Han-Peng Chai

Certain bright soliton interactions of the Sasa-Satsuma equation in a monomode optical fiber

Under investigation in this paper is the Sasa-Satsuma equation, which describes the propagation of ultrashort pulses in a monomode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Based on the known bilinear forms, through the modified expanded formulas and symbolic computation, we construct the bright two-soliton solutions. Through classifying the interactions under different parameter conditions, we reveal six cases of interactions between the two solitons via an asymptotic analysis. With the help of the analytic and graphic analysis, we find that such interactions are different from those of the nonlinear Schrödinger equation and Hirota equation. When those solitons interact with each other, the singular-I soliton is shape-preserving, while the singular-II and nonsingular solitons may be shape preserving or shape changing. Such elastic and inelastic interaction phenomena in a scalar equation might enrich the knowledge of soliton behavior, which could be expected to be experimentally observed.

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.

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.

Rogue Waves and Lump Solitons of the (3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves*

In this paper, the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order, higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.

Darboux transformation and vector solitons for a variable-coefficient coherently coupled nonlinear Schrödinger system in nonlinear optics

Efforts have been put into investigating a variable-coefficient coherently coupled nonlinear Schrodinger system with the alternate signs of nonlinearities, describing the propagation of the waves in the nonlinear birefringent optical fiber. Via the Lax pair, Darboux transformation for the system is derived. Then, we derive the vector one- and two-soliton solutions. Figures are displayed to help us study the properties of the vector solitons: with the strength of the four-wave mixing terms γ(t) as a constant, the vector soliton propagates with the unvarying velocity and amplitude; with γ(t) being a time-dependent function, amplitude and velocity of the vector soliton keep varying during the propagation; bell- and M-shaped solitons can both be observed in q2 mode, while we just observe the bell-shaped soliton in q1 mode, where q1 and q2 are the two slowly varying envelopes of the propagating waves; head-on and overtaking interactions between the vector two solitons are both presented.

Vector semirational rogue waves for a coupled nonlinear Schrödinger system in a birefringent fiber

Abstract Under investigation in this paper is a coupled nonlinear Schrodinger system with the four-wave mixing term, which describes the propagation of optical waves in a birefringent fiber. Via the Darboux dressing transformation, the semirational solutions which give rise to the vector rogue waves and breathers are obtained. We display the vector rogue waves and the interaction between the rogue waves and bright–dark solitons. During the interaction, breather-like structures arise because of the interference between the dark and bright components of the soliton. Besides, it can be observed that the rogue wave and soliton merge together. Interactions between the breathers and bright–dark solitons are shown graphically. Keeping | α 1 | 2 a + | α 2 | 2 c + b α 1 α 2 ∗ + b ∗ α 1 ∗ α 2 invariant, we find that the smaller value of a c − | b | 2 yields the more obvious breather-like structure, with a and c representing the self- and cross-phase modulations, respectively, b representing the four-wave mixing effect, α 1 and α 2 being two constants. Similarly, keeping a c − | b | 2 invariant, we find that the smaller value of | α 1 | 2 a + | α 2 | 2 c + b α 1 α 2 ∗ + b ∗ α 1 ∗ α 2 yields the more obvious breather-like structure. Bound state forming between the Kuznetsov-Ma soliton and breather-like structure is illustrated.

Solitonic properties for a forced generalized variable-coefficient Korteweg-de Vries equation for the atmospheric blocking phenomenon

Abstract In this paper, investigation is given to a forced generalized variable-coefficient Korteweg-de Vries equation for the atmospheric blocking phenomenon. Based on the Lax pair, under certain variable-coefficient-dependent constraints, we present an infinite sequence of the conservation laws. Through the Riccati equations obtained from the Lax pair, a Wahlquist-Estabrook-type Bäcklund transformation (BT) is derived, based on which the nonlinear superposition formula as well as one- and two-soliton-like solutions are obtained. Via the truncated Painlevé expansion, we give a Painlevé BT, along with the one-soliton-like solutions. With the Painlevé BT, bilinear forms are constructed, and we get a bilinear BT as well as the corresponding one-soliton-like solutions. Bell-type bright and dark soliton-like waves and kink-type soliton-like waves are observed, respectively. Graphic analysis shows that (1) the velocities of the soliton-like waves are related to h(t), d(t), f(t) and R(t), while the soliton-like wave amplitudes just depend on f(t), and (2) with the nonzero f(t) and R(t), soliton-like waves propagate on the varying backgrounds, where h(t), d(t) and f(t) are the dispersive, dissipative and line-damping coefficients, respectively, R(t) is the external-force term, and t is the scaled time coordinate.

Semi-rational solutions for the (3+1)-dimensional Kadomtsev–Petviashvili equation in a plasma or fluid

Abstract In this paper, we investigate the ( 3 + 1 ) -dimensional Kadomtsev–Petviashvili equation in a plasma or fluid. For the amplitude of the electrostatic wave potential in the plasma or shallow-water wave in the fluid, via the Kadomtsev–Petviashvili hierarchy reduction, we obtain the semi-rational solutions in determinant form for such an equation. Interactions between the first-order lump (or rogue wave) and soliton are illustrated. We find that the lump arises and then separates from the soliton on the x – y and x – z planes; and that the rogue wave possesses a line profile and arises from the soliton (or constant background) on the y – z plane, where x , y and z are the scaled spatial coordinates. Interactions between the two lumps (or rogue waves) and two solitons are presented. Interactions between the second-order lump (or rogue wave) and one soliton are also presented.

Bilinear Forms and Soliton Solutions for the Reduced Maxwell-Bloch Equations with Variable Coefficients in Nonlinear Optics*

Investigation in this paper is given to the reduced Maxwell-Bloch equations with variable coefficients, describing the propagation of the intense ultra-short optical pulses through an inhomogeneous two-level dielectric medium. We apply the Hirota method and symbolic computation to study such equations. With the help of the dependent variable transformations, we present the variable-coefficient-dependent bilinear forms. Then, we construct the one-, two- and N-soliton solutions in analytic forms for them.

Composite rogue waves and modulation instability for the three-coupled Hirota system in an optical fiber

Abstract. We investigate the three-coupled Hirota system, which is applied to model the long distance communication and ultrafast signal routing systems governing the propagation of light pulses. With the aid of the Darboux dressing transformation, composite rogue wave solutions are derived. Spatial–temporal structures, including the four-petaled structure for the three-coupled Hirota system, are exhibited. We find that the four-petaled rogue waves occur in two of the three components, whereas the eye-shaped rogue wave occurs in the other one. The composite rogue waves can split up into two or three single rogue waves. The corresponding conditions for the occurrence of such phenomena are discussed and presented. We find that the relative position of every single rogue wave is influenced by the ratios of certain parameters. Besides, the linear instability analysis is performed, and our results agree with those from the baseband modulation instability theory.