Parallel Solitons Research Articles

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries (KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

Soliton dynamics to the Higgs equation and its multi-component generalization

In this paper, we study the coupled Higgs equation and its multi-component generalization based on the Hirota’s direct method. One and two-soliton solutions of the coupled Higgs equation are derived by the perturbation approach. We express the N-soliton solutions in the form of Pfaffians and demonstrate that the N-component coupled Higgs equation turns out to be the Pfaffian identity. One and two-soliton solutions of the multi-component coupled Higgs equation are obtained from the Pfaffians. Starting from the explicit solutions, parallel solitons, periodic and nearly periodic interactions, elastic and inelastic collisions are investigated.

Bilinear forms, N-soliton solutions, breathers and lumps for a (2+1)-dimensional generalized breaking soliton system

In this paper, a ([Formula: see text])-dimensional generalized breaking soliton system, for the interactions of the Riemann wave with a long wave, is investigated. Via the Hirota method, bilinear forms different from those in the existing literatures are derived. [Formula: see text]-soliton solutions are constructed via the Wronskian technique. Solitons with the crest curves being curvilineal are constructed, whose shape changes with the propagation. Parallel solitons have been obtained. Directions of the soliton propagation change, and speeds of the solitons are different: The higher the amplitude of the soliton is, the faster the soliton propagates. Breathers are constructed. Solutions consisting of a lump and two solitons are derived: Two solitons propagate in the same direction and the lump occurs in the region of the interaction between the two solitons.

Dynamics of soliton interaction solutions of the Davey-Stewartson I equation.

In this paper, we first modify the binary Darboux transformation to derive three types of soliton interaction solutions of the Davey-Stewartson I equation, namely the higher-order lumps, the localized rogue wave on a solitonic background, and the line rogue wave on a solitonic background. The uniform expressions of these solutions contain an arbitrary complex constant, which plays a key role in obtaining diverse interaction scenarios. The second-order dark-lump solution contains two hollows that undergo anomalous scattering after a head-on collision, and the minimum values of the two hollows evolve in time and reach the same asymptotic constant value 0 as t→±∞. The localized rogue wave on a solitonic background describes the occurrence of a waveform from the solitonic background, quickly evolving to a doubly localized wave, and finally retreating to the solitonic background. The line rogue wave on the solitonic background does not create an extreme wave at any instant of time, unlike the one on a constant background, which has a large amplitude at the intermediate time of evolution. For large t, the solitonic background has multiple parallel solitons possessing the same asymptotic velocities and heights. The obtained results improve our understanding of the generation mechanisms of rogue waves.

Soliton interaction of a generalized nonlinear Schrödinger equation in an optical fiber

Under investigation in this work is a generalized nonlinear Schrödinger (NLS) equation with high-order dispersion and quintic nonlinear terms, which can describe a subpicosecond pulse propagation in optical fibers. By developing Hirota method, the bilinear form and analytical one- and two-soliton solutions are derived. Based on the analytical solutions, we give the corresponding soliton patterns to analyse dynamics of the pulse solitons. It shows that high order dispersion term may change the periodicity of propagation. The interaction of two pulse solitons is an elastic collision. By choosing suitable parameter values, we can obtain the two parallel solitons. It can be applied to improve transmission quality and capacity of information in optical fiber.

Higher-order rogue wave solutions to the Kadomtsev–Petviashvili 1 equation

We construct a family of eigenfunction solutions of the Lax pair of the Kadomtsev–Petviashvili 1 (KP1) equation, which is expressed in terms of the Taylor coefficients of a fundamental exponential function associated with the Lax pair. Using the binary Darboux transformation, the higher-order rogue waves on a solitonic background of the KP1 equation are obtained by multiple soliton interactions. In fact, these solitons can evolve to significant and strongly localized transient waves during directional evolution, similarly to KP2 shallow-water interactions. We conjecture that the nth-order rogue wave solution evolves in the form of a triangular extreme wave pattern that consists of n(n+1)2 solitonic lumps in the intermediate time, while only n+1 parallel line solitons possessing equal height are present before and after the initiation of collision dynamics. Such fascinating higher-order rogue waves have not yet been reported in the context of this 2+1 dimensional integrable system. These exact solutions are particularly relevant for the fundamental understanding of extreme wave events in a variety of physical systems, such as plasma and solids.

Multi-soliton solutions and interaction for a (2+1)-dimensional nonlinear Schrödinger equation

In this paper, a (2+1)-dimensional nonlinear Schrödinger (NLS) equation with variable coefficients is studied. Base on the symmetry of the equation, we rotate the original coordinate for 45 degrees to get a simple one in the new coordinate. This method is quite easy to compare with solving it directly, and it is never been used before. By the Hirota method with an auxiliary function, the soliton solutions are derived. We revealed the structures between soliton interaction and discussed the interaction properties: Influences of the coefficients and wave numbers on interaction shapes of solitons are presented, i.e. parabolic, cubic, quasi-periodical, and polyline solitons are presented; Solitonic amplitudes are inversely proportional to the wave numbers. Besides, if the coefficients small enough, two-soliton solutions are degenerate into two parallel solitons; Soliton solutions do not only depend on the wave numbers, but also affected by the coefficients; To demonstrate those soliton solutions, we propose different methods to control those parameters, we also discuss a new form and draw some good conclusions. In the end, bound-state solitons are obtained.

Soliton solutions to the reverse-time nonlocal Davey–Stewartson III equation

This study derives solitons solutions of the reverse-time nonlocal Davey–Stewartson III equation by the Kadomtsev–Petviashvili hierarchy reduction method and Hirota’s bilinear method. The solutions are expressed as N×N Gram-type determinants with different parametric reduction conditions. N-soliton and line breather solutions on both constant and periodic backgrounds are derived. The dynamics of these solutions are discussed. All possible configurations of these solutions are illustrated for 1≤N≤4. Both intersecting and parallel solitons are presented. In particular, the elastic and inelastic collisions of the two parallel-soliton solutions are determined. In the inelastic case, the amplitudes of the solitons change after collision. Moreover, the parametric conditions for determining the inelastic collisions are derived, and all possible types of inelastic behaviors are obtained and displayed.

Lax pairs, infinite conservation laws, Darboux transformation, bilinear forms and solitonic interactions for a combined Calogero-Bogoyavlenskii-Schiff-type equation

Calogero-Bogoyavlenskii-Schiff-type (CBS-type) equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Under investigation in this paper is a combined CBS-type equation. Lax pairs in the differential and matrix forms are derived respectively. Infinite conservation laws, which are different from those in the existing literatures, and n-fold Darboux transformation are constructed through the matrix-form Lax pair, where n is a positive integer. Bilinear forms are constructed. Parallel solitons can be derived from the bilinear forms, which are caused by a constraint in the linearization process. Waveforms for the parallel solitons have the no superposition, nonlinear superposition and linear superposition forms. Bell-to-anti-bell-shaped solitons and oblique solitonic interactions are discovered via the n-fold Darboux transformation. Each bell-shaped asymptotic soliton of the bell-to-anti-bell-shaped soliton evolves into the anti-bell-shaped one.

Propagation characteristics of parallel dark solitons in silicon-on-insulator waveguide**Project supported by the National Natural Science Foundation of China (Grant No. 61741509).

The propagation characteristic of two identical and parallel dark solitons in a silicon-on-insulator (SOI) waveguide is simulated numerically using the split-step Fourier method. The parallel dark solitons imposed by the initial chirp are investigated mainly by changing their power, their relative time delay. The simulation shows that the time delay deforms the parallel dark soliton pulse, forming a bright-like soliton in the transmission process and making the transmission quality down. By increasing the power of one dark soliton, the energy of the other dark soliton can be increased, and larger increase in a soliton’s power leads to larger increase in the energy of the other. When the initial chirp is introduced into one of the dark solitons, higher energy consumption is observed. In particular, positive chirps resulting in pulse broadening width while negative chirps narrowing, with an obvious compression effect on the other dark soliton. Finally, large negative chirps are found to have a profound impact on parallel and nonparallel dark solitons.

Gramian solutions and soliton interactions for a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in a plasma or fluid

Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation for the nonlinear phenomena in a plasma or fluid. Based on the existing bilinear form, N-soliton solutions in the Gramian are derived, where N = 1, 2, 3…. With N = 3, three-soliton solutions are constructed. Fission and fusion for the three solitons are presented. Effects of the variable coefficients, i.e. h(t), l(t), q(t), n(t) and m(t), on the soliton fission and fusion are revealed: soliton velocity is related to h(t), l(t), q(t), n(t) and m(t), while the soliton amplitude cannot be affected by them, where t is the scaled temporal coordinate, h(t), l(t) and q(t) give the perturbed effects, and m(t) and n(t), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We show the three parallel solitons with the same direction.

