In this paper, a generalized variable-coefficient fifth-order Korteweg–de Vries equation is investigated. Based on the Hirota bilinear method and symbolic computation, the N-soliton solutions, Bäcklund transformation and Lax pair are presented. Furthermore, the characteristic-line method is applied to discuss the solitonic propagation and collision under the effects of the variable coefficients, from which the following conclusions can be derived: (i) solitonic amplitude decreases as the positive coefficient of the line-damping term increases; (ii) coefficients of the dispersive and dissipative terms determine the solitonic direction and speed by changing the sign and absolute value of the solitonic velocity; (iii) the appearances of the characteristic lines depend on the forms of the variable coefficients.
Dynamic features describing the collisions of the bound vector solitons and soliton complexes are investigated for the coupled nonlinear Schrödinger (CNLS) equations, which model the propagation of the multimode soliton pulses under some physical situations in nonlinear fiber optics. Equations of such type have also been seen in water waves and plasmas. By the appropriate choices of the arbitrary parameters for the multisoliton solutions derived through the Hirota bilinear method, the periodic structures along the propagation are classified according to the relative relations of the real wave numbers. Furthermore, parameters are shown to control the intensity distributions and interaction patterns for the bound vector solitons and soliton complexes. Transformations of the soliton types (shape changing with intensity redistribution) during the collisions of those stationary structures with the regular one soliton are discussed, in which a class of inelastic properties is involved. Discussions could be expected to be helpful in interpreting such structures in the multimode nonlinear fiber optics and equally applied to other systems governed by the CNLS equations, e.g., the plasma physics and Bose-Einstein condensates.
Two types of amplification of solitons during the interactions for a nonautonomous nonlinear Schrödinger model with the time-and space-dependent dispersion, nonlinearity, and external potentials have been investigated with the similarity transformations and Hirota bilinear method. Type 1 refers to the generation of the large central amplitudes with small wings for two collisional solitons, and Type 2 refers to the generation of the amplified peaks for the bound solitons. Both amplifications in our study could be applied to generate the solitons with large maximum amplitudes in such systems as those with the soliton managed by the Feshbach resonance in Bose-Einstein condensates and with the dispersion-managed soliton in nonlinear optics.
Under investigation is a generalized variable-coefficient forced Korteweg-de Vries equation in fluids and other fields. From the bilinear form of such equation, the N-soliton solution and a type of analytic solution are constructed with symbolic computation. Analytic analysis indicates that: (1) dispersive and dissipative coefficients affect the solitonic velocity; (2) external-force term affects the solitonic velocity and background; (3) line-damping coefficient and some parameters affect the solitonic velocity, background, and amplitude. Solitonic propagation and interaction can be regarded as the combination of the effects of various variable coefficients. According to a constraint among the nonlinear, dispersive, and line-damping coefficients in this paper, the possible applications of our results in the real world are also discussed in three aspects, i.e., solution with the constraint, solution without the constraint, and approximate solution.
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