Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schr?dinger equation

We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.

Four-soliton solution and soliton interactions of the generalized coupled nonlinear Schrodinger equation

Based on the generalized coupled nonlinear Schr¨odinger equation,we obtain the analytic four-bright–bright soliton solution by using the Hirota bilinear method.The interactions among four solitons are also studied in detail.The results show that the interaction among four solitons mainly depends on the values of solution parameters;k1 and k2 mainly affect the two inboard solitons while k3 and k4 mainly affect the two outboard solitons;the pulse velocity and width mainly depend on the imaginary part of ki(i=1,2,3,4),while the pulse amplitude mainly depends on the real part of ki(i=1,2,3,4).

Effective regulation of the interaction process among three optical solitons

The interaction between three optical solitons is a complex and valuable research direction,which is of practical application for promoting the development of optical communication and all-optical information processing technology.In this paper,we start from the study of the variable-coefficient coupled higher-order nonlinear Schodinger equation(VCHNLSE),and obtain an analytical three-soliton solution of this equation.Based on the obtained solution,the interaction of the three optical solitons is explored when they are incident from different initial velocities and phases.When the higher-order dispersion and nonlinear functions are sinusoidal,hyperbolic secant,and hyperbolic tangent functions,the transmission properties of three optical solitons before and after interactions are discussed.Besides,this paper achieves effective regulation of amplitude and velocity of optical solitons as well as of the local state of interaction process,and interaction-free transmission of the three optical solitons is obtained with a small spacing.The relevant conclusions of the paper are of great significance in promoting the development of high-speed and large-capacity optical communication,optical signal processing,and optical computing.

Influence of the initial parameters on soliton interaction in nonlinear optical systems

In nonlinear optical systems,optical solitons have the transmission properties of reducing error rate,improving system security and stability,and have important research significance in future research on all optical communication.This paper uses the bilinear method to obtain the two-soliton solutions of the nonlinear Schrödinger equation.By analyzing the relevant physical parameters in the obtained solutions,the interaction between optical solitons is optimized.The influence of the initial conditions on the interactions of the optical solitons is analyzed in detail,the reason why the interaction of the optical solitons is sensitive to the initial condition is discussed,and the interactions of the optical solitons are effectively weakened.The relevant results are beneficial for reducing the error rate and promoting the communication quality of the system.

Exotic interactions between solitons of the （2＋1）-dimensional asymmetric Nizhnik-Novikov-Veselov system

Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semifoldon, are investigated.

Restudy of Structures and Interactions of Solitons in （2＋1）-Dimensional Nizhnik-Novikov-Veselov Equations

Some new structures and interactions of solitons for the （2＋1）-dimensional Nizhnik-Novikov-Veselov equation are revealed with the help of the idea of the bilinear method and variable separation approach. The solutions to describe the interactions between two dromions, between a line soliton and a y-periodic soliton, and between two y-periodic solitons are included in our results. Detailed behaviors of interaction are illustrated both analytically and in graphically. Our analysis shows that the interaction properties between two solitons are related to the form of interaction constant. The form of interaction constant and the dispersion relationship are related to the form of the seed solution （u0, v0, w0 ） in Backlund transformation.

Interactions between bright solitons in different species of Bose-Einstein condensates

The dynamics of a bright bright vector soliton in a cigar-shaped Bose-Einstein condensate trapping in a harmonic potential is studied. The interaction between bright solitons in different species with small separation is derived. Unlike the interaction between solitons of the same species, it is independent of the phase difference between solitons. It may be of attraction or repulsion. In the former case, each soliton will oscillate about and pass through each other around the mass-center of the system, which will also oscillate harmonically due to the harmonic trapping potential.

Structures and Interactions of Soliton in （2＋1）-Dimensional Generalized Nizhnik-Novikov-Veselov Equation

A variable separation approach is used to obtain exact solutions of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov equation. Two of these exact solutions are analyzed to study the interaction between a line soliton and a y-periodic soliton (i.e. the array of the localized structure in the y direction, which propagates in the x direction) and between two dromions. The interactions between a line soliton and a y-periodic soliton are classified into several types according to the phase shifts due to collision. There are two types of singular interactions. One is the resonant interaction that generates one line soliton while the other is the extremely repulsive or long-range interaction where two solitons interchange each other infinitely apart. Some new phenomena of interaction between two dromions are also reported in this paper, and detailed behaviors of interactions are illustrated both analytically and graphically.

High-order effect on the transmission of two optical solitons

For optical solitons with the pulse width in the subpicosecond and femtosecond scales in optical fibers,a modified model containing higher-order effects such as third-order dispersion and third-order nonlinearity is needed.In this paper,in order to study the dynamic mechanism of femtosecond solitons in different media,we take the nonlinear Schr?dinger equation considering higher-order effects as the theoretical model,discuss the propagation of solitons in single-mode fibers,and explore the third-order dispersion and third-order nonlinear effects on the generation of optical solitons.The exact solution of the theoretical model is obtained through the bilinear method,and the transmission characteristics of two solitons with exact soliton solutions in actual fiber systems are analyzed and studied.The influence of various conditions on the transmission and interaction of optical solitons is explored.Methods for optimizing the transmission characteristics of optical solitons in optical communication systems are suggested.The relevant conclusions of this paper have guiding significance for improving the quality of fiber optic communication and increasing bit rates.

