Complex Soliton Research Articles

Two improved techniques for the perturbed nonlinear Biswas–Milovic equation and its optical solitons

In this article, abundant optical soliton solutions of the Biswas–Milovic equation with Kudryashov’s law and nonlinear perturbation terms in polarization preserving fibers is constructed by employing two improved analytical schemes, namely, the improved Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM). As a result of these improved approaches, many constraint conditions emerge that are required for the existence of soliton solutions. These solutions are the rational, exponential, trigonometric hyperbolic, and trigonometric which can be classified into the bright soliton, W-shaped soliton, dark soliton, singular soliton, periodic, and the mixed complex soliton. All solutions obtained have been verified using symbolic computations. Moreover, the dynamics of some of the achieved solutions are presented by plotting two and three-dimensional graphs.

Dynamic behaviors of multiple-soliton pulsation in an L-band passively mode-locked fiber laser with anomalous dispersion.

As a universal phenomenon in nonlinear optical systems, the soliton pulsating behavior is useful to achieve high pulse energy and can further enrich the complex soliton dynamics. To the best of our knowledge, herein we have demonstrated the observation of multiple-soliton pulsations in an L-band mode-locked fiber laser based on a nonlinear amplifying loop mirror with anomalous dispersion for the first time. Based on the dispersive Fourier transform method, we find that the pulsations in the multi-soliton regime are accompanied with the pulse width breathing and spectrum oscillation. In addition, the corresponding number of pulsating solitons increases linearly from 8 to 16 with the pump power. Our findings can facilitate a better understanding of the complex mechanism of soliton pulsations.

Oscillatory self-organization dynamics between soliton molecules induced by gain fluctuation.

In passively mode-locked fiber lasers (PMLFLs), the dissipative solitons (DSs) can self-organize to form complex structures through delicate interactions. However, it is still elusive to control these soliton structures by external influences. We here find that at a certain critical power, the location between two soliton molecules can be controlled by a slow modulated pump power. After applying the pump power with periodic fluctuation, two soliton molecules oscillate from the state of soliton molecular complex to stable distribution with maximum inter-molecular separation. During this process, the internal structure of each soliton molecule keeps steady. The slow gain depletion and recovery mechanism which plays a dominant role affects the motion of soliton molecules. These results could further expand the molecular analogy of spectroscopy and stimulate the development of optical information storage and processing.

On complex wave solutions depicted by the variable coefficients coupled Hirota equation

In this paper, the variable coefficients coupled Hirota equation is investigated by virtue of the unified method. Playing an important role in optics, the coupled Hirota equation describes the transmission process when high intensity ultra-short pulses pass through an optical glass fiber. Two types of complex wave solutions, polynomial function and rational function solutions, are gained. The polynomial function solutions incorporate complex solitary, complex soliton and complex elliptic wave solutions, while the rational function solutions involve complex periodic and complex soliton wave solutions. Furthermore, by choosing different values of the parameters, we illustrate the dynamical behaviors and the physical signification of these solutions.

Complex soliton patterns formation in a multi-wavelength Er-doped fiber laser

We report on the emission of complex soliton patterns from a multi-wavelength mode-locked Er-doped fiber laser through nonlinear polarization rotation (NPR). The optical spectrum exhibits three distinct well-separated spectral peaks centred at 1567 nm, 1585 nm, and 1616 nm. It is mainly attributed to the linear losses and the nonlinear birefringence filtering inside the cavity. Each wavelength in the spectrum contributes by its own soliton dynamics to a composite-state soliton regime. This is verified by using an external optical tunable filter with 0.5 nm filter bandwidth to filter out the lasing at each wavelength. By controlling the cavity parameters, this regime still can be operated in harmonic states.

