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Under investigation in this paper is a generalized variable-coefficient fifth-order Korteweg-de Vries equation, which describes the interaction between a water wave and a floating ice cover or the gravity-capillary waves. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear forms, N -soliton solutions, Bäcklund transformation and Lax pair are derived. Infinitely-many conservation laws are obtained based on the Bell-polynomial-typed Bäcklund transformation. Soliton fusion and fission, and influence of the variable coefficients from the equation are analyzed: Both variable coefficients c ( t ) and n ( t ) are in direct proportion to the soliton velocities but have no effect on the amplitudes, while another constant coefficient α can affect the types of the interactions, in the sense of the elastic or inelastic. Elastic–inelastic interactions among the three solitons are presented as well. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the N -coupled generalized nonlinear Schrödinger ( N -CGNLS) equations with cubic-quintic nonlinearity, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in N fields of an optical fiber. Lax pair and Darboux transformation (DT) for such equations are obtained. Via the modified DT, the first- and second-order vector rogue waves are analytically presented. Through the variation of the coefficient ρ 1 , which is relevant to the quintic nonlinearity, time-retarded induced Raman process and nonlinear dispersion, rogue-wave rotation happens, but ρ 1 has no effect on the wave peaks and cores. Two different patterns of the second-order vector rogue waves for the 3-CGNLS equations are analyzed graphically: One pattern with one peak surrounded by four tails, the other pattern including three separated equal peaks. Finally, condition for the modulation instability is given. [ABSTRACT FROM AUTHOR]

In this article, the generalised nonlinear Schrödinger-Maxwell-Bloch system is investigated, which describes the propagation of the optical solitons in an optical fibre doped with two-level resonant impurities like erbium with the fourth-order dispersion taken into account. Bilinear forms are derived via the Hirota method, symbolic computation, and the auxiliary function. Bright solitons can be obtained for the complex envelope of the field and the measure of the polarisation for the resonant medium, while the dark ones have been deduced for the extant population inversion. Propagation of the one and two solitons is analysed with the results that the solitons keep their shapes unchanged after the interaction, except for the phase shifts, which means that the interaction is elastic. Velocities of the solitons decrease when the effect of discreteness and higher-order dispersion increases. For the bound-state solitons, which can be formed among the solitons at the same velocity, the period decreases when the effect of discreteness and higher-order dispersion increases. [ABSTRACT FROM AUTHOR]

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg-de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV-sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz-Kaup-Newell-Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is an inhomogeneous nonlinear Schrödinger equation, which describes the propagation of a large-wavelength small-amplitude electron plasma wave in a parabolic-distributed and constant-interactional-damping inhomogeneous plasma. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear form, Bäcklund transformation and -soliton solutions are obtained. Influence of the linear density coefficient and damping coefficient on the soliton envelopes is also discussed, i.e. can affect the soliton position, while is related to the soliton intensity, velocity and phase shift. Periodically attractive and repulsive interactions are shown. Asymptotic analysis shows that the interactions between/among the solitons are elastic. [ABSTRACT FROM AUTHOR]

The Korteweg-de Vries (KdV)-type equations can describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics, while the (2+1)-dimensional Nizhnik-Novikov-Vesselov equations are the isotropic extensions of KdV-type equations. In this paper, we investigate the (2+1)-dimensional modified Nizhnik-Novikov-Vesselov equations. By virtue of the binary Bell polynomials, bilinear forms, multi-soliton solutions and Bäcklund transformations are derived. Effects of some parameters on the solitons and monotonic function are graphically illustrated. We can observe the coalescence of the two solitons in their collision region, where their shapes change after the collision. [ABSTRACT FROM AUTHOR]

Studied in this paper is a ( $$3\,+ 1$$ )-dimensional nonlinear Schrödinger equation with the group velocity dispersion, fiber gain-or-loss and nonlinearity coefficient functions, which describes the evolution of a slowly varying wave packet envelope in the inhomogeneous optical fiber. With the Hirota method and symbolic computation, the bilinear form and dark multi-soliton solutions under certain variable-coefficient constraint are derived. Interactions between the different-type dark two solitons have been asymptotically analyzed and presented. Both velocities and amplitudes of the two linear-type dark solitons do not change before and after the interaction. The two parabolic-type dark solitons propagating with the opposite directions both change their directions after the interaction. Interaction between the two periodic-type dark solitons is also presented. Interactions between the linear-, parabolic- and periodic-type dark two solitons are elastic. [ABSTRACT FROM AUTHOR]

Abstract: Under investigation in this paper is a discrete integrable Ablowitz–Ladik equation, which has certain applications in the electrical and optical systems. Via the Hirota method and symbolic computation, the bilinear forms and -bright soliton solutions are obtained. Propagation and interaction behaviors of the discrete solitons are analyzed through the one- and two-soliton solutions. The discreteness effects on soliton and soliton interaction are discussed. Asymptotic analysis shows that the interactions between two solitons are elastic. Head-on and overtaking interactions can be obtained with the choices of the directions of the velocities. Two kinds of overtaking interactions are investigated analytically and graphically, respectively. [Copyright &y& Elsevier]

Under investigation in this paper are the coupled Gross-Pitaevskii equations, which describe the dynamics of two-component Bose-Einstein condensates. Infinitely many conservation laws are obtained based on the Lax pair. Via the Hirota method, Bell-polynomial approach and symbolic computation, bilinear forms, Bell-polynomial-typed transformation, and bilinear-typed Bäcklund transformation are also derived. One- and two-soliton-like solutions are expressed explicitly. The gain/loss coefficient G( t) can influence the velocity of the solitonic envelopes. Head-on and overtaking elastic interactions are shown and analyzed. Inelastic interactions between two soliton-like envelopes are presented as well. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients, which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a non-Kerr medium, or in the twin-core nonlinear optical fiber or waveguide. Under certain constraints on the variable coefficients in such equations, mixed-type (bright–dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation, and such vector-soliton solutions are only related to the delayed nonlinear response effect and nonlinearity. Through the graphic analysis, we find that the delayed nonlinear response effect and nonlinearity can both affect the vector-soliton amplitude, while the vector-soliton velocity merely depends on the delayed nonlinear response effect. With the choice on the variable coefficients representing the delayed nonlinear response effect and nonlinearity, interactions between the amplitude- and velocity-unchanging, amplitude-changing, velocity-changing and amplitude- and velocity-changing vector two solitons are obtained. We see that the interaction between the vector two solitons is elastic. We also find that the interaction period of the bound vector solitons decreases as the increase of the delayed nonlinear response effect or increases as the decrease of the delayed nonlinear response effect, but is independent of the nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract: For describing the dynamics of alpha helical proteins with internal molecular excitations, nonlinear couplings between lattice vibrations and molecular excitations, and spin excitations in one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole–dipole interactions, we consider an inhomogeneous generalized fourth-order nonlinear Schrödinger equation. Based on the Ablowitz–Kaup–Newell–Segur system, infinitely many conservation laws for the equation are derived. Through the auxiliary function, bilinear forms and N-soliton solutions for the equation are obtained. Interactions of solitons are discussed by means of the asymptotic analysis. Effects of linear inhomogeneity on the interactions of solitons are also investigated graphically and analytically. Since the inhomogeneous coefficient of the equation h=α x+β, the soliton takes on the parabolic profile during the evolution. Soliton velocity is related to the parameter α, distance scale coefficient and biquadratic exchange coefficient, but has no relation with the parameter β. Soliton amplitude and width are only related to α. Soliton position is related to β. [Copyright &y& Elsevier]