All-optical switches based on three-soliton inelastic interaction and its application in optical communication systems

In high-speed optical communication systems,in order to improve the communication rate,the distance between pulses must be compressed,which will cause the problem of the interaction between optical pulses in optical communication systems,which has been widely concerned by researches.In this paper,the bilinear method will be used to analyze the coupled high-order nonlinear Schro¨dinger equations and obtain their three-soliton solutions.Then,the influence of the relevant parameters in the three-soliton solution on the soliton inelastic interaction is studied.In addition,the constraint conditions of each parameter in the three-soliton solution are analyzed,the inelastic interaction properties of optical solitons under different parameter conditions are obtained,and the relevant laws of the inelastic interaction of solitons are studied.The results will have potential applications in the soliton control,all-optical switching and optical computing.

Interaction Between Line Soliton and Algebraic Soliton for Asymmetric Nizhnik-Novikov-Veselov Equation

Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for （2＋1）-dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton： 1） the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2） the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.

The interaction of in-band and in-gap lattice soliton trains in optically induced two-dimensional photonic lattices

We demonstrate the coherent interactions of lattice soliton trains, including in-band solitons （IBSs） and gap soliton trains （GSTs）, in optically induced two-dimensional photonic lattices with self-defocusing nonlinearity. It is revealed that the π-staggered phase structures of the lattice soliton trains will lead to anomalous interactions. Solely by changing their initial separations, the transition between attractive and repulsive interaction forces or reversion of the energy transfer can be obtained. The ‘negative refraction＇ effect of the soliton trains on the interaction is also discussed. Moreover, two interacting IBSs can merge into one GST when attraction or energy transfer happens.

Two-Dimensional Soliton Interaction for Multi-component Bose-Einstein Condensates

For two-component disk-shaped Bose-Einstein condensates with repulsive atom-atom interaction, the small amplitude, finite and long wavelength nonlinear waves can be described by a Kadomtsev-Petviashvili-Ⅰ equation at the lowest order from the originai coupled Gross-Pitaevskii equations. One- and two-soliton solutions of the Kadomtsev- Petviashvili-1 equation are given, therefore, the wave functions of both atomic gases are obtained as well. The instability of a soliton under higher-order long wavelength disturbance has been investigated. It is found that the instability depends on the angle between two directions of both soliton and disturbance.

Incoherent interaction between one- and two-dimensional solitons in noncentrosymmetric photorefractive media

The incoherent interaction between solitons with different transverse dimensions in a noncentrosymmetric photorefractive crystal is studied both in theory and in experiment. An anomalous incoherent interaction between one- and two-dimensional solitons, whose attractive and repulsive effects depend on the soliton separation, is numerically demonstrated by employing an anisotropic model. By launching a one-dimensional green beam and a two-dimensional red beam into a biased SBN：60 crystal, the hybrid-dimensional soliton interaction is performed. The experimental results are in good agreement with the numerical ones.

Interaction properties of solitons for a couple of nonlinear evolution equations

We study one-and two-soliton solutions for the Cahn–Allen(CA) equation and the Brethorton equation. The CA equation has broad spectrum of applications especially in anti-phase boundary motion and it is used in phase-field models.While the Brethorton equation is a model for dispersive wave systems, it is used to find the resonant nonlinear interaction among three linear modes. We use the Hirota bilinear method to obtain one-and two-soliton solutions to the CA equation and the Brethorton equation.

Ultrashort optical solitons in the dispersion-decreasing fibers

We derive analytical bright and dark solitons of the modified nonlinear Schroedinger equations with variable coefficients. Under constraint conditions between system parameters, the optical soliton transmission in the dispersiondecreasing fibers can be exactly controlled by proper dispersion management. The analytical description of the interactions between the bright and dark solitons are first obtained.

Non-completely elastic interactions in a(2+1)-dimensional dispersive long wave equation

With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the （24-1）-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.

Transcription's bubble under the influence of long-range interactions and helicoidal coupling

The influence of power-low long-range interactions （LRI） and helicoidal coupling （HC） on the properties of localized solitons in a DNA molecule when a ribonucleic acid polymerase （RNAP） binds to it at the physiological temperature is analytically and numerically investigated in this paper. We have made an analogy with the Heisenberg model Hamiltonian of an anisotropic spin ladder with ferromagnetic legs and anti-ferromagnetic rung coupling. When we limit ourselves to the second-order terms in the Taylor expansion, the DNA dynamics is found to be governed by a completely integrable nonlinear Schr？dinger （NLS） equation. In this case, results show that increasing the value of HC force or LRI parameter enhances the bubble height and reduces the number of base pairs which form the bubble. For the fourth-order terms in a Taylor expansion, results are closely resembling those of second-order terms, and are confirmed by numerical investigation. These results match with some experimental data and thus provide a better representation of the base pairs opening in DNA which is essential for the transcription process.

Computational Methods for Three Coupled Nonlinear Schrödinger Equations

In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.

Exact Solutions and Coherence Structures of Generalized Breor-Kaup Equations

Using the mapping method based on q-deformed hyperbolic functions,the exact solutions of generalizedBreor-Kaup equations are obtained.Based on the solutions,two coherent structures,periodic-branch kink and non-propagating kink,have been obtained.Moreover,one solitonal interaction form,two line solitons interaction on the kinkbackground,has been found.