Analytical construction of soliton families in one‐ and two‐dimensional nonlinear Schrödinger equations with nonparity‐time‐symmetric complex potentials

AbstractThe existence of soliton families in nonparity‐time‐symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one‐ and two‐dimensional nonlinear Schrödinger equations with localized Wadati‐type nonparity‐time‐symmetric complex potentials. By utilizing the conservation law of the underlying non‐Hamiltonian wave system, we convert the complex soliton equation into a new real system. For this new real system, we perturbatively construct a continuous family of low‐amplitude solitons bifurcating from a linear eigenmode to all orders of the small soliton amplitude. Hence, the emergence of soliton families in these nonparity‐time‐symmetric complex potentials is analytically explained. We also compare these analytically constructed soliton solutions with high‐accuracy numerical solutions in both one and two dimensions, and the asymptotic accuracy of these perturbation solutions is confirmed.

Survey of third- and fourth-order dispersions including ellipticity angle in birefringent fibers on W-shaped soliton solutions and modulation instability analysis

In this work, the modified Sardar sub-equation method is used to construct optical soliton solutions to the coupled nonlinear Schrodinger equation (CNLSE) having third-order and fourth-order dispersions describing short-pulse propagation in two-core fibers. Several solutions are determined including W-shaped bright, dark soliton solutions, singular soliton solutions, periodic function solutions and the combined complex soliton solutions. It should be noted that this integration scheme is very powerful mathematical tools for obtaining exact optical soliton solutions of nonlinear evolution equations. Under suitable values for the physical parameters, some representative wave structures are graphically displayed. In addition, the linear stability technic is used to analyze the modulation gain spectra in the birefringence associated with an ellipticity angle. In our knowledge, these obtained results are new in the context of nonlinear birefringent optical fibers.

Vector kink-dark complex solitons in a three-component Bose–Einstein condensate

We investigate kink-dark complex solitons (KDCSs) in a three-component Bose–Einstein condensate (BEC) with repulsive interactions and pair-transition (PT) effects. Soliton profiles critically depend on the phase differences between dark solitons excitation elements. We report a type of kink-dark soliton profile which shows a droplet-bubble-droplet with a density dip, in sharp contrast to previously studied bubble-droplets. The interaction between two KDCSs is further investigated. It demonstrates some striking particle transition behaviours during their collision processes, while soliton profiles survive after the collision. Additionally, we exhibit the state transition dynamics between a kink soliton and a dark soliton. These results suggest that PT effects can induce more abundant complex solitons dynamics in multi-component BEC.

Instability modulation properties of the (2 + 1)-dimensional Kundu–Mukherjee–Naskar model in travelling wave solutions

This paper focuses on the instability modulation and new travelling wave solutions of the (2 + 1)-dimensional Kundu–Mukherjee–Naskar equation via the tanh function method. Dark, mixed dark–bright, complex solitons and periodic wave solutions are archived. Strain conditions for the validity of results are also reported. Instability modulation properties of the governing model are also extracted. Various wave simulations in 2D, 3D and contour graphs under the strain conditions are presented.

Analytical survey of the predator–prey model with fractional derivative order

This work addresses the analytical investigation of the prey–predator behavior modeled by nonlinear evolution equation systems with fractional derivative order. Through the New Extended Algebraic Method (NEAM), we unearthed diverse types of soliton solutions including bright, dark solitons, combined trigonometric function solutions, and singular solutions. Besides the results obtained in the work of Khater, some new complex soliton solutions are also unearthed. The NEAM can also be used like the synthesis of the two mathematical tools.

Comparing Performance of Deep Convolution Networks in Reconstructing Soliton Molecules Dynamics from Real-Time Spectral Interference

Deep neural networks have enabled the reconstruction of optical soliton molecules with more complex structures using the real-time spectral interferences obtained by photonic time-stretch dispersive Fourier transformation (TS-DFT) technology. In this paper, we propose to use three kinds of deep convolution networks (DCNs), including VGG, ResNets, and DenseNets, for revealing internal dynamics evolution of soliton molecules based on the real-time spectral interferences. When analyzing soliton molecules with equidistant composite structures, all three models are effective. The DenseNets with layers of 48 perform the best for extracting the dynamic information of complex five-soliton molecules from TS-DFT data. The mean Pearson correlation coefficient (MPCC) between the predicted results and the real results is about 0.9975. Further, the ResNets in which the MPCC achieves 0.9906 also has the better ability of phase extraction than VGG which the MPCC is about 0.9739. The general applicability is demonstrated for extracting internal information from complex soliton molecule structures with high accuracy. The presented DCNs-based techniques can be employed to explore undiscovered mechanisms underlying the distribution and evolution of large numbers of solitons in dissipative systems in experimental research.

Lie symmetry analysis for complex soliton solutions of coupled complex short pulse equation

The current work is devoted for operating the Lie symmetry approach, to coupled complex short pulse equation. The method reduces the coupled complex short pulse equation to a system of ordinary differential equations with the help of suitable similarity transformations. Consequently, these systems of nonlinear ordinary differential equations under each subalgeras are solved for traveling wave solutions. Further, with the help of similarity variable, similarity solutions and traveling wave solutions of nonlinear ordinary differential equation, complex soliton solutions of the coupled complex short pulse equation are obtained which are in the form of sinh, cosh, sin and cos functions.

Investigation of exact soliton solutions in magneto-optic waveguides and its stability analysis

This paper deals the propagation of waves through magneto-optic waveguides which are modeled by the generalized vector nonlinear Schrodinger’s equation (NLSE). Two types of nonlinearities namely quadratic-cubic (QC) and dual-power laws are studied by implementing authentic mathematical technique called extended Fan-sub equation (EFSE) method. We secure the exact solutions in the forms of Jacobi’s elliptic functions, trigonometric, hyperbolic, including solitary wave solutions like bright, dark, complex, singular, and mixed complex solitons. Moreover, singular periodic wave solutions are recovered and the constraint conditions for valid soliton solutions are also reported. We also discuss the modulation instability (MI) analysis of the governing models. 3D and their corresponding 2D graphs are sketched for better understanding about the derived solutions with the values of unknown parameters. The most significant achievement of this approach is that it offers all the solutions that can be found by the use of other methods such as processes using the Riccati equation, an elliptic equation of the first kind, an auxiliary ordinary equation, or the generalized Riccati equation as mapping equation as well as we have succeeded in a single move to get and organize various types of new solutions. The obtained outcomes show that the applied integration technique is concise, direct, efficient, and can be applied in more complex phenomena with the assistant of symbolic computations.

Diverse exact solutions for modified nonlinear Schrödinger equation with conformable fractional derivative

In this article, our focus is to extract the diverse exact solutions to the conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE) that describes the propagation of water waves in the ocean engineering. Diverse exact solutions like trigonometric, hyperbolic and exponential function solutions are extracted. We also secure some other special wave solutions in the forms of shock wave, singular, multiple and mixed complex solitons. The generalized exponential rational function method (GERFM) is used to explain the dynamics of soliton to CTFMNLSE. Furthermore, the constraint conditions for the existence of solutions are reported also singular periodic wave solutions are recovered. Besides, the accomplished solutions are beneficial to interpretation of the wave propagation study and also important for numerical and experimental verifications in ocean engineering

Multiple soliton, fusion, breather, lump, mixed kink-lump and periodic solutions to the extended shallow water wave model in (2+1)-dimensions

In this paper, we consider the shallow water wave model in the (2+1)-dimensions. The Hirota simple method is applied to construct the new dynamics one-, two-, three-, [Formula: see text]-soliton solutions, complex multi-soliton, fusion, and breather solutions. By using the quadratic function, the one-lump, mixed kink-lump and periodic lump solutions to the model are obtained. The Hirota bilinear form variable of this model is derived at first via logarithmic variable transform. The physical phenomena to this model are explored. The obtained results verify the proposed model.

Dark-bright soliton interactions in coupled nonautonomous nonlinear Schrödinger equation with complex potentials

We investigate dark-bright soliton dynamics in a coupled nonlinear Schrödinger equation in presence of complex potentials and the associated spectral problem. Using asymptotic analysis and graphical analysis, we study the interactions of soliton complexes under different potentials. We present the results of elastic and inelastic interaction between solitons in presence of the potentials. The external potential brings about notable changes in the dynamics of soliton by changing it’s velocity, frequency and the wave number. The changes are however, same on the all the solitons and hence the external potential do not alters the conditions of the elastic interaction. We also present interesting features of complex three soliton interactions under the complex potentials. The shape change may occur even when there is no relative velocity between solitons. We hope that the present analysis would be useful for a better understanding of soliton interactions in an inhomogeneous nonlinear fiber and the dynamics of condensates in atomic physics.

Complex wave solutions described by a (3+1)-dimensional coupled nonlinear Schrödinger equation with variable coefficients

In this article, a (3+1)-dimensional variable coefficients coupled nonlinear Schrö-dinger equation is analysed with the unified method, the improved F-expansion method with Riccati equation and the modified Kudryashov method. The polynomial solutions are investigated and classified into three categories including complex solitary wave solutions, complex soliton wave solutions and complex elliptic wave solutions by applying the unified method. Besides, the physical insights of polynomial solutions are graphically discussed with suitable parameters. The complex wave solutions are derived by the improved F-expansion method with Riccati equation which contain the complex hyperbolic trigonometric solutions, complex trigonometric solutions and complex rational solutions. Lastly, the modified Kudryashov method is applied to obtain the complex wave solutions.

Modulation instability analysis and optical solitons in birefringent fibers to RKL equation without four wave mixing

In this present work, we retrieve the optical solitons in birefringent fibers modeled by Radhakrishnan-Kundu-Lakshmanan (RKL) equation in coupled vector form. The system is composed of self-phase, cross-phase modulation terms, self-steepening terms and the effect of four-wave mixing is discarded. In this article, by employing innovative integration norms known as the extended rational sine-cosine/sinh-cosh techniques different forms of optical solitons like dark, singular, combined and complex soliton solutions are recovered. Furthermore, singular periodic wave solutions are also observed. The constraint conditions which ensure the formation of soliton solutions are also specified. By using the suitable values of the parameters involved, the 2-, 3-dimensional and the contour graphs of some of the reported solutions are plotted. The stability of the given model is studied by exercising the modulation instability analysis which confirms that the model is stable and guarantees that all extracted solutions are stable and exact.

Analysis of real-time spectral interference using a deep neural network to reconstruct multi-soliton dynamics in mode-locked lasers

The dynamics of optical soliton molecules in ultrafast lasers can reveal the intrinsic self-organized characteristics of dissipative systems. The photonic time-stretch dispersive Fourier transformation (TS-DFT) technology provides an effective method to observe the internal motion of soliton molecules real time. However, the evolution of complex soliton molecular structures has not been reconstructed from TS-DFT data satisfactorily. We train a residual convolutional neural network (RCNN) with simulated TS-DFT data and validate it using arbitrarily generated TS-DFT data to retrieve the separation and relative phase of solitons in three- and six-soliton molecules. Then, we use RCNNs to analyze the experimental TS-DFT data of three-soliton molecules in a passive mode-locked laser. The solitons can exhibit different phase evolution processes and have compound vibration frequencies simultaneously. The phase evolutions exhibit behavior consistent with single-shot autocorrelation results. Compared with autocorrelation methods, the RCNN can obtain the actual phase difference and analyze soliton molecules comprising more solitons and almost equally spaced soliton pairs. This study provides an effective method for exploring complex soliton molecule dynamics.

Complete integrability and complex solitons for generalized Volterra system with branched dispersion

In this paper, we show that complete integrability is preserved in a multicomponent differential-difference Volterra system with branched dispersion relation. Using the Hirota bilinear formalism, we construct multisoliton solutions for a system of coupled [Formula: see text] equations. We also show that one can obtain the same solutions through a periodic reduction starting from a two-dimensional completely integrable generalized Volterra system. For some particular cases, graphical representations of solitons are displayed and stability is discussed using an asymptotic analysis.