Parallel Solitons Research Articles

Discrete Solitons and Bäcklund Transformation for the Coupled Ablowitz–Ladik Equations

Abstract Under investigation in this paper are the coupled Ablowitz–Ladik equations, which are linked to the optical fibres, waveguide arrays, and optical lattices. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation. Bright/dark one- and two-soliton solutions are also obtained. Asymptotic analysis indicates that the interactions between the bright/dark two solitons are elastic. Amplitudes and velocities of the bright solitons increase as the value of the lattice spacing increases. Increasing value of the lattice spacing can lead to the increase of both the bright solitons’ amplitudes and velocities, and the decrease of the velocities of the dark solitons. The lattice spacing parameter has no effect on the amplitudes of the dark solitons. Overtaking interaction between the unidirectional bright two solitons and a bound state of the two equal-velocity solitons is presented. Overtaking interaction between the unidirectional dark two solitons and the two parallel dark solitons is also plotted.

Dark solitons and Bäcklund transformation for the (2+1)-dimensional coupled nonlinear Schrödinger equation with the variable coefficients in a graded-index waveguide

Under investigation in this paper is a set of the (2 + 1)-dimensional coupled nonlinear Schrödinger equations, which describes the propagation of an optical beam inside a (2 + 1)-dimensional graded-index waveguide with the polarization effects. Employing the Hirota method and symbolic computation, we obtain the dark one-, two- and N-soliton solutions under the variable-coefficient constraints. Bäcklund transformation and the corresponding soliton solutions are derived. We graphically study the dark solitons with the influence of the profile function and nonlinearity coefficient. Parallel and period solitons are observed. Amplitudes of the dark solitons increase with the profile function increasing; When the profile function is a periodic function, amplitudes of the dark solitons change periodically. Decrease of the nonlinearity coefficient leads to the larger solitons' velocities, but does not affect the solitons' amplitudes.

Solitons and dynamic analysis for a (2 + 1)-dimensional breaking soliton equation

In this paper, we study a (2 + 1)-dimensional breaking soliton equation, which describes the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y axis with a long wave along the x axis, where x and y are the scaled space coordinates. Grammian N-soliton solutions for the equation are derived. With N = 1 and 2, the one- and two-soliton solutions are given. Graphic analysis shows that the soliton amplitude and velocity are related to the dispersion. An overtaking interaction between the two parallel solitons is shown. We find that the two solitons always have the same soliton direction. Then, we investigate the equation from a planar-dynamic-system viewpoint. That equation is reduced to a two-dimensional planar dynamic system, which is proved to be a Hamiltonian system. Through the qualitative analysis, we give the phase portraits of the dynamic system, based on which the relation among the Hamiltonian, orbits of the dynamic system and types of the analytic solutions are discussed. The analysis shows that the solitary- and periodic-wave solutions for that equation correspond to the homoclinic and periodic orbits of the dynamic system, respectively.

Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

Under investigation in this paper is a (2 + 1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Via the symbolic computation and Hirota method, the bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interaction for the dark solitons are illustrated graphically: Amplitude and shape of the dark one soliton keep invariant during the propagation, which imply the transport of the energy is stable in the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain. Through the asymptotic analysis, elastic and inelastic interactions between the dark two solitons are discussed. For the elastic interaction, oblique, head-on and overtaking interactions between the dark two solitons are displayed, where the amplitudes and shapes remain unchanged after interaction except for certain phase shifts. However, in the area of the inelastic interaction, amplitudes of the dark two solitons vanish after interaction. For the elastic interaction among the dark three solitons, oblique, overtaking and interaction among the dark two parallel solitons and a single one are presented, whose characteristics are similar to those of the dark two solitons. Inelastic-elastic interaction is investigated as well. Linear stability analysis is used to analyze modulation instability and prove the dark solitons are stable.

Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional B-type Kadomtsev–Petviashvili equation in the fluid/plasma mechanics

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili equation for the shallow water wave in a fluid or electrostatic wave potential in a plasma. Bilinear form, Bäcklund transformation and Lax pair are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota’s method. Propagation and interaction of the solitons are illustrated graphically: (i) Through the asymptotic analysis, elastic and inelastic interactions between the two solitons are discussed analytically and graphically, respectively. The elastic interaction, amplitudes, velocities and shapes of the two solitons remain unchanged except for a phase shift. However, in the area of the inelastic interaction, amplitudes of the two solitons have a linear superposition. (ii) Elastic interactions among the three solitons indicate that the properties of the elastic interactions among the three solitons are similar to those between the two solitons. Moreover, oblique and overtaking interactions between the two solitons are displayed. Oblique interactions among the three solitons and interactions among the two parallel solitons and a single one are presented as well. (iii) Inelastic–elastic interactions imply that the interaction between the inelastic region and another one is elastic.

Dark solitons and rouge waves for a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential in the Bose–Einstein condensate

Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential, which describes the dynamics of the (2+1)-dimensional Bose–Einstein condensate. Employing the Hirota method and symbolic computation, we obtain the dark one-soliton, two-soliton, three-soliton, breather-wave and rouge-wave solutions, respectively. We graphically study the dark solitons with the time-varying harmonic potential and scaled scattering length. Parallel and period solitons are observed. We obtain that when the external trapping potential increases with time, amplitudes of the dark solitons increase and widths of those solitons become narrower; when the external trapping potential is a periodic function, amplitudes and widths of the dark solitons periodically change. Decrease in the scaled scattering length leads to the narrower solitons’ widths, but does not affect the solitons’ amplitudes. Breather waves and rouge waves are also displayed: Rouge waves emerge when the period of the breather waves go to the infinity.

Bright–dark vector soliton solutions for the coupled complex short pulse equations in nonlinear optics

Under investigation in this paper are the coupled complex short pulse equations, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the Hirota method, bright–dark one- and two-soliton solutions are obtained. Interactions between two bright or two dark solitons are verified to be elastic through the asymptotic analysis. With different parameter conditions, the oblique interactions, bound states of solitons and parallel solitons are analyzed.

Dark soliton control in inhomogeneous optical fibers

Analytic two-dark soliton solutions for a variable–coefficient nonlinear Schrödinger equation are obtained via modified Hirota method. Parallel solitons are observed and soliton control such as the soliton compression is realized with different group velocity dispersion profiles. Besides, soliton interactions are investigated with the interaction distance being adjusted. In addition, soliton repulsive structures as well as attractive ones are obtained with exponential dispersion profile. Results in our research may be useful for the soliton control in inhomogeneous optical fibers, which will be a benefit to the realistic optical communication systems.

Soliton dynamics to the multi-component complex coupled integrable dispersionless equation

The generalized coupled integrable dispersionless (CID) equation describes the current-fed string in a certain external magnetic field. In this paper, we propose a multi-component complex CID equation. The integrability of the multi-component complex equation is confirmed by constructing Lax pairs. One-soliton and two-soliton solutions are investigated to exhibit rich evolution properties. Especially, similar as the multi-component short pulse equation and the first negative AKNS equation, periodic interaction, parallel solitons, elastic and inelastic interaction, energy re-distribution happen between two solitons. Multi-soliton solutions are given in terms of Pfaffian expression by virtue of Hirota’s bilinear method.

Mixed-type solitons and soliton interaction for the (2+1)-dimensional two-component long wave–short wave resonance interaction equations in a two-layer fluid through the Bell polynomials

In this work, we investigate the ( 2 + 1 )-dimensional two-component long wave–short wave resonance interaction equations in a two-layer fluid. Through the Bell polynomials, bilinear forms and mixed-type soliton solutions are derived. Such soliton phenomena as the V-type solitons, two parallel solitons with the periodic interaction, breather-type solitons and semi-breather-type solitons are observed, and the relevant parameter conditions are presented.

