Weiguo Zhang

Survival and stationary distribution of a SIR epidemic model with stochastic perturbations

Abstract In this paper, the dynamics of a SIR epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, when R 0 > 1 , we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. When R 0 ⩽ 1 , a result on fluctuation of the solution around the disease-free equilibrium of deterministic system is established under suitable conditions. Finally, numerical simulations are carried out to illustrate the theoretical results.

Generalized Darboux transformation and rogue wave solutions for the higher-order dispersive nonlinear Schrödinger equation

In this paper, rogue wave solutions of the higher-order dispersive nonlinear Schrodinger equation are investigated, which describe the propagation of ultrashort optical pulse in optical fibers. The Nth-order rogue wave solutions with 2N + 1 free complex parameters are constructed via the generalized Darboux transformation method. As applications, rogue waves from the first to the fifth order are calculated according to different combinations of parameters. In particular, rogue waves dynamics and several new spatial–temporal structures are also discussed and exhibited to make a comparison with those of the nonlinear Schrodinger equation.

New explicit solutions for the Klein–Gordon equation with cubic nonlinearity

Abstract In this paper, we investigate Klein–Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.

Generalized Darboux transformation and rational soliton solutions for Chen–Lee–Liu equation

In this paper, the Chen–Lee–Liu (C–L–L) equation is investigated by the Darboux transformation (DT) method. A specific construction of the N-fold DT for C–L–L equation is derived in a simple way. The form of the N-fold DT is a matrix polynomial and each element of the matrix can be expressed by a ratio of two determinants. Furthermore, by choosing suitable eigenvalues and eigenfunctions under the reduction conditions, we can obtain the determinant solution of the reduced C–L–L equation. Moreover, the generalized DT (gDT) for C–L–L equation is also constructed through the limiting technique. As applications of the gDT method, the first and the second-order rational solitons with vanished background (VBC) and non-vanished background (NVBC) from different seeds are calculated for C–L–L equation. We hope our results can be realized by experiments in plasma physics and optical fibers.

Qualitative analysis to traveling wave solutions of Zakharov–Kuznetsov–Burgers equation and its damped oscillatory solutions

Abstract The theory of planar dynamical systems is applied in this paper to carry out a qualitative analysis to the planar dynamical system corresponding to the bounded traveling wave solution of the Zakharov–Kuznetsov–Burgers equation, and obtain the existence and uniqueness of the bounded traveling wave solutions. According to the discussions on the relationships between the shapes of bounded traveling wave solutions and the dissipation coefficient d, a critical value d 0 is found for arbitrary traveling wave speed v and integral constant g. This equation has a unique monotone kink profile solitary wave solution as the dissipation coefficient d satisfies | d ¯ | > d 0 ; while it has a unique damped oscillatory solution as | d ¯ | d 0 . This paper also presents the exact bell profile solitary wave solution as d = 0 . Furthermore, we appropriately design the structure of the damped oscillatory solution in light of the evolution relationships of the solution orbit in the global phase portraits to which the damped oscillatory solution corresponds and obtain its approximate solution by means of the undetermined coefficients method. Finally, based on the integral equation that reflects the relationships between the approximate damped oscillatory solution and its exact solution, the error estimate is given for the approximate damped oscillatory solution. The error is infinitesimal decreasing in exponential form.

Survival and Stationary Distribution in a Stochastic SIS Model

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: when , we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; when , we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.

Strong global convergence of an adaptive nonmonotone memory gradient method

In this paper, we develop an adaptive nonmonotone memory gradient method for unconstrained optimization. The novelty of this method is that the stepsize can be adjusted according to the characteristics of the objective function. We show the strong global convergence of the proposed method without requiring Lipschitz continuous of the gradient. Our numerical experiments indicate the method is very encouraging.

The elastic-fusion-coupled interaction for the Boussinesq equation and new soliton solutions of the KP equation

In this paper, by using of the classical Darboux transformation method, we obtain explicit solutions of the Boussinesq equation and the Kadomtsev-Petviashvili (KP) equation. Especially, for the Boussinesq equation we get explicit solutions describing the elastic-fusion-coupled interaction, which is a new phenomenon for the Boussinesq equation. For the KP equation, we obtain new soliton solutions.

Orbital Stability of Solitary Waves for Generalized Symmetric Regularized-Long-Wave Equations with Two Nonlinear Terms

This paper investigates the orbital stability of solitary waves for the generalized symmetric regularized-long-wave equations with two nonlinear terms and analyzes the influence of the interaction between two nonlinear terms on the orbital stability. Since is not onto, Grillakis-Shatah-Strauss theory cannot be applied on the system directly. We overcome this difficulty and obtain the general conclusion on orbital stability of solitary waves in this paper. Then, according to two exact solitary waves of the equations, we deduce the explicit expression of discrimination and give several sufficient conditions which can be used to judge the orbital stability and instability for the two solitary waves. Furthermore, we analyze the influence of the interaction between two nonlinear terms of the equations on the wave speed interval which makes the solitary waves stable.

Qualitative analysis and solutions of bounded travelling waves for the fluidized-bed modelling equation

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients e and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥e + δυ∥ λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.