

Coupled sn-cn, sn-dn, cn-dn Jacobi elliptic functions and solitons solutions in magneto-optic waveguides with quadratic-cubic nonlinearity

The current paper is dedicated to the study of the propagation of progressive solitons in magneto-optical waveguides that carry the quartic-cubic nonlinearity described by a coupled system of of nonlinear Schrödinger equations. In this direction, we take the help of a modified F-expansion method which allows us to widen the class of solutions to these coupled equations and permit the propagation of different waves in the two channels. First of all, we constructed the solutions in terms of Jacobi elliptic functions in the forms of and After that, we obtained the progressive soliton solutions when these functions approached their limiting values with regards to the modulus of ellipticity. The progressive solutions include bright-dark soliton, bright-bright soliton, dark-dark soliton, bright-front soliton, dark-front soliton. In addition, we present the figures 1–6 which graphically exhibit the representative structures of each explicit solution for some special parameter values.

Unidirectional segregation of bright-bright soliton through a parity-time-symmetric potential.

We study the dynamics of two-component vector solitons, namely, bright-bright (BB) solitons interacting with parity-time (PT)-symmetric potentials. We employ direct numerical simulations to demonstrate the unidirectional segregation of the BB soliton. Using a modified perturbed dynamical variational Lagrangian approximation, we develop an analytical model to verify the results obtained from numerical calculations. Simplified variational equationsof motion suggest that the splitting of BB solitons can be explained by considering the effective force between the two components.

Soliton-mediated ionic pulses and coupled ionic excitations in a dissipative electrical network model of microtubules

Many cellular activities are mediated by microtubules (MTs), with most electrophysiological processes depending on ionic flow through MT cylinders. This paper addresses such conductive features by representing the MT as a nonlinear electrical transmission line composed of capacitive and dissipative properties. Thanks to the semi-discrete approximation near the continuum limit, coupled ionic pulses flowing along cellular microtubules are described by a set of coupled cubic complex Ginzburg-Landau (CGL) equations whose one of the solutions is a plane wave. The stability of the latter is checked, under weak modulation, using an explicit analytical expression for the modulational instability (MI) growth rate. The parametric analysis of the instability growth rate allows detecting parameter regions where ionic conductivity, through modulated waves, in the MT network is likely to occur. Dissipative bright-bright pulse soliton solutions for the nonlinearly coupled cubic CGL equations are constructed using a modified Hirota's bilinear method. A generalized coupled mode for the discrete ionic signal is proposed and used as the initial condition to be propagated in the nonlinear electrical transmission lattice of MT under direct numerical simulations. The ionic pulse transfer, mediated by the nonlinear interaction between oscillatory modes, is manifested by the formation of trains of modulated waves whose behaviors depend on the right choice of system parameters. Those results theoretically suggest that coupled ionic signals may facilitate information processing involving MTs.

Modulated wave and modulation instability gain brought by the cross-phase modulation in birefringent fibers having anti-cubic nonlinearity

In this paper, we investigate the modulated wave and W-shaped profile in birefringent fibers having the anti-cubic nonlinearity terms. We use the traveling wave hypothesis to show out the velocity of the soliton and the constraint relation on the anti-cubic nonlinear terms. We use the Jacobi elliptic function solutions to point out two types of combined solutions. After some assumption on the modulus of the Jacobi elliptic function, we have shown out the combined bright-bright soliton and dark-dark soliton-like solutions. We use the linearizing algorithm to analyze the modulation instability (MI) growth rate. We have shown that the anti-cubic nonlinear terms and cross-phase modulation (XPM) can increase MI bands and the amplitude of the MI growth rate. To corroborate the prediction made on analytical results, we use the numerical investigation to show the propagation of the modulated wave and W-shaped profile in terms of cell index. We exhibited through the numerical results that the modulated wave can conserve high energy during its propagation in birefringent fibers. The obtained results will certainly open new perspectives in optical fibers during the transmission of huge data.

