Bidirectional Solitons Research Articles

Bi-directional solitons of dual-mode Gardner equation derived from ideal fluid model

In this article, the main goal is to find the exact solution to the dual-mode Gardner equation derived from the ideal fluid model. To achieve this, two distinct methods will be employed: the tan/cot method and the tanh/coth method. Using these methods, the problem can be seen through different angles, providing a full understanding of how the equation works and the various solutions it provides. Furthermore, the periodic, kink and singular solutions will be illustrated through the use of 3D and 2D graphs. This visualization will clearly show how the phase velocity parameter affects the solutions, improving our understanding of wave dynamics within the dual-mode Gardner equation. The findings from this study help in understanding how both solitons and periodic solutions influence wave behavior in shallow water and how light travels through optical fibers.

Uni- and Bi-directional Er3+-doped fiber soliton laser mode-locked by GIMF-based SA

In this paper, we have experimentally demonstrated an ultrafast Er-doped fiber laser mode-locked by the GIMF-based saturable absorber without an optical isolator for various soliton behaviors. The adjustment of the polarization controller results in different losses for the counter-propagating lasers. And thus, not only the bidirectional solitons, but also the quasi-unidirectional solitons are realized. For the bidirectional operation, the counterpropagating conventional solitons with the pulse energy of >1.12 nJ demonstrates different central wavelengths, repetition rates, and pulse widths. For the unidirectional operation, the laser operates in either the clockwise or the counter-clockwise directions with an emission strengths difference of larger than 31 dB. Interestingly, the bidirectional operations could show different soliton state of the forward conventional solitons and backward bound solitons. Our research would be beneficial for the further understanding of the formation and transformation of the bidirectional solitons and can find vital applications in gyroscopic and other dual-channel applications.

Transient switching dynamics between asynchronous and synchronous modes in a bidirectional ultrafast fiber laser

Continuous interest in bidirectional ultrafast fiber lasers not only reinforces the practical applications in coherent dual-comb spectroscopy but also enriches the understanding of soliton dynamics in nonlinear systems. However, the transient dynamics of bidirectional solitons are still unexplored. Here, we report on an observation of the switching dynamics in a net-anomalous-dispersion bidirectional ultrafast fiber laser which can operate in both asynchronous and synchronous operations. Transient switching dynamics between the asynchronous and synchronous states are demonstrated, which is accompanied by switchable and weak interaction between bidirectional solitons. The dynamical evolution and output properties of the asynchronous and synchronous bidirectional solitons are investigated respectively. Stable breathing solitons in the synchronous operation show opposite behaviors in both spectra and pulse energy dynamics, which vary with different pump powers. Our findings provide insight into the transient dynamics in ultrafast lasers and contribute to the understanding of nonlinear systems in general.

Mutual dynamics between synchronous solitons in a bidirectional mode-locked fiber laser.

In this Letter, the mutual dynamics between synchronous solitons in a bidirectional mode-locked fiber laser are studied via dispersive Fourier transform methodology. We explore the spectral evolution and the statistical correlations between solitons with bidirectional propagation, indicating the low and high mutual linear dependences of the spectral energy jitters in stable and breathing mode-locking states, respectively. Moreover, to the best of our knowledge, the oscillating and sliding phase dynamics are experimentally revealed by the interference between bidirectional breathing solitons in ultrafast fiber lasers for the first time. Our findings enrich the understanding of the internal mutual dynamics between bidirectional solitons, which guides the extension of their potential applications, such as Sagnac-effect-based optical sensing.

Mutually induced soliton polarization instability in a bidirectional ultrafast fiber laser.

The bidirectional ultrafast fiber laser is a promising light source for dual-comb applications. The counter-propagating geometry could lead to soliton interaction through gain sharing, as well as the possible outcome of polarization instability. However, the polarization dynamics hidden behind the soliton interaction process in bidirectional fiber lasers were rarely investigated. Herein, we report on the polarization instability induced by the mutual soliton interactions through fiber gain in a bidirectional mode-locked fiber laser. Depending on the adjustment of the intracavity birefringence, the polarization states of two counter-propagating solitons can exhibit similar periodical polarization switching behaviors with a polarization-rotating transition state. The successive interactions of the bidirectional solitons mediated by the polarization cross-saturation effect of gain fiber could be responsible for the soliton polarization instability. These findings, in addition to the fundamental interest of the soliton nonlinear dynamics in dissipative optical systems, also open up new possibilities for creating dynamical control of the soliton polarization state and performance improvement in bidirectional ultrafast fiber lasers.

Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion

A new nonlinear integrable fifth-order equation with temporal and spatial dispersion is investigated, which can be used to describe shallow water waves moving in both directions. By performing the singularity manifold analysis, we demonstrate that this generalized model is integrable in the sense of Painleve for one set of parametric choices. The simplified Hirota method is employed to construct the one-, two-, three-soliton solutions with non-typical phase shifts. Subsequently, an extended projective Riccati expansion method is presented and abundant travelling wave solutions are constructed uniformly. Furthermore, several new interaction solutions between periodic waves and kinky waves are also derived via a direct method. The rich interactions including overtaking collision, head-on collision and periodic-soliton collision are analyzed by some graphs.

