De-Yin Liu

Analytic study on a ( 2 + 1 )-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetism

In this paper, a ( 2 + 1 )-dimensional nonlinear Schrodinger equation for a ( 2 + 1 )-dimensional Heisenberg ferromagnetic spin chain with the bilinear and anisotropic interactions is investigated. Via the Hirota method and symbolic computation, bilinear forms and multi-soliton solutions are derived. The one, two and three solitons are analyzed graphically and we find the amplitudes and widths of the two and three solitons keep invariant after each interaction. The bell-shape one soliton as well as parallel, crossed two and three solitons are respectively observed. Through the asymptotic analysis, expressions which denote the two solitons before and after the interactions are obtained and interactions between the two solitons are proved to be elastic.

Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers

We report the existence and properties of vector breather and semirational rogue-wave solutions for the coupled higher-order nonlinear Schrödinger equations, which describe the propagation of ultrashort optical pulses in birefringent optical fibers. Analytic vector breather and semirational rogue-wave solutions are obtained with Darboux dressing transformation. We observe that the superposition of the dark and bright contributions in each of the two wave components can give rise to complicated breather and semirational rogue-wave dynamics. We show that the bright-dark type vector solitons (or breather-like vector solitons) with nonconstant speed interplay with Akhmediev breathers, Kuznetsov-Ma solitons, and rogue waves. By adjusting parameters, we note that the rogue wave and bright-dark soliton merge, generating the boomeron-type bright-dark solitons. We prove that the rogue wave can be excited in the baseband modulation instability regime. These results may provide evidence of the collision between the mixed ultrashort soliton and rogue wave.

Nonautonomous Matter-Wave Solitons in a Bose-Einstein Condensate with an External Potential

Nonautonomous matter-wave solitons in a Bose–Einstein condensate with an external potential are reported. Via the non-isospectral Ablowitz–Kaup–Newell–Segur system, the Gross–Pitaevskii equation is found to have the double Wronskian solutions. The verification of such solutions is finished through some double Wronskian identities. With the zero-potential Lax pair, the double Wronskian solutions give the nonautonomous N-soliton solutions that contain 2N parameters. For characterizing the asymptotic behavior of the bright two-soliton solutions, the explicit expressions of asymptotic solitons are given. Effects of the linear and harmonic potentials on the bound states between two matter-wave solitons are discussed.

Soliton collisions for a generalized variable-coefficient coupled Hirota–Maxwell–Bloch system for an erbium-doped optical fiber

In this paper, we construct soliton solutions for a generalized variable-coefficient coupled Hirota–Maxwell–Bloch system, which can describe the ultrashort optical pulse propagation in a nonlinear, dispersive fiber doped with two-level resonant atoms. Under certain transformations and constraints, one- and two-soliton solutions are obtained via the Hirota method and symbolic computation, and soliton collisions are graphically presented and analyzed. One soliton is shown to maintain its amplitude and shape during the propagation. Soliton collision is elastic, while bright two-peak solitons and dark two-peak solitons are also observed. We discuss the influence of the coefficients for the group velocity, group-velocity dispersion (GVD), self-phase modulation, distribution of the dopant, and Stark shift on the soliton propagation and collision features, with those coefficients are set as some constants and functions, respectively. We find the group velocity and self-phase modulation can change the solitons’ amplitudes and widths, and the solitons become curved when the GVD and distribution of the dopant are chosen as some functions. When the Stark shift is chosen as a certain constant, the two peaks of bright two-peak solitons and dark two-peak solitons are not parallel. In addition, we observe the periodic collision of the two solitons.

