In this paper, the Toda equation and the discrete nonlinear Schrdinger equation with a saturable nonlinearity via the discrete ＂ （G′/G＂）-expansion method are researched. As a result, with the aid of the symbolic computation, new hyperbolic function solution and trigonometric function solution with parameters of the Toda equation are obtained. At the same time, new envelop hyperbolic function solution and envelop trigonometric function solution with parameters of the discrete nonlinear Schro¨dinger equation with a saturable nonlinearity are obtained. This method can be applied to other nonlinear differential-difference equations in mathematical physics.
We propose a new fractional two-dimensional triangle function combination discrete chaotic map(2D-TFCDM)with the discrete fractional difference.Moreover,the chaos behaviors of the proposed map are observed and the bifurcation diagrams,the largest Lyapunov exponent plot,and the phase portraits are derived,respectively.Finally,with the secret keys generated by Menezes-Vanstone elliptic curve cryptosystem,we apply the discrete fractional map into color image encryption.After that,the image encryption algorithm is analyzed in four aspects and the result indicates that the proposed algorithm is more superior than the other algorithms.
A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s/（4） has been presented （Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613）. Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl（4）. In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.
The symmetry constraint and binary nonlinearization of Lax pairs for the super classical-Boussinesq hierarchy is obtained. Under the obtained symmetry constraint, the n-th flow of the super classical-Boussinesq hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.
The coupled modified nonlinear Schrodinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann-Hilbert problem for the coupled modified nonlinear Schrodinger equations is formulated. And then, through solving the obtained Riemann-Hilbert problem under the conditions of irregularity and reflectionless case, N-soliton solutions for the equations are presented. Furthermore, the localized structures and dynamic behaviors of the one-soliton solution are shown graphically.
