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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 (N = 2 , 3 , ...) 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. [ABSTRACT FROM AUTHOR]

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, wheretis the normalized time,is the strength of the four wave mixing terms, andis 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: Whenis 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); Whenis at-varying function, i.e., amplitude and velocity of the vector soliton both vary withtincreasing, whileaffects the vector soliton’s amplitude and velocity. With the differentor, 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. [ABSTRACT FROM PUBLISHER]

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 thexaxis, increase with wave numbersincreasing. Withincreasing,increase when, while decrease when., the velocities in the y axis, increase withincreasing anddecreasing. [ABSTRACT FROM PUBLISHER]

Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger 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. [ABSTRACT FROM AUTHOR]

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, Bäcklund 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 , and during the propagation, kink- and anti-bell-shape solitons are obtained when , anti-kink- and bell-shape solitons are obtained when ; Head-on interaction between the two bidirectional solitons, overtaking interaction between the two unidirectional solitons are presented; interactions between the two solitons are elastic. [ABSTRACT FROM AUTHOR]

In this article, the propagation and collision of vector solitons are investigated from the 3-coupled variable-coefficient nonlinear Schrödinger equations, which describe the amplification or attenuation of the picosecond pulses in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations. On the basis of the Lax pair, infinitely-many conservation laws are obtained. Under an integrability constraint among the variable coefficients for the group velocity dispersion (GVD), nonlinearity and fibre gain/loss, and two mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation. Influence of the variable coefficients for the GVD and nonlinearity on the vector soliton amplitudes and velocities is analysed. Through the asymptotic and graphic analysis, bound states and elastic and inelastic collisions between the vector two solitons are investigated: Not only the elastic but also inelastic collision between the 2-bright-1-dark vector two solitons can occur, whereas the collision between the 1-bright-2-dark vector two solitons is always elastic; for the bound states, the GVD and nonlinearity affect their types; with the GVD and nonlinearity being the constants, collision period decreases as the GVD increases but is independent of the nonlinearity. [ABSTRACT FROM AUTHOR]

Energy transfer through a (2+1)-dimensional α -helical protein can be described by a (2+1)-dimensional fourth-order nonlinear Schrödinger equation. For such an equation, a Lax pair and the infinitely-many conservation laws are derived. Using an auxiliary function and a bilinear formulation, we get the one-, two-, three- and N -soliton solutions via the Hirota method. The soliton velocity is linearly related to the lattice parameter γ , while the soliton’ direction and amplitude do not depend on γ . Interactions between the two solitons are elastic, while those among the three solitons are pairwise elastic. Oblique, head-on and overtaking interactions between the two solitons are displayed. Oblique interaction among the three solitons and interactions among the two parallel solitons and a single one are presented as well. [ABSTRACT FROM AUTHOR]