In nonlinear optical fibers, the vector solitons can be governed by the systems of coupled nonlinear Schrödinger from polarized optical waves in an isotropic medium. Based on the Ablowitz–Kaup–Newell–Segur technology, the Darboux transformation method is successfully applied to two coupled nonlinear Schrödinger systems. With the help of symbolic computation, the bright vector one- and two-soliton solutions including one-peak and two-peak solitons are further constructed via the iterative algorithm of Darboux transformation. Through the figures for several sample solutions, the stable propagation and elastic collisions for these kinds of bright vector solitons are discussed and the possible applications are pointed out in optical communications and relevant optical experiments.In addition, the conserved quantities of such two systems, i.e., the energy, momentum and Hamiltonian, are also presented.

To provide an analytical scheme for the dynamical behavior of nonlinear Alfvén waves in inhomogeneous plasmas, this paper investigates a generalized variable-coefficient derivative nonlinear Schrödinger equation. In the sense of admitting the Lax pair and infinitely many conservation laws, the integrability of this equation is established under certain coefficient constraint which suggests which inhomogeneities support stable Alfvén solitons. The Hirota method is adopted to construct the one- and multi-Alfvén-soliton solutions. The inhomogeneous soliton features are also discussed through analyzing some important physical quantities. A sample model is treated with our results, and graphical illustration presents two energy-radiating Alfvén soliton structures.

By using the Darboux transformation, we obtain two new types of exponential-and-rational mixed soliton solutions for the defocusing nonlocal nonlinear Schrödinger equation. We reveal that the first type of solution can display a large variety of interactions among two exponential solitons and two rational solitons, in which the standard elastic interaction properties are preserved and each soliton could be either the dark or antidark type. By developing the asymptotic analysis method, we also find that the second type of solution can exhibit the elastic interactions among four mixed asymptotic solitons. But in sharp contrast to the common solitons, the mixed asymptotic solitons have the t-dependent velocities and their phase shifts before and after interaction also grow with |t| in the logarithmical manner. In addition, we discuss the degenerate cases for such two types of mixed soliton solutions when the four-soliton interaction reduces to a three-soliton or two-soliton interaction. √ 15 7 i. (b) MD-MAD-V-V interaction with ρ = 1, b = 1 4 , φ = 0, s 1 = 0, s 2 = 3 25 i and γ 1 = 1 − √ 15 7 i. (c) V-V-MAD-MD interaction with ρ = 1, b = 1 4 , φ = 0, s 1 = 0, s 2 = 0 and γ 1 = 1. (d) V-V-MD-MAD interaction with ρ = 1, b = 1 4, φ = 0, s 1 = 0, s 2 = 3 25 i and γ 1 = −1. Here, "V" represents the vanishment of an asymptotic soliton as |t| → ∞.

In this paper, we construct the general Darboux transformation on the Sasa-Satsuma equation and represent the iterated solutions in terms of the three-component Wronskian. From the onceiterated solution, we derive the breather as well as the single-and double-hump solitons. We also analyze three types of collisions: soliton-soliton, breather-breather and soliton-breather collisions. The surprising result is that the soliton-breather collision may exhibit the shapechanging phenomena, that is, one breather (or soliton) may change into a soliton (or breather) when interacting with another breather. Such novel collision phenomena may be applied in alloptical information processing, optical switching and routing of optical signals.

For the long-distance communication and manufacturing problems in optical fibers, the propagation of subpicosecond or femtosecond optical pulses can be governed by the variable-coefficient nonlinear Schrödinger equation with higher order effects, such as the third-order dispersion, self-steepening and self-frequency shift. In this paper, we firstly determine the general conditions for this equation to be integrable by employing the Painlevé analysis. Based on the obtained 3 × 3 Lax pair, we construct the Darboux transformation for such a model under the corresponding constraints, and then derive the nth-iterated potential transformation formula by the iterative process of Darboux transformation. Through the one- and two-soliton-like solutions, we graphically discuss the features of femtosecond solitons in inhomogeneous optical fibers.

In this letter, via the Darboux transformation method we construct new analytic soliton solutions for the Sasa-Satsuma equation which describes the femtosecond pulses propagation in a monomode fiber. We reveal that two different types of femtosecond solitons, i.e., the anti-dark (AD) and Mexican-hat (MH) solitons, can form on a continuous wave (CW) background, and numerically study their stability under small initial perturbations. Different from the common bright and dark solitons, the AD and MH solitons can exhibit both the resonant and elastic interactions, as well as various partially/completely inelastic interactions which are composed of such two fundamental interactions. In addition, we find that the energy exchange between some interacting soliton and the CW background may lead to one AD soliton changing into an MH one, or one MH soliton into an AD one.

In this paper, a generalized variable-coefficient Gardner equation arising in nonlinear lattice, plasma physics and ocean dynamics is investigated. With symbolic computation, the Lax pair and Bäcklund transformation are explicitly obtained when the coefficient functions obey the Painlevé-integrable conditions. Meanwhile, under the constraint conditions, two transformations from such an equation either to the constant-coefficient Gardner or modified Korteweg-de Vries (mKdV) equation are proposed. Via the two transformations, the investigations on the variable-coefficient Gardner equation can be based on the constant-coefficient ones. The N-soliton-like solution is presented and discussed through the figures for some sample solutions. It is shown in the discussions that the variable-coefficient Gardner equation possesses the right-and left-travelling soliton-like waves, which involve abundant temporally-inhomogeneous features.

Considering the transverse perturbation and axially non-planar geometry, the cylindrical Kadomtsev–Petviashvili (KP) equation is investigated in this paper, which can describe the propagation of dust-acoustic waves in the dusty plasma with two-temperature ions. Through imposing the decomposition method, such a (2+1)-dimensional equation is decomposed into two variable-coefficient (1+1)-dimensional integrable equations of the same hierarchy. Furthermore, three kinds of Darboux transformations (DTs) for these two (1+1)-dimensional equations are constructed. Via the three DTs obtained, the multi-soliton-like solutions of the cylindrical KP equation are explicitly presented. Especially, the one- and two-parabola-soliton solutions are discussed by several figures and some effects resulting from the physical parameters in the dusty plasma and transverse perturbation are also shown.