Parabolic Solitons Research Articles

Elastic interaction of second-order rogue matter waves for the modified Gross–Pitaevskii equation with time-dependent trapping potential and gain/loss

In this work, we are interested in the one-dimensional modified Gross–Pitaevskii equation with the presence of a confining harmonic time-dependent trap and gain/loss atoms. Starting from Hirota bilinear method, we build the one- and two-soliton solutions in the context of trapping Bose–Einstein condensates. From this set, were generated various second-order rogue matter solitary waves including parabolic solitons, line-like solitons, and dromion-like structures. In addition, great emphasis was put on the influence of higher-order interactions over the amplification and stabilization processes of solitons as well as the effects of the gain/loss whose impact beyond amplification is the creation of areas of collapse and revival of solitons during propagation. Finally, we build the multi-soliton solution then making it possible to describe the elastic-type interaction processes of the various types of rogue matter waves obtained. The theoretical results obtained hence enrich the study of non-linear phenomena within Bose–Einstein condensates and more.

The Investigation of Nonlinear Time-Fractional Models in Optical Fibers and the Impact Analysis of Fractional-Order Derivatives on Solitary Waves

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using a powerful technique named the extended generalized Kudryashov approach, we extract different rich structured soliton solutions to these models, including bell-shaped, cuspon, parabolic soliton, singular soliton, and squeezed bell-shaped soliton. We also study the impact of fractional-order derivatives on these solutions, providing new insights into the dynamics of nonlinear models. The results are compared with the existing literature, revealing novel and distinct solutions that offer a deeper understanding of these fractional models. The results show that the implemented approach is useful, reliable, and compatible for examining fractional nonlinear evolution equations in applied science and engineering.

Spatiotemporal modulated solitons in a quasi-one-dimensional spin-1 Bose–Einstein condensates

In this paper, we investigate the nonautonomous bright/dark solitons in a quasi-one-dimensional spin-1 Bose–Einstein condensates through a three coupled Gross–Pitaevskii (GP) system with space–time-dependent external potential and temporally modulated gain/loss distributions. Based on the Hirota bilinear method, we analytically construct the bright soliton solutions when the coupled GP system exhibits attractive interactions, while we obtain the dark soliton solutions when the coupled GP system exhibits repulsive interactions. The influence of spatiotemporal modulated external potentials, such as the gain/loss distribution Γ(t), on bright/dark soliton dynamics is analyzed in detail via the analytical solutions. By taking different Γ(t), we obtain different types of bright solitons, including periodic, dromion-like and parabolic solitons, and also derive dark solitons on different backgrounds, such as periodic, parabolic and kink backgrounds. We analyze the regulatory effects of different wavenumber ratios on the attraction and squeezing of bound-state solitons. Through the asymptotic analysis, we find that the interactions between two solitons are elastic. In addition, we conduct research on the forward and inverse problems of the above results via the parallel hard-constraint physical informed neural network (phPINN) method. The predicted solitons and potential functions are in good agreement with the exact solitons and potential in the system.

Multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations via Riemann-Hilbert approach

In this paper, we study multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fibers. We analyze the spectrum of the Lax pair to establish the Riemann-Hilbert problem. Using such Riemann-Hilbert problem, we calculate various multi-soliton solutions without reflection, including breather-like and mixed solitons. We illustrate the propagation and interaction dynamics of the solitons through appropriate parameter selection and asymptotic analysis. We find that the interaction between solitons is elastic, the amplitudes of solitons are only determined by the initial velocity and interaction, and the soliton with lower energy always yields a position shift when elastic interaction occurs. In addition, we observe that the existence time of the wave changes with energy and that multiple elastic interactions between solitons can be obtained when we choose appropriate variable coefficients. Then, we investigate the influences of group velocity dispersions and fourth-order dispersions on the interactions of solitons through parameter modulation mode and asymptotic analysis. Furthermore, we present several new types of nonlinear phenomena graphically, including elastic interactions between parabolic solitons and hump-type solitons, elastic interactions between cubic solitons and hump-type solitons, and periodic-changing propagations.

Novel solutions of (2+1) dimensional modified Bogoyavlenskii’s breaking soliton equation with variable coefficients

In this paper, by using the homogeneous equilibrium method, the exact solutions of a modified Bogoyavlenskii’s breaking soliton equation are derived and the soliton solutions with arbitrary functions are constructed. Then, the basic law of interaction between the different solitons are revealed and some new local structures are addressed and discussed. The periodic solitons, parabolic solitons and folded solitons of arbitrary shape propagating with variable speed are considered. It is helpful not only to verify the numerical solution and analyze the stability of the solution, but also to understand the dynamics of the high dimensional nonlinear wave field.

