Jun Chai

Solitons and rouge waves for a generalized ( 3 + 1 )-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics

Evolution of the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in fluid mechanics in three spatial dimensions can be described by a generalized ( 3 + 1 )-dimensional variable-coefficient Kadomtsev-Petviashvili equation, which is studied in this paper with symbolic computation. Via the truncated Painleve expansion, an auto-Backlund transformation is derived, based on which, under certain variable-coefficient constraints, one-soliton, two-soliton, homoclinic breather-wave and rouge-wave solutions are respectively obtained via the Hirota method. Graphic analysis shows that the soliton propagates with the varying soliton direction. Change of the value of any one of g ( t ) , m ( t ) , n ( t ) , h ( t ) , q ( t ) and l ( t ) in the equation can cause the change of the soliton shape, while the soliton amplitude cannot be affected by that change, where g ( t ) represents the dispersion, m ( t ) and n ( t ) respectively stand for the disturbed wave velocities along the y and z directions, h ( t ) , q ( t ) and l ( t ) are the perturbed effects, y and z are the scaled spatial coordinates, and t is the temporal coordinate. Soliton direction and type of the interaction between the two solitons can vary with the change of the value of g ( t ) , while they cannot be affected by m ( t ) , n ( t ) , h ( t ) , q ( t ) and l ( t ) . Homoclinic breather wave and rouge wave are respectively displayed, where the rouge wave comes from the extreme behaviour of the homoclinic breather wave.

Bright optical solitons or light bullets for a (3 + 1)-dimensional generalized nonlinear Schrödinger equation with the distributed coefficients

Under investigation in this paper is a (3 + 1)-dimensional generalized nonlinear Schrodinger equation with the distributed coefficients for the spatiotemporal optical solitons or light bullets. Through the symbolic computation and Hirota method, one- and two-soliton solutions are derived. We also present the Backlund transformation, through which we derive the soliton solutions. When the gain/loss coefficient is the monotonically decreasing function for the propagation coordinate z, amplitude for the spatiotemporal optical soliton or light bullet decreases along z, while when the gain/loss coefficient is the monotonically increasing function for z, amplitude for the spatiotemporal optical soliton or light bullet increases along z. Directions of the solitons are different because the signs of imaginary parts of the frequencies are adverse. Based on the two-soliton solutions, elastic and inelastic collisions between the two spatiotemporal optical solitons or light bullets are derived under different conditions presented in the paper.

Solitons, bilinear Bäcklund transformations and conservation laws for a -dimensional Bogoyavlenskii-Kadontsev-Petviashili equation in a fluid, plasma or ferromagnetic thin film

Abstract In this paper, we investigate a -dimensional Bogoyavlenskii–Kadontsev–Petviashili equation in a fluid, plasma or ferromagnetic thin film. Through the Bell polynomials, Hirota method and symbolic computation, the one- and two-kink-soliton solutions are derived. Bäcklund transformation, Lax pair and conservation laws are presented. Elastic collisions including the oblique, parallel, unidirectional and bidirectional collisions between the two-kink solitons are discussed. In addition, the relation between the velocities and wave numbers of the two-kink solitons are analysed. When wave numbers , the velocities in the x axis, increase with wave numbers increasing. With increasing, increase when , while decrease when . , the velocities in the y axis, increase with increasing and decreasing.

Analytic study on certain solitons in an erbium-doped optical fibre

Abstract ()-dimensional non-linear optical waves through the coherently excited resonant medium doped with the erbium atoms can be described by a -dimensional non-linear Schrödinger equation coupled with the self-induced transparency equations. For such a system, via the Hirota method and symbolic computation, linear forms, one-, two- and N-soliton solutions are obtained. Asymptotic analysis is conducted and suggests that the interaction between the two solitons is elastic. Bright solitons are obtained for the fields E and P, while the dark ones for the field N, with E as the electric field, P as the polarization in the resonant medium induced by the electric field, and N as the population inversion profile of the dopant atoms. Head-on interaction between the bidirectional two solitons and overtaking interaction between the unidirectional two solitons are seen. Influence of the averaged natural frequency on the solitons are studied: (1) can affect the velocities of all the solitons; (2) Amplitudes of the solitons for the fields P and N increase with decreasing, and decrease with increasing; (3) With decreasing, for the fields P and N, one-peak one soliton turns into the two-peak one, as well as interaction type changes from the interaction between two one-peak ones to that between a one-peak one and a two-peak one; (4) For the field E, influence of on the solitons cannot be found. The results of this paper might be of potential applications in the design of optical communication systems which can produce the bright and dark solitons simultaneously.