Characteristics of integrability, bidirectional solitons and localized solutions for a ( $$3+1$$ 3 + 1 )-dimensional generalized breaking soliton equation

The characteristics of integrability, bidirectional solitons and localized solutions are investigated for a ( $$3+1$$ )-dimensional breaking soliton (GBS) equation with general forms. Firstly, starting from the GBS equation, we perform the singularity manifold analysis and obtain a new integrable model in the sense of Painleve property. Secondly, taking advantage of the Bell polynomial approach, we construct the Backlund transformation, Lax pair and an infinite sequence of conservation laws. Subsequently, this new equation is also found to allow bidirectional soliton solutions, and the head-on and overtaking collisions between solitons are illustrated by some illustrative graphs. Finally, some localized excitations, such as lump solution, multi-dromions, periodic solitary waves solution, are obtained.

Solitons, Lax pair and infinitely-many conservation laws for a higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a higher-order nonlinear Schrödinger equation in an optical fiber. Lax pair and infinitely-many conservation laws are derived via the symbolic computation. By virtue of the Darboux transformation, one-, two- and three-soliton solutions are derived. Propagation and interaction of the solitons are illustrated graphically: Velocity of the soliton is linearly related to the higher-order dispersion coefficients, while amplitude of the soliton does not depend on them at all. Width of the one soliton increases with the decrease of the intensity of the complex eigenvalue parameter. Head-on interaction between the two bidirectional solitons as well as overtaking and oscillating interaction between the two unidirectional solitons are presented. For the interactions among the three solitons, we display two head-on and one overtaking interactions along with three overtaking interactions. Graphical analysis shows that each interaction between the two solitons is elastic, and each interaction among the three solitons is pairwise elastic.

Solitons, Bäcklund transformation and Lax pair for a generalized variable-coefficient Boussinesq system in the two-layered fluid flow

Under investigation in this paper is a generalized variable-coefficient Boussinesq system, which describes the propagation of the shallow water waves in the two-layered fluid flow. Bilinear forms, Bäcklund transformation and Lax pair are derived by virtue of the Bell polynomials. Hirota method is applied to construct the one- and two-soliton solutions. Propagation and interaction of the solitons are illustrated graphically: kink- and bell-shape solitons are obtained; shapes of the solitons are affected by the variable coefficients [Formula: see text], [Formula: see text] and [Formula: see text] during the propagation, kink- and anti-bell-shape solitons are obtained when [Formula: see text], anti-kink- and bell-shape solitons are obtained when [Formula: see text]; Head-on interaction between the two bidirectional solitons, overtaking interaction between the two unidirectional solitons are presented; interactions between the two solitons are elastic.

A New Simplified Bilinear Method for theN-Soliton Solutions for a Generalized FmKdV Equation with Time-Dependent Variable Coefficients

AbstractIn this paper a generalized fractional modified Korteweg–de Vries (FmKdV) equation with time-dependent variable coefficients, which is a generalized model in nonlinear lattice, plasma physics and ocean dynamics, is investigated. With the aid of a simplified bilinear method, fractional transforms and symbolic computation, the correspondingN-soliton solutions are given and illustrated. The characteristic line method and graphical analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons and elastic interactions. Finally, the resonance phenomenon for the equation is examined.

Applications of a simplified bilinear method to ion-acoustic solitary waves in plasma

In this paper, we propose a computational method for nonlinear partial differential equations modeling ion-acoustic waves as well as dusty plasmas in laboratory and space sciences. Many types of solitary waves including soliton solutions, N-soliton solutions and singular N-soliton solutions are derived. The characteristic line method and graphical analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons and elastic interactions. Furthermore, the effects of inhomogeneities of media and nonuniformities of boundaries, depicted by the variable coefficients, on the soliton behavior are discussed.

N-soliton solutions for the [formula omitted]-dimensional Hirota–Maccari equation in fluids, plasmas and optical fibers

Under investigation in this paper is the Hirota–Maccari equation, which is a generalized ( 2 + 1 ) -dimensional model in fluid dynamics, plasma physics and optical fiber communication. With the aid of the Hirota bilinear method and symbolic computation, the corresponding N-soliton solutions are given and illustrated. The characteristic face method and asymptotic analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons, elastic interactions and inelastic interactions. Finally, a kind of special phenomenon with the parameters varying is investigated, which might provide people with useful information on the dynamics of the relevant fields.

N-Fold Darboux Transformation and Bidirectional Solitons for Whitham–Broer–Kaup Model in Shallow Water

Under investigation in this paper is the Whitham–Broer–Kaup (WBK) system, which describes the dispersive long wave in shallow water. Through a variable transformation, the WBK system is casted into a general Broer–Kaup system whose Lax pair can be derived by the Ablowitz–Kaup–Newell–Segur technology. With symbolic computation, based on the aforementioned Lax pair, the N-fold Darboux transformation is constructed with a gauge transformation and the multi-soliton solutions are obtained. Finally, the elastic interactions of the two-soliton solutions (including the head-on and overtaking collisions) for the WBK system are graphically studied. Those multi-soliton collisions can be used to illustrate the bidirectional propagation of the waves in shallow water.

Some Water Wave Equations and Integrability

A theory of bidirectional solitons on water is developed by using the classical Boussinesq equation. Moreover, analytical multi-solitons of Camassa-Holm equation are presented.

Bidirectional solitons on water.

A theory of bidirectional solitons on water is developed by using an integrable Boussinesq surface-variable equation. We present an explicit transformation between the system and a member of the Ablowitz-Kaup-Newell-Segur system, and derive an exact multisoliton solution by using a Darboux transformation. The phase shifts and the maximum wave heights during the interaction are studied for two-soliton overtaking and head-on collisions. They agree with the Korteweg-de Vries solution for overtaking collision and the perturbation solution for head-on collision.