Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

Effects of P+Cr complex modification and solidification conditions on microstmcture of hypereutectic Al-Si alloys by wedge-shaped copper mould casting

Large and segregated primary Si particles may drastically decrease the mechanical properties of AI-Si alloys. To solve this problem, a P-Cr complex modifier was added into the alloy, and the effects of P-Cr complex modification and solidification conditions on the microstructure of hypereutectic Al-Si alloys casting produced in wedge-shaped copper mould were studied. The thermal analysis technique was applied to calculate the cooling rate during solidification. The microstructures were observed by means of optical and scanning electron microscopies. Results showed that the primary Si segregates in the as-cast hypereutectic AI-Si alloys. The segregation of primary Si can be inhibited by adding a P＋Cr complex modifier and increasing the cooling rate during solidification. The refinement of primary Si particles by P＋Cr complex modification is due to the formation of CrSi2 and AlP particles which act as the heterogeneous nuclei for the primary Si phase. The segregation of Si was also inhibited through the adherence of heavier CrSi2 particles to the primary Si particles.

ELECTROCHEMISTRY OF OI-μ-OXO-DIMANGANESE COMPLEX AT THIOURACIL MODIFIED GOLD ELECTRODE

The electrochemistry of di-μ-oxo-dimanganese complex was investigated. It was found that no redox peak was observed in the cyclic voltammogram (CV) of the complex at the bare gold electrode, but at thiouracil-modified gold electrode, a pair of redox peaks were observed showing that thiouracil can promote the proton-coupled electron transfer reaction of the complex.

Modified projective synchronization with complex scaling factors of uncertain real chaos and complex chaos

To increase the variety and security of communication, we present the definitions of modified projective synchronization with complex scaling factors （CMPS） of real chaotic systems and complex chaotic systems, where complex scaling factors establish a link between real chaos and complex chaos. Considering all situations of unknown parameters and pseudo-gradient condition, we design adaptive CMPS schemes based on the speed-gradient method for the real drive chaotic system and complex response chaotic system and for the complex drive chaotic system and the real response chaotic system, respectively. The convergence factors and dynamical control strength are added to regulate the convergence speed and increase robustness. Numerical simulations verify the feasibility and effectiveness of the presented schemes.

■-dressing method for the complex modified KdV equation

The dressing method based on the 2×2 matrix■-problem is generalized to study the complex modified KdV equation(cmKdV).Through two linear constraint equations,the spatial and time spectral problems related to the cmKdV equation are derived.The gauge equivalence between the cmKdV equation and the Heisenberg chain equation is obtained.Using a recursive operator,a hierarchy of cmKdV with source is proposed.On the basis of the■-equation,the N-solition solutions of the cmKdV equation are obtained by selecting the appropriate spectral transformation matrix.Furthermore,we get explicit one-soliton and two-soliton solutions.

Complex Modified Projective Synchronization for Fractional-order Chaotic Complex Systems

The aim of this paper is to study complex modified projective synchronization（CMPS） between fractional-order chaotic nonlinear systems with incommensurate orders. Based on the stability theory of incommensurate fractional-order systems and active control method, control laws are derived to achieve CMPS in three situations including fractional-order complex Lorenz system driving fractional-order complex Chen system, fractional-order real Rssler system driving fractional-order complex Chen system, and fractionalorder complex Lorenz system driving fractional-order real Lü system. Numerical simulations confirm the validity and feasibility of the analytical method.

Numerical simulation of the soliton solutions for a complex modified Korteweg—de Vries equation by a finite difference method

In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with the vanishing boundary condition.It is proved that such a numerical scheme has the second order accuracy both in space and time,and conserves the mass in the discrete level.Meanwhile,the resuling scheme is shown to be unconditionally stable via the von Nuemann analysis.In addition,an iterative method and the Thomas algorithm are used together to enhance the computational efficiency.In numerical experiments,this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation.The numerical accuracy,mass conservation and linear stability are tested to assess the scheme's performance.

Abundant general solitary wave solutions to the family of KdV type equations

This work explores the construction of more general exact traveling wave solutions of some nonlinear evolution equations(NLEEs)through the application of the(G/G,1/G)-expansion method.This method is allied to the widely used(G/G)-method initiated by Wang et al.and can be considered as an extension of the(G/G)-expansion method.For effectiveness,the method is applied to the family of KdV type equations.Abundant general form solitary wave solutions as well as periodic solutions are successfully obtained through this method.Moreover,in the obtained wider set of solutions,if we set special values of the parameters,some previously known solutions are revived.The approach of this method is simple and elegantly standard.Having been computerized it is also powerful,reliable and effective.