Analytical soliton solutions of the perturbed fractional nonlinear Schrödinger equation with space–time beta derivative by some techniques

Nonlinear fractional-order evolution equations are fundamental strategies for simulating nonlinear phenomena on a large scale in technology, science, and engineering. This article considers the nonlinear space–time fractional perturbed nonlinear Schrödinger (NLS) equation defined in the sense of the beta-derivative, a widely used model to simulate nonlinear optics, ultra-short pulse lasers, optical communication systems, plasmas, and other domains. The model is converted into a nonlinear equation defined by fractional wave transformation. We construct diverse sorts of soliton solutions in the integration of rational, trigonometric, exponential, and hyperbolic functions with open parameters using the typical generalized exponential rational function (GERF) and the improved Bernoulli sub-equation function (IBSEF) schemes. Some standard shapes of waveforms, including parabolic soliton, kink, flat kink, bright-dark soliton, periodic soliton, breather, breathing periodic, singular periodic, breathing-like soliton, flat parabolic, and several other types of solitons, are determined. Periodic solitons under perturbations are required to understand ultra-short pulse lasers and long-distance optical communication systems. To validate the physical characteristics of the established solitons, we sketch 3D, 2D, and contour plots of the solutions using consistent values of the parameters. The techniques used in this study to extract inclusive and standard solutions are approachable, efficient, and speedier to compute the soliton solutions for nonlinear models in communication engineering and optics.

Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation

IntroductionThe Gross-Pitaevskii equation is a class of the nonlinear Schrödinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. ObjectivesWe give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. MethodsBy constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. ResultsThe results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. ConclusionsThe study enriches the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation.

A (2+1)-dimensional generalized Hirota–Satsuma–Ito equations: Lie symmetry analysis, invariant solutions and dynamics of soliton solutions

This paper investigates the exact invariant solutions and the dynamics of soliton solutions to the (2+1)-dimensional generalized Hirota–Satsuma–Ito (g-HSI) equations. By applying the Lie symmetry technique, infinitesimal vectors, the commutation relations, and various similarity reductions are derived from the g-HSI equations. Using the two stages of Lie symmetry reductions, the equation is transformed into various nonlinear ordinary differential equations (NLODEs). After that, by solving the various resulting ODEs, we obtain abundant explicit exact solutions in terms of the involved functional parameters. These closed-form invariant solutions are successfully presented in the form of distinct complex wave-structures of solutions like combo-form solitons, dark-bright solitons, W-shaped solitons, the interaction between multiple solitons, parabolic wave solitons, multi-wave structures, and curved-shaped parabolic solitons. Furthermore, using computerized symbolic computation and numerical simulation, the physical behaviors of some obtained solutions are displayed in three-dimensional graphics. The resulting solutions are found to be useful for understanding the dynamics of the exact closed-form solutions of this model and show the authenticity as well as the effectiveness of the proposed method. Therefore, the gained solutions and their dynamical wave structures are quite significant for understanding the propagation of the excitation waves in shallow water wave models. Furthermore, using the resulting symmetries, the conservation laws of g-HSI equations have been obtained by applying Ibragimov’s theorem.

Accessible solitons in three-dimensional parabolic cylindrical coordinates

A class of self-similar beams, named three-dimensional (3D) spatiotemporal parabolic accessible solitons, are introduced in the 3D highly nonlocal nonlinear media. We obtain exact solutions of the 3D spatiotemporal linear Schrödinger equation in parabolic cylindrical coordinates by using the method of separation of variables. The 3D localized structures are constructed with the help of the confluent hypergeometric Tricomi functions and the Hermite polynomials. Based on such an exact solution, we graphically display three different types of 3D beams: the Gaussian solitons, the ring necklace solitons, and the parabolic solitons, by choosing different mode parameters. We also perform direct numerical simulation to discuss the stability of local solutions. The procedure we follow provides a new method for the manipulation of spatiotemporal solitons.

Phase shift, amplification, oscillation and attenuation of solitons in nonlinear optics

In nonlinear optics, the soliton transmission in different forms can be described with the use of nonlinear Schrödinger (NLS) equations. Here, the soliton transmission is investigated by solving the NLS equation with the reciprocal of the group velocity β1(z), the group velocity dispersion coefficient β2(z) and nonlinear coefficient γ(z). Two-soliton solutions for the NLS equation are obtained through the Hirota method. According to the solutions obtained, β1(z) and γ(z) with different function forms are taken to study the characteristics of solitons. The effect of the phase shift on the soliton interaction is discussed, and the non-oscillating soliton amplification, which is transmitted in a bound state, is explored. Parabolic solitons with oscillations are analysed. Moreover, parabolic solitons can be reduced to dromion-like structures. Results indicate that the transmission of solitons can be adjusted with the group velocity dispersion and Kerr nonlinearity coefficients. The phase shift, amplification, oscillation and attenuation of solitons can also be controlled by other related parameters. This work accomplishes the theoretical study of transmission characteristics of optical solitons in spatially dependent inhomogeneous optical fibres. The conclusions of this research have theoretical guidance for the research of optical amplifier, all-optical switches and mode-locked lasers.