Coupled Gerdjikov-Ivanov System and its Exact Solutions through Darboux Transformation

The coupled Gerdjikov-Ivanov (GI) system is discussed and its elementary Darboux transformation (DT) is constructed. From elementary (DT), we constructed 2N-fold (DT). The symmetrical properties of the Lax pair and iteration of 2N-fold (DT) give different types of solutions for both zero seed and non-zero seed. Zero seed solutions include brightbright (one and two) Soliton solutions. In non-zero seed solutions, we obtained breather, Ma breather, dark-bright Soliton, breather fission, and dark-bright rogue wave solutions are obtained and plot it.

Chirped optical soliton propagation in birefringent fibers modeled by coupled Fokas-Lenells system

The propagation of ultrashort light pulses in a birefringent optical fiber exhibiting spatiotemporal dispersion, cross- and self-phase modulation, self-steepening, and group-velocity dispersion effects is addressed. The evolution of light pulses in such system is described by the coupled Fokas-Lenells equations which offer an accurate description of pulse dynamics in the femtosecond range when certain terms of the next asymptotic order beyond those necessary for the nonlinear Schrödinger equation are sustained. We report the first analytical demonstration of the propagation of chirped solitons in a birefringent fiber medium governed by the coupled Fokas-Lenells equations. The formation of those nonlinearly chirped solitons in the optical material may be attributed to the presence of self-steepening process. The results show that the frequency chirp associated with each of the two field components is directly proportional to the total intensity of the pulse. The chirped solitons for the system including the dark-dark and bright-bright soliton pairs in the presence of all fiber parameters are retrieved. The chirps accompanying the soliton pairs are also determined. The existence constraints of these chirped solitons are presented. In addition, the stability of the chirped solutions with respect to the finite perturbations is studied numerically.

Influence of the next-nearest neighbor and the boson–boson interactions on U-shaped, W-shaped profile and modulation instability gain spectra in a zig–zag optical lattice

In this paper, we employ the auxiliary equation method (AEM) to show the leverage of the nearest neighbors hopping and the boson–boson interaction in a zig zag optical lattice on the stressed optical solitons (os) and the Modulation Instability (MI) gain of cold bosonic atoms. The pedestal of the work is set on the continuum approximation of a cold bosonic atoms in a zig zag optical lattice which has demonstrated in detailed in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589]. The results collected have enclosed bright, dark optical solitons and the combined bright-dark and bright-bright optical solitons. Compared the acquired results with Tala-Tebue et al. [Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44], further the os including U-shaped and W-shaped profile have been drop in and the powerful of the nearest neighbors hopping and the boson–boson interaction has been exposed across the spatiotemporal and plot evolution of the bright and dark os. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the model parameters. The collected results are new in literature in our knowledge and will improve the previous results acquired in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589; Zhao XH, Tian B, Liu DY, et al. Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattices Microstruct 100(2016):587–595; Uddin MF, Hafez MG, Hammouch Z, et al. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media 2020. https://doi.org/10.1080/17455030.2020.1722331; Inc M, Aliyu AI, Yusuf A. Optical solitons to the nonlinear Shrdingers equation with spatio-temporal dispersion using complex amplitude ansatz. J Mod Opt 2017;64(21):2273–2280].

Soliton solutions of coupled complex modified Korteweg-de Vries system through Binary Darboux transformation

In this article, we find various kind of solutions of coupled complex modified (KdV) system by using very interesting method binary Darboux transformation. Generally the solutions are classified into zero seed and non-zero seed. In zero seed solutions, we find breather solution and one soliton solution. While in non-zero seed solutions, we obtain bright-bright solitons, w-shaped solitons, bright-dark solitons, periodic and rouge waves solutions. The behavior of these solutions can easily examine from figures.

Unidirectional flow of composite bright-bright solitons through asymmetric double potential barriers and wells

We investigate the dynamics of two component bright-bright (BB) solitons through reflectionless double barrier and double well potentials in the framework of a Manakov system governed by the coupled nonlinear Schrödinger equations. The objective is to achieve unidirectional flow and unidirectional segregation/splitting, which may be used in the design of optical data processing devices. We observe how the propagation of composite BB soliton is affected by the presence of interaction coupling between the two components passing through the asymmetric potentials. We consider Gaussian and Rosen-Morse double potential barriers in order to achieve the unidirectional flow. Moreover, we observe a novel phenomenon which we name ”Polarity Reversal” in the unidirectional flow. In this situation, the polarity of the diode is reversed. To understand the physics underlying these phenomena, we perform a variational calculation where we also achieve unidirectional segregation/splitting using an asymmetric double square potential well. Our comparative study between analytical and numerical analysis lead to an excellent agreement between the two methods.