Optical solitons and their shaping in a monomode optical fiber with some inhomogeneous dispersion profiles

In the real fiber optic communication system, the soliton transmission can be signified by nonlinear Schrödinger (NLS) equation with inhomogeneous coefficients. Here, the two-soliton propagation have been investigated by solving the inhomogeneous NLS equation with the process of Darboux transformation. According to obtained soliton solutions, variable coefficients are considered with different forms to study the features of solitons. Since soliton structures are mainly influenced by the controlling parameters, various soliton structures are obtained for specific choices of variable coefficients. Specifically, we adopt various exponential profiles for dispersion coefficient. Through properly manipulating exponential profiles, non-oscillating soliton amplification, dromion-like structure, bounded soliton and parallel soliton transmissions are explored. Results indicate that the characteristics of optical solitons are influenced by variable coefficients. In this work, we report the analytical study on transmission features of optical solitons in inhomogeneous optical fiber under various dispersion profiles. Obtained results are have some theoretical guidance for experimental research on optical amplifier, soliton based optical switches and achievement of parallel transmission of solitons.

Multi-breather wave solutions for a generalized (3＋1)-dimensional Yu–Toda–Sasa–Fukuyama equation in a two-layer liquid

Two-layer fluid models are proposed to describe certain nonlinear phenomena in fluid mechanics, thermodynamics and medical sciences. For a generalized(3＋1)-dimensional Yu–Toda–Sasa–Fukuyama equation for the interfacial waves in a two-layer liquid, in which h0 is the constant coefficient of the development term, h1 is the constant coefficient of the dispersion term, h2 and h3 are the constant coefficients of the nonlinear terms, and h4 is the constant coefficient of the linear term. Based on the Pfaffian technique and certain constraint on h3, we obtain the multi-breather solutions in terms of the Gramian. For the one-breather waves, amplitudes are proportional to h1h2, velocity components along the x and z directions for the characteristic lines are proportional to h1h0, and velocity component along the y direction for the characteristic line is proportional to h1h43h02, where x, y and z are the spatial coordinates and t is the temporal coordinate. Interaction between the two-breather waves implies that the two-breather waves can evolve periodically along two straight lines on the x-y and y-z planes, period for the two-breather waves with t is proportional to h0h1, period for the two-breather waves along the y direction is proportional to h43h1, while the two-breather waves are not periodic along the x and z directions. Amplitudes, velocities and widths of the two-breather waves keep unchanged after the interaction between the two-breather waves on the x-y and y-z planes, which means that the interaction is elastic. On the x-z plane, the two-breather waves appear as two parallel solitons at certain angles with the x and z axes.

Temporal analysis of Airy beam propagation in photorefractive media

We theoretically examine the temporal behavior of Airy beams and their interactions in photorefractive medium with the split-step Fourier method. Simulations indicate that Airy beam propagations are effectively controlled by the illumination time. Under saturated nonlinearity, the beam intensity maintains the soliton profile at the output section, and its propagation direction is modulated by the beam amplitude. Moreover, the temporal behavior of Airy beam interactions varies greatly with illumination time, especially for the in-phase case. Breathing solitons and parallel soliton pairs are produced in-phase and out-of-phase, respectively. These results are potentially useful in all-optical device applications.

Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication

ABSTRACT Under investigation in this paper is a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication. Lax pair, one/N-fold binary Darboux transformations and the limit forms of the one-fold binary Darboux transformation for such an equation are obtained. Based on one/N-fold binary Darboux transformations and the limit forms of the one-fold binary Darboux transformation, one- and N-dark soliton solutions are obtained. Soliton amplitude is not affected by the coefficients of the nonlinear-Schrödinger operator, Hirota operator, Lakshmanan-Porsezian-Daniel operator and quintic operator, but soliton velocity is linearly related to them. Overtaking interaction between the two solitons is illustrated. Parallel solitons are formed: soliton width and interval between the parallel solitons are decreased with the increasing value of each and every coefficient of the nonlinear-Schrödinger operator, Hirota operator, Lakshmanan-Porsezian-Daniel operator and quintic operator respectively. For the interactions among the three solitons, e.g. interaction among three overtaking solitons, and interaction between the parallel solitons and a single soliton are displayed. We find that the interactions between the two solitons and among the three solitons are elastic.