Attosecond soliton shaping through dispersion tailoring technique in a monomode optical fiber

Analytical attosecond multi-soliton solutions for the variable coefficient sixth order nonlinear Schrödinger equation is investigated by means of Darboux transformation technique. Through the tailoring of group velocity dispersion (GVD) coefficient of inhomogeneous optical fibers, various abundant soliton structures are constructed. As few examples, V-shaped soliton, S-shaped soliton, dromion structure and parabolic soliton are attained by modulating appropriate GVD parameters in an inhomogeneous monomode optical fiber. Moreover, for the first time, switching behavior of attosecond soliton have been achieved through dispersion management. Soliton control offers novel intangible opportunities for all-optical shaping, switching and routing which might be of potential applications in the design of ultrafast optical switches and understanding the essential role of variable coefficients.

Interactions of solitons, dromion-like structures and butterfly-shaped pulses for variable coefficient nonlinear Schrödinger equation

Investigation on interactions of various types of solitons plays an important role in the analysis of various physical mechanisms. In this paper, analytic two- and three-soliton solutions for the variable coefficient nonlinear Schrödinger (vcNLS) equation are obtained with the help of the Hirota method. Through choosing the related parameters of solutions, we present various types of soliton interactions, and analyze the influence of corresponding parameters on soliton interactions. Periodic and oscillatory interactions between parabolic solitons are observed firstly. The method to control the butterfly-shaped pulse interaction is provided. Those results have guiding effect on controlling the interaction of solitons in nonlinear optics and Bose–Einstein condensation.

Bound-state solitons for the coupled variable-coefficient higher-order nonlinear Schrödinger equations in the inhomogeneous optical fiber

Bound-state vector soliton solutions for the coupled variable-coefficient higher-order nonlinear Schrödinger equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fiber, are investigated. Introducing auxiliary functions, we derive the bilinear forms and corresponding constraints on the variable coefficients. Through symbolic computation, we construct the one- and two-soliton solutions. We see that the variable coefficients in the equations affect the soliton structures. With different choices of the variable coefficients, we obtain the cubic, periodic, and parabolic solitons. Bound-state solitons and interactions are analyzed graphically.

Nonautonomous solitons and interactions for a variable-coefficient resonant nonlinear Schrödinger equation

A variable-coefficient resonant nonlinear Schrödinger (vc-RNLS) equation is considered in this paper. Binary Bell polynomials are employed to obtain the bilinear form and multi-soliton solutions under the integrable conditions. Four types of nonautonomous solitons are derived including the parabolic soliton, compressed soliton, phase-shifted soliton and periodic soliton. Propagation dynamics for each type is analyzed in detail. Nonautonomous resonant and intermediate-state soliton interactions are found to be existent under certain conditions. Specially, periodic soliton interactions are discussed, which shows that the periodic dispersion has no effect on the generation of resonance and intermediate-state solitons. Those analysis might have the applications in optical communication systems with the black hole physics flavor.

Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction

We have investigated the femtosecond soliton propagation in inhomogeneous fiber, which is described by the modified inhomogeneous Hirota equation with variable coefficient (MIH-vc). With the aid of AKNS method, corresponding Lax pair is constructed. By virtue of the Darboux transformation method and symbolic computation, the analytic one- and two-soliton solutions are explicitly obtained. Using obtained solutions, we graphically discuss the features of femtosecond solitons in modified inhomogeneous Hirota system by changing the profile of variable coefficients. We analyze various form of group velocity dispersion, third order dispersion and nonlinearity parameter for periodic amplification system, exponentially distributed system, parabolic solitons, periodic exponentially modulated system, which will be observable in the future experiments. These results are potentially useful in future experiments and soliton control for long-distance optical communication. Finally, the soliton solutions of the MIH-vc equation in double Wronskian form is constructed and further verified using the Wronskian technique by substitute in bilinear equations.

Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system with symbolic computation

In an inhomogeneous nonlinear light guide doped with two-level resonant atoms, the generalized coupled variable-coefficient nonlinear Schrodinger-Maxwell-Bloch system can be used to describe the propagation of optical solitons. In this paper, the Lax pair and conservation laws of that model are derived via symbolic computation. Furthermore, based on the Lax pair obtained, the Darboux transformation is constructed and soliton solutions are presented. Figures are plotted to reveal the following dynamic features of the solitons: (1) Periodic mutual attractions and repulsions of four types of bound solitons: of two one-peak bright solitons; of two one-peak dark solitons; of two two-peak bright solitons and of two two-peak dark solitons; (2) Two types of elastic interactions of solitons: of two bright solitons and of two dark solitons; (3) Two types of parallel propagations of parabolic solitons: of two bright solitons and of two dark solitons. Those results might be useful in the study of optical solitons in some inhomogeneous nonlinear light guides.

Characteristics of Soliton Evolution in the Wave-Breaking-Free Regime in a Passively Mode-Locked Yb-Doped Fiber Laser

We focus on several aspects concerning the numerical simulation of a passively mode-locked Yb-doped fiber laser by a non-distributed model. The characteristics of soliton evolution in a wave-breaking-free regime are numerically investigated with the split-step Fourier method. Based on the model, a parabolic-shaped soliton with a nearly linear chirp and bound soliton pairs are obtained by controlling the intra-cavity average dispersion of the fiber laser. A phenomenon is observed that by keeping the system parameters unchanged, linearly chirped parabolic soliton and bound soliton pairs are attainable under different initial conditions in the transient region between these two kinds of solitons.