Solitons and bilinear Bäcklund transformations for a (3+1)-dimensional Yu–Toda–Sasa–Fukuyama equation in a liquid or lattice

Abstract In this paper, we investigate a ( 3 + 1 ) -dimensional Yu–Toda–Sasa–Fukuyama equation for the interfacial wave in a two-layer liquid or elastic quasiplane wave in a lattice. Through the Bell polynomials, symbolic computation and Hirota method, the one and two bell-soliton solutions are derived. Backlund transformation is presented. Parallel collision between the two solitons exists when the soliton directions are the same. Oblique collision appears between the two solitons with different soliton directions.

Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid

In this paper, we investigate a two-mode Korteweg-de Vries equation, which describes the one-dimensional propagation of shallow water waves with two modes in a weakly nonlinear and dispersive fluid system. With the binary Bell polynomial and an auxiliary variable, bilinear forms, multi-soliton solutions in the two-wave modes and Bell polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton propagation and collisions between the two solitons are presented. Based on the graphic analysis, it is shown that the increase in s can lead to the increase in the soliton velocities under the condition of , but the soliton amplitudes remain unchanged when s changes, where s means the difference between the phase velocities of two-mode waves, and are the nonlinearity parameter and dispersion parameter respectively. Elastic collisions between the two solitons in both two modes are analyzed with the help of graphic analysis.

Solitons, Bäcklund transformation and Lax pair for a generalized variable-coefficient Boussinesq system in the two-layered fluid flow

Under investigation in this paper is a generalized variable-coefficient Boussinesq system, which describes the propagation of the shallow water waves in the two-layered fluid flow. Bilinear forms, Backlund transformation and Lax pair are derived by virtue of the Bell polynomials. Hirota method is applied to construct the one- and two-soliton solutions. Propagation and interaction of the solitons are illustrated graphically: kink- and bell-shape solitons are obtained; shapes of the solitons are affected by the variable coefficients α1, α3 and α4 during the propagation, kink- and anti-bell-shape solitons are obtained when α3 > 0, anti-kink- and bell-shape solitons are obtained when α3 < 0; Head-on interaction between the two bidirectional solitons, overtaking interaction between the two unidirectional solitons are presented; interactions between the two solitons are elastic.

Akhmediev breathers, Kuznetsov–Ma solitons and rogue waves in a dispersion varying optical fiber

Dispersion varying fibres have applications in optical pulse compression techniques. We investigate Akhmediev breathers, Kuznetsov–Ma (KM) solitons and optical rogue waves in a dispersion varying optical fibre based on a variable-coefficient nonlinear Schrodinger equation. Analytical solutions in the forms of Akhmediev breathers, KM solitons and rogue waves up to the second order of that equation are obtained via the generalised Darboux transformation and integrable constraint. The properties of Akhmediev breathers, KM solitons and rogue waves in a dispersion varying optical fibre, e.g. dispersion decreasing fibre (DDF) or a periodically distributed system (PDS), are discussed: in a DDF we observe the compression behaviours of KM solitons and rogue waves on a monotonically increasing background. The amplitude of each peak of the KM soliton increases, while the width of each peak of the KM soliton gradually decreases along the propagation distance; in a PDS, the amplitude of each peak of the KM soliton varies periodically along the propagation distance on a periodic background. Different from the KM soliton, the Akhmediev breather and rogue waves repeat their behaviours along the propagation distance without the compression.

Lax pair and vector solitons for a variable-coefficient coherently-coupled nonlinear Schrödinger system in the nonlinear birefringent optical fiber

Abstract Nowadays, with respect to the nonlinear birefringent optical fibers, efforts have been put into investigating the coupled nonlinear Schrödinger (NLS) systems. In this paper, symbolic computation on a variable-coefficient coherently-coupled NLS system with the alternate signs of nonlinearities is performed. Under a variable-coefficient constraint , the system is shown to be integrable in the Lax sense with a Lax pair constructed, where t is the normalized time, is the strength of the four wave mixing terms, and is the strength of the anti-trapping parabolic potential. With an auxiliary function, bilinear forms, vector one- and two-soliton solutions are obtained. Figures are displayed to help us study the vector solitons: When is a constant, vector soliton propagates stably with the amplitude and velocity unvarying (vector soliton’s amplitude changes with the change of that constant, while its velocity can not be affected by that constant); When is a t-varying function, i.e. , amplitude and velocity of the vector soliton both vary with t increasing, while affects the vector soliton’s amplitude and velocity. With the different or , interactions between the amplitude- and velocity-unvarying vector two solitons and those between the amplitude- and velocity-varying vector two solitons are displayed, respectively. By virtue of the system and its complex-conjugate system, conservation laws for the vector solitons, including the total energy and momentum, are constructed.

Dark solitons, Lax pair and infinitely-many conservation laws for a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in the inhomogeneous Heisenberg ferromagnetic spin chain

Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrodinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.