Generation of single or double parallel breathing soliton pairs, bound breathing solitons, moving breathing solitons, and diverse composite breathing solitons in optical fibers.

Interactions of two truncated Airy pulses with arbitrarily initial relative phases, initial pulse intervals, and different soliton order are numerically investigated in optical fibers. When the soliton order is 1, depending on different initial pulse intervals, the initial in-phase Airy pulses may evolve to single breathing solitons, bound breathing solitons, and single parallel breathing solitons. While the out-of-phase Airy pulses may evolve to parallel or repulsive soliton pairs with breathing or weak breathing, after radiating away some dispersive waves. When the initial relative phases take arbitrary values except 0 and π, moving single breathing solitons and repulsive or parallel soliton pairs will form. Moreover, the whole temporal profiles may become asymmetric. The repulsive soliton pairs consist of two moving breathing solitons with different intensities, moving velocities, and breathing periods. The most interestingly is that, when the soliton order is larger than one, we observe double bound breathing solitons, double parallel breathing soliton pairs, and diverse composite breathing solitons which consist of two or more different breathing solitons. one can effectively manipulate and select the soliton expected and its evolution dynamics by adjusting the soliton order, initial pulse intervals, and initial relative phases.

Multi-soliton solutions for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation

In this paper, a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation is investigated, which can describe an inhomogeneous Heisenberg ferromagnetic spin chain with the octupole–dipole interaction or alpha helical protein with higher-order excitations and interactions under the continuum approximation. Bilinear forms, one- and two-soliton solutions are derived by virtue of the Hirota method and symbolic computation. Propagation and interaction properties of the solitons are discussed: Parabolic, cubic and periodic solitons are presented. Amplitudes of the solitons are only related to the wave numbers, while the velocities are related to both the wave numbers and variable coefficients. Interactions between the two parallel solitons are discussed.

Dark solitons behaviors for a (2+1)-dimensional coupled nonlinear Schrödinger system in an optical fiber

In this paper, we investigate a (2+1)-dimensional coupled nonlinear Schrödinger system, which describes the transverse effects in an optical fiber, time-independent copropagation and field of optical soliton. Bilinear forms, dark one- and two-soliton solutions are derived by virtue of the Hirota method. Propagation and interaction properties of the dark solitons are discussed: (i) Amplitudes and velocities of the dark solitons are affected by the values of the wave numbers μ, λ and θ. (ii) Head-on and overtaking interactions between the two parallel dark solitons are discussed, where the amplitudes of the dark solitons remain unchanged after each interaction, implying that the interactions are elastic. (iii) Stationary dark solitons are depicted in this paper. (iv) Through the asymptotic analysis, elastic interaction between the two solitons is discussed analytically.

Bright and dark wirelike spatiotemporal solitons of a partially nonlocal nonlinear Schrödinger equation

A (3+1)-dimensional nonautonomous partially nonlocal nonlinear Schrödinger equation with different diffractions is reduced to an autonomous nonlinear Schrödinger equation. Based on solutions of the autonomous nonlinear Schrödinger equation, and using the Darboux transformation method, spatiotemporal bright and dark soliton solutions are obtained. Dynamical evolution of wirelike spatiotemporal bright and dark single soliton, separated and interactive wirelike wave bright and dark double soliton structures are discussed in the sinusoidal diffraction system. For specified value of the parameter n in the Hermite polynomial, bright and dark single soliton, parallel separated bright and dark double soliton and interactive bright and dark double soliton structures all exist n + 1 columns in z -direction.

Bäcklund transformation and soliton solutions in terms of the Wronskian for the Kadomtsev–Petviashvili-based system in fluid dynamics

In this paper, investigation is made on a Kadomtsev–Petviashvili-based system, which can be seen in fluid dynamics, biology and plasma physics. Based on the Hirota method, bilinear form and Backlund transformation (BT) are derived. N-soliton solutions in terms of the Wronskian are constructed, and it can be verified that the N-soliton solutions in terms of the Wronskian satisfy the bilinear form and Backlund transformation. Through the N-soliton solutions in terms of the Wronskian, we graphically obtain the kink-dark-like solitons and parallel solitons, which keep their shapes and velocities unchanged during the propagation.