Conservation laws and dark-soliton solutions of an integrable higher-order nonlinear Schrödinger equation for a density-modulated quantum condensate

In this paper, an integrable higher-order nonlinear Schrodinger equation for a density-modulated quantum condensate is investigated. Based on the Ablowitz–Kaup–Newell–Segur system, an infinite number of conservation laws are obtained. Introducing an auxiliary function, we derive the bilinear forms and construct the dark-soliton solutions with the help of the Hirota method and symbolic computation. Dark one, two, and three solitons are analyzed graphically. Via asymptotic analysis, interactions between the two dark solitons are proved to be elastic. We see that the coefficients in the equation only affect the soliton velocity. We analyze the linear stability of the plane wave solutions in the presence of a small perturbation.

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.

Lump Solutions for the (3+1)-Dimensional Kadomtsev–Petviashvili Equation

Abstract In this article, we investigate the lump solutions for the Kadomtsev–Petviashvili equation in (3+1) dimensions that describe the dynamics of plasmas or fluids. Via the symbolic computation, lump solutions for the (3+1)-dimensional Kadomtsev–Petviashvili equation are derived based on the bilinear forms. The conditions to guarantee analyticity and rational localisation of the lump solutions are presented. The lump solutions contain eight parameters, two of which are totally free, and the other six of which need to satisfy the presented conditions. Plots with particular choices of the involved parameters are made to show the lump solutions and their energy distributions.

Soliton interactions, Bäcklund transformations, Lax pair for a variable-coefficient generalized dispersive water-wave system

Abstract Under investigation in this paper is a variable-coefficient generalized dispersive water-wave system, which can simulate the propagation of the long weakly non-linear and weakly dispersive surface waves of variable depth in the shallow water. Under certain variable-coefficient constraints, by virtue of the Bell polynomials, Hirota method and symbolic computation, the bilinear forms, one- and two-soliton solutions are obtained. Bäcklund transformations and new Lax pair are also obtained. Our Lax pair is different from that previously reported. Based on the asymptotic and graphic analysis, with different forms of the variable coefficients, we find that there exist the elastic interactions for u, while either the elastic or inelastic interactions for v, with u and v as the horizontal velocity field and deviation height from the equilibrium position of the water, respectively. When the interactions are inelastic, we see the fission and fusion phenomena.

Bound-state solutions, Lax pair and conservation laws for the coupled higher-order nonlinear Schrödinger equations in the birefringent or two-mode fiber

For describing the propagation of ultrashort pulses in the birefringent or two-mode fiber, we consider the coupled nonlinear Schrodinger equations with higher-order effects. According to the Ablowitz–Kaup–Newell–Segur system, 3×3 Lax pair for such equations is derived. Based on the Lax pair, we construct the conservation laws and Darboux transformation (DT). One- and two-soliton solutions are obtained via the DT, and we graphically present the one soliton and bound-state two solitons.

Bright and dark solitons and Bäcklund transformations for the coupled cubic-quintic nonlinear Schrödinger equations with variable coefficients in an optical fiber

Effects of the quintic nonlinearity for the ultrashort optical pulse propagation in a non-Kerr medium, or in the twin-core nonlinear optical fiber or waveguide, can be described by the coupled cubic-quintic nonlinear Schr?dinger equations with variable coefficients. For such equations, a Lax pair and the infinitely-many conservation laws are constructed. Under certain variable-coefficient constraints, bilinear forms, bilinear B?cklund transformations, bright/dark one-, two- and N-soliton solutions are derived via the Hirota method and symbolic computation. Those soliton solutions are merely related to the delayed nonlinear response effect, b(z). Graphical analyses on the soliton propagation and interaction suggest that b(z) can affect the bright/dark-soliton velocities but has no effect on their amplitudes. With the different choices of b(z), propagation of the linear-, periodic-, parabolic- and S-type bight and dark one solitons are seen, and different types of the interaction between the bright two solitons are displayed, such as the interactions between the bidirectional bright two ones, between the unidirectional bright two ones and between a moving bright one and a stationary one. Interactions between the linear-, S-, and parabolic-type dark two solitons are asymptotically analyzed and graphically illustrated as well, and those interactions are all elastic.