Multiple double-pole bright-bright and bright-dark solitons and energy-exchanging collision in the M-component nonlinear Schrödinger equations.

Multiple double-pole bright-bright and bright-dark soliton solutions for the multicomponent nonlinear Schrödinger (MCNLS) system comprising three types of nonlinearities, namely, focusing, defocusing, and mixed (focusing-defocusing) nonlinearities, arising in different physical settings are constructed. An interesting type of energy-exchanging phenomenon during collision of these double-pole solitons is unraveled. To explore the objectives, we consider the general solutions of a set of generalized MCNLS equations and by taking the long-wavelength limit with proper parameter choices of single-pole bright-bright and bright-dark soliton pairs, the multiple double-pole bright-bright and bright-dark soliton solutions are constructed in terms of determinants. The regular double-pole bright-bright solitons exist in the focusing and focusing-defocusing MCNLS equations and undergo a particular type of energy-sharing collision for M≥2 in addition to the usual elastic collisions. A striking feature observed in the process of energy-sharing collisions is that the double-pole two-soliton possessing unequal intensities before collision indeed exactly exchange their intensities after collision. Further, the existence of double-pole bright-dark solitons in the MCNLS equations with focusing, defocusing, and mixed (focusing-defocusing) nonlinearities is analyzed by constructing explicit determinant form solutions, where the double-pole bright solitons exhibit elastic and energy-exchanging collisions while the double-pole dark solitons undergo mere elastic collision. The double-pole bright-dark solitons possess much richer localized coherent patterns than their counterpart double-pole bright-bright solitons. For particular choices of parameters, we demonstrate that the solitons would degenerate into the background, resulting in a lower number of solitons. Another important observation is the formation of doubly localized rogue waves with extreme amplitude, in the case of double-pole bright-dark four-solitons. Our results should stimulate interest in such special multipole localized structures and are expected to have ramifications in nonlinear optics.

Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber

Optical fiber communication plays an important role in the modern communication. In this paper, we investigate a variable-coefficient coupled Hirota system which describes the vector optical pulses in an inhomogeneous optical fiber . With respect to the complex wave envelopes, we construct a Lax pair and a Darboux transformation, both different from the existing ones. Infinitely-many conservation laws are derived based on our Lax pair. We obtain the one/two-fold bright-bright soliton solutions, one/two-fold bright-dark soliton solutions and one/two-fold breather solutions via our Darboux transformation. When α ( t ) , β ( t ) and δ ( t ) are the trigonometric functions, we present the bright-bright soliton, bright-dark soliton and breather which are all periodic along the propagation direction, where α ( t ) , β ( t ) and δ ( t ) represent the group velocity dispersion, third-order dispersion and nonlinear terms of the self-phase modulation and cross-phase modulation. Interactions between the two bright-bright soliton, two bright-dark solitons and two breathers are presented. Bound state of the two bright-bright solitons is formed. Widths and velocities of the two bright-bright solitons do not change but their amplitudes change after their interaction via the asymptotic analysis . Periods of the bright-dark solitons decrease when the periods of the trigonometric α ( t ) , β ( t ) and δ ( t ) decrease.

Evolution from Periodic Intensity Modulations to Dissipative Vector Solitons in A Single-Mode Fiber Laser

We investigated—both experimentally and numerically—the operation of a weakly birefringent cavity fiber laser under different net cavity dispersion values. Experimentally, we found that under coherent cross-polarization coupling, either in-phase or anti-phase low frequency intensity modulations between the two orthogonal polarization components of the laser emission could be obtained. The evolution of the periodic intensity modulations in the fiber laser under different operation conditions was studied. In this paper, we show that under suitable conditions, they can be shaped into a train of bright-bright, dark-dark, or dark-bright vector solitons.