Hai-Qiang Zhang

Optical soliton solutions for two coupled nonlinear Schrödinger systems via Darboux transformation

In nonlinear optical fibers, the vector solitons can be governed by the systems of coupled nonlinear Schrodinger 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 Schrodinger 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.

Integrability aspects with optical solitons of a generalized variable-coefficient N-coupled higher order nonlinear Schrödinger system from inhomogeneous optical fibers

For describing the long-distance communication and manufacturing problems of N fields propagation in inhomogeneous optical fibers, we consider a generalized variable-coefficient N-coupled nonlinear Schrodinger system with higher order effects such as the third-order dispersion, self-steepening and self-frequency shift. Using the Painleve singularity structure analysis, we obtain two cases for this system to admit the Painleve property. Then for case (1) we derive the optical dark solitons via solving the Hirota bilinear equations; and based on the obtained (2N+1)×(2N+1) Lax pair, we construct the Darboux transformation to obtain the optical bright solitons (including the multisoliton profiles) for case (2). Finally, the features of optical solitons (both dark and bright ones) in inhomogeneous optical fibers are analyzed and graphically discussed.

Lax pair and Darboux transformation for multi-component modified Korteweg–de Vries equations

In this paper, through generalizing the 2 × 2 matrix Ablowitz–Kaup–Newell–Segur linear eigenvalue problem to the 2N × 2N case, a new Lax pair associated with the multi-component modified Korteweg–de Vries equations is derived in the form of block matrices. Furthermore, the Darboux transformation is applied to this integrable multi-component system, and the n-times iterative potential formula is presented by applying the Darboux transformation successively. This formula enables us to construct a series of explicit solutions of multi-component modified Korteweg–de Vries equations. In illustration, starting from the zero background, we construct the multi-soliton solutions by performing the symbolic computation.

Interactions of bright solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations from optical fibres with symbolic computation

In nonlinear optical fibres, the evolution of two polarization envelopes is governed by a system of coupled nonlinear Schrodinger (CNLS) equations. In this paper, with the aid of symbolic computation, the analytical bright one- and two-soliton solutions of the (2+1)-dimensional CNLS equations under certain constraints are presented by employing the Hirota method. We have discussed the head-on and overtaking interactions which include elastic and inelastic collisions between two parallel bright solitons. In the interaction process, the intensities of solitons can exhibit various redistributions. We also point out that these properties have important physical applications in constructing various logic gates and nonlinear optical fibers.

Soliton-like solutions of a generalized variable-coefficient higher order nonlinear Schrödinger equation from inhomogeneous optical fibers with symbolic computation

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 Schrodinger 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 Painleve 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.

Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system with symbolic computation

In an inhomogeneous nonlinear light guide doped with two-level resonant atoms, the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system can be used to describe the propagation of optical solitons. In this paper, the Lax pair and conservation laws of that model are derived via symbolic computation. Furthermore, based on the Lax pair obtained, the Darboux transformation is constructed and soliton solutions are presented. Figures are plotted to reveal the following dynamic features of the solitons: (1) Periodic mutual attractions and repulsions of four types of bound solitons: of two one-peak bright solitons; of two one-peak dark solitons; of two two-peak bright solitons and of two two-peak dark solitons; (2) Two types of elastic interactions of solitons: of two bright solitons and of two dark solitons; (3) Two types of parallel propagations of parabolic solitons: of two bright solitons and of two dark solitons. Those results might be useful in the study of optical solitons in some inhomogeneous nonlinear light guides.

Darboux transformation and new solutions for the Whitham–Broer–Kaup equations

In this paper, we derive the Lax pair associated with the Whitham–Broer–Kaup equations. Based on the Lax pair obtained, we construct the Darboux transformation with multi-parameters and obtain the one- and multi-soliton solutions. In addition, we qualitatively analyze various types of interaction behavior between two solitary waves along with graphical demonstration, in order to provide valuable information on the shallow water motion.

Solitonic excitations and interactions in an α-helical protein modeled by three coupled nonlinear Schrödinger equations with variable coefficients

Governing the molecular excitations associated with bioenergy transport in an α-helical protein through the soliton modes, three-coupled higher-order nonlinear Schrodinger equations with variable coefficients are investigated via symbolic computation. Using Bell polynomials, a bilinear form and Backlund transformation are derived for this model. Furthermore, explicit N-soliton solutions are constructed in terms of the double Wronskian determinant. Solitonic excitations are found to remain stable against the disorders of the parameters under certain constraints. Finally, propagation characteristics and interactions of the solitonic excitations are discussed. Soliton amplitudes are related to the energy modulation coefficients, and both the soliton properties and energy distribution during the interactions are affected by the inhomogeneity coefficients of the protein with time-dependent disorders.

Integrability study on a generalized (2+1)-dimensional variable-coefficient Gardner model with symbolic computation

Gardner model describes certain nonlinear elastic structures, ion-acoustic waves in plasmas, and shear flows in ocean and atmosphere. In this paper, by virtue of the computerized symbolic computation, the integrability of a generalized (2+1)-dimensional variable-coefficient Gardner model is investigated. Painlevé integrability conditions are derived among the coefficient functions, which reduce all the coefficient functions to be proportional only to γ(t), the coefficient of the cubic nonlinear term u(2)u(x). Then, an independent transformation of the variable t transforms the reduced γ(t)-dependent equation into a constant-coefficient integrable one. Painlevé test shows that this is the only case when our original generalized (2+1)-dimensional variable-coefficient Gardner model is integrable.

Symbolic computation on the multi-soliton-like solutions of the cylindrical Kadomtsev–Petviashvili equation from dusty plasmas

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.