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A generalized auxiliary equation method is proposed to construct more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including soliton-like solutions, triangular-like solitions, single and combined non-degenerate Jacobi elliptic doubly-like periodic solutions, and Weierstrass elliptic doubly-like periodic solutions.

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.

In this paper, we use the Fokas method to analyze the complex Sharma-Tasso-Olver(cSTO) equationAssuming that the solution u(x, y) of the cSTO equation is exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem(RHP) formulated in the complex λ −plane (the Fourier plane), which has a jump matrix with explicit (x, y)−dependence involving four scalar functions of λ , called spectral functions. The spectral functions {a(λ ), b(λ )} and {A(λ ), B(λ )} are obtained from the initial data u 0 (x) = u(x, 0) and the boundary data g 0 (y) = u(0, y), g 1 (y) = u x (0, y), g 2 (y) = u xx (0, y), respectively. The problem has the jump across {Imλ 6 = 0}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.Given initial and boundary values u 0 (x), g 0 (y), g 1 (y) and g 2 (y) such that there exist spectral functions satisfying the global relation, we show that the function u(x, y) defined by the above Riemann-Hilbert problem exists globally and solves the cSTO equation with the prescribed initial and boundary values. Key word: Riemann-Hilbert problem; the cSTO equation; initial-boundary value problem; jump matrix 2010 MR Subject Classification 35G31; 35Q15 IntroductionA unified method for analyzing boundary value problems with decaying initial data, extending ideas of the so-called inverse scattering transform(IST) method, was discovered in 1967 [1]. This method can be thought of as a nonlinear Fourier transform(FT) method. However, this nonlinear FT is not the same for every nonlinear evolution equation, but it is constructed from the x part of the Lax pair. Furthermore, neither the direct nonlinear FT of the initial data, nor the inverse nonlinear FT can be expressed in closed form: the former involves a linear Volterra integral equation and the latter involves a matrix Riemann-Hilbert problem(RHP). The y part is used only to determine the evolution of the direct nonlinear FT [2,3]. However, in many laboratory and field situations, the wave motion is initiated by what corresponds to the imposition of boundary conditions rather than initial conditions. This naturally leads to the formulation of an initial-boundary value (IBV) problem instead of a pure initial value problem.In 1997, a new unified approach based on the Riemann-Hilbert factorization problem to solve IVB problems for linear and nonlinear integrable PDEs was presented by Fokas [4], we call that Fokas method. The Fokas

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The unified transform method is used to analyze the initial-boundary value problem for the coupled derivative nonlinear Schrödinger(CDNLS) equations on the half-line. In this paper, we assume that the solution u(x, t) and v(x, t) of CDNLS equations are exists, and we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ.