Yu-Qiang Yuan

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.

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.

Bright-dark solitons for a set of the general coupled nonlinear Schrödinger equations in a birefringent fiber

Under investigation in this paper is the coupled nonlinear Schrodinger equations with the four-wave mixing term, which describe the optical solitons in a birefringent fiber. Via the Kadomtsev-Petviashvili hierarchy reduction, we obtain the N -bright-dark soliton solutions in terms of the Gram determinant. Propagation and interaction of the solitons corresponding to the electric fields in the two orthogonal polarizations are discussed and presented graphically. We find that the one bright-dark soliton possesses the periodic oscillation and exhibits the breather-like profile, which is different from that in the previous literature. Besides, for the one soliton, we observe that the larger velocity leads to the fiercer oscillation. Elastic interactions including the head-on and overtaking interactions between the two bright-dark solitons are demonstrated. Particularly, we find the oblique inelastic interaction between the two bright-dark solitons, which possess the V-shape profile in the zero background component and the Y-shape profile in the nonzero background component. Besides, we present two cases of the bound-state solitons. For the one case, the two solitons interact with each other all the time along a direction and for the other case, the resonance phenomenon is raised.

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.

Wronskian and Grammian solutions for a (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

Abstract In this paper, a ( 2 + 1 ) -dimensional Date–Jimbo–Kashiwara–Miwa equation is investigated. Based on the Hirota method and auxiliary variable, Backlund transformation is obtained. Under certain conditions, the Hirota bilinear forms can be reduced to the Plucker and Jacobi identities via the Wronskian and Grammian solutions. Through the Wronskian technique and Pfaffian derivative formulae, N -soliton solutions in the Wronskian and Grammian are given. Graphically presented, soliton collisions are elastic both in the Wronskian and Grammian, and after each collision, the solitons keep their amplitudes unchanged except for the phase shifts.

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.

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.