Lax pair, conservation laws and solitons for a ([formula omitted])-dimensional fourth-order nonlinear Schrödinger equation governing an [formula omitted]-helical protein

Energy transfer through a (2+1)-dimensional α-helical protein can be described by a (2+1)-dimensional fourth-order nonlinear Schrödinger equation. For such an equation, a Lax pair and the infinitely-many conservation laws are derived. Using an auxiliary function and a bilinear formulation, we get the one-, two-, three- and N-soliton solutions via the Hirota method. The soliton velocity is linearly related to the lattice parameter γ, while the soliton’ direction and amplitude do not depend on γ. Interactions between the two solitons are elastic, while those among the three solitons are pairwise elastic. Oblique, head-on and overtaking interactions between the two solitons are displayed. Oblique interaction among the three solitons and interactions among the two parallel solitons and a single one are presented as well.

Soliton interactions for coupled nonlinear Schrödinger equations with symbolic computation

Soliton interactions for the coupled nonlinear Schrodinger equations, governing the propagation of envelopes of electromagnetic waves in birefringent optical fibers, are investigated with symbolic computation. Based on the Hirota method, analytic two- and three-soliton solutions for this model are derived. Relevant interaction properties are discussed. Stationary bound vector solitons with the periodic attraction and repulsion are obtained. Soliton intensity could be reduced if the nonlinearity in optical fibers is enlarged, while the soliton period could be prolonged as the group velocity dispersion in the anomalous dispersion regime of optical fibers increases. Through the asymptotic analysis for the two-soliton solutions, interactions between two solitons are proven to be elastic. Besides, parallel soliton transmission systems without soliton interactions are presented. Moreover, interactions between the regular and bound vector solitons are studied. Dual complex structures and triple-soliton bound states are presented. Results could be of certain value to the studies on the soliton control and optical switching technologies.

Bäcklund transformation and N-soliton solutions for a (2+1)-dimensional nonlinear evolution equation in nonlinear water waves

Under investigation in this paper is a (2+1)-dimensional nonlinear evolution equation generated via the Jaulent–Miodek hierarchy for nonlinear water waves. With the aid of binary Bell polynomials and symbolic computation, bilinear forms and a Bäcklund transformations are derived. -soliton solutions are obtained through the Hirota method. Soliton propagation is discussed analytically. The bell-shaped soliton, anti-bell-shaped soliton and shock wave can be seen with some parameters selected. Soliton interactions are analyzed graphically: four kinds of elastic interactions are presented: two parallel bell-shaped solitons, two parallel anti-bell-shaped solitons, three parallel bell-shaped solitons and three parallel anti-bell-shaped solitons. We see that (1) the solitons maintain their original amplitudes, widths and directions except for some phase shifts after each interaction, and (2) the smaller the soliton amplitude is, the faster the soliton travels.

All-Optical EXOR for Cryptographic Application Based on Spatial Solitons

The purpose of this paper is to present an all-optical EXOR for cryptographic application based on spatial soliton beams. The device is based on the propagation and interactions properties of spatial soliton in a Kerr nonlinear material. The interaction force between parallel soliton beam is analyzed from the analytical point of view and an exact solution is presented.

N-fold Darboux transformation and double-Wronskian-typed solitonic structures for a variable-coefficient modified Kortweg-de Vries equation

Under investigation in this paper is a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model describing certain situations from the fluid mechanics, ocean dynamics and plasma physics. N-fold Darboux transformation (DT) of a variable-coefficient Ablowitz–Kaup–Newell–Segur spectral problem is constructed via a gauge transformation. Multi-solitonic solutions in terms of the double Wronskian for the vc-mKdV model are derived by the reduction of the N-fold DT. Three types of the solitonic interactions are discussed through figures: (1) Overtaking collision; (2) Head-on collision; (3) Parallel solitons. Nonlinear, dispersive and dissipative terms have the effects on the velocities of the solitonic waves while the amplitudes of the waves depend on the perturbation term.

Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation

Under investigation in this paper is an inhomogeneous nonlinear system, which describes the marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics under inhomogeneous media. Through symbolic computation, the Painlevé integrable condition, Lax pair and conservation laws are derived for this system. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Parallel propagation of solitons: separation distance of the two parallel solitons depends on the value of |Im(λ1)|-|Im(λ2)| (where λ1 and λ2 are the spectrum parameters); (2) Periodic propagation of bound solitons: periodic bound solitons taking on contrary trends, and mutual attractions and repulsions of two bright bound solitons; (3) Elastic interactions of two one-peak bright solitons and of two one-peak dark solitons.

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

In this paper, an infinite sequence of conservation laws for a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids are constructed based on the Bäcklund transformation. Hirota bilinear form and symbolic computation are applied to obtain three kinds of solutions. Variable coefficients can affect the conserved density, associated flux, and appearance of the characteristic lines. Effects of the wave number on the soliton structures are also discussed and types of soliton structures, e.g., the double-periodic soliton, parallel soliton and soliton complexes, are presented.

Alfven soliton and multisoliton dynamics perturbed by nonlinear Landau damping

The evolution of weakly dispersive nonlinear Alfven waves propagating either parallel or oblique to the ambient magnetic field is investigated through the derivative nonlinear Schrödinger equation (DNLS) perturbed by nonlinear Landau damping. The dynamics is analyzed with the aid of a numeric algorithm based on the inverse scattering transform (IST) and an adiabatic model that takes advantages of the perturbed DNLS invariants. Both techniques are applied to five types of DNLS soliton and multisoliton solutions: (i) the parallel Alfven soliton, (ii) the bright and dark one-parameter oblique, (iii) the breather two-parameter oblique, (iv) two parallel Alfven solitons, and (v) the combination of a dark and a bright oblique solitons. For the parallel solitons, the adiabatic model describes correctly the dynamics and it also recovers the well-known result given by the perturbed IST. Due to the radiation emission and the formation of dark solitons, the behavior of oblique solitons is more complicated and multisoliton solutions are required in the adiabatic model. The analysis shows that parallel solitons develop into the normal regime, whereas the oblique waves leads to the formation of dark solitons and breathers with a wavepacket form.

Long-Range Interaction between Two Periodic Solitons through Growing-and-Decaying Mode in the Davey–Stewartson I Equation

where r is constant. One important feature of the solutions to the DS I equation is the reversion of the unstable wave train to its initial state. The analytical solution for the nonlinear evolution of a modulational unstable mode is given by the growing-and-decaying (GD) mode solution. This shows that an unstable mode grows exponentially in its early stage. After reaching a maximummodulation, the unstable mode vanishes with time to reproduce the initial unmodulated state. The DS equation has a periodic soliton solution, which represents the spatial structure of the inclined sequence of algebraic solitons in addition to the dark line soliton and algebraic soliton solutions. In previous studies, we investigated the interactions between two periodic solitons, and between the periodic soliton and other types of solitons. Note that there are two types of singular interaction: the resonant interaction, in which two solitons interact so as to generate a single soliton, and the extremely long-range interaction, in which two solitons interchange each other infinitely apart through the messenger soliton, which is similar to a periodic soliton. Recently, the interaction between a soliton and a GD mode has been investigated, and the existence of resonant interactions has been demonstrated. We reported that the existence of the periodic soliton changed the evolution of the modulational instability markedly, as if the soliton dominated the evolution of instability. These results suggest the possible existence of an interaction between two solitons through the GD mode. The purpose of the present note is to show that a long-range interaction exists between two parallel periodic solitons through the GD mode. Using the N-soliton solution of Satsuma and Ablowitz, the solution that describes the interaction between two periodic solitons, which have the same real part ð ; Þ and different imaginary parts ð 1; 1Þ and ð 2; 2Þ of complex wave numbers, is written as

Coherent interactions between discrete spatial solitons in light-induced planar waveguide arrays

The coherent interactions between one-dimensional discrete spatial optical solitons, derived by Petviashvili iteration method, are numerically investigated with the split-step beam propagation method in light-induced planar waveguide arrays. The influences of the initial phase difference, the soliton peak intensities, the periods and intensities of the writing beams for waveguide arrays, and the amplitudes of the externally bias fields on the interactions between two discrete solitons are analyzed in detail. It is found that the interactions between two parallel discrete solitons with different initial phase differences behave in a way similar to that encountered in continuous media, i.e., in-phase (out-of-phase) discrete solitons attract (repel) each other, and the intermediates are always accompanied with energy transfer. In addition, the interaction processes are influenced by the variations of soliton peak intensities, the configurations of the light-induced waveguide arrays, and the amplitudes of the external bias fields.