Abundant travelling wave solutions for nonlinear Kawahara partial differential equation using extended trial equation method

In this paper, we use the extended trial equation method (ETEM) to construct the exact solutions in many different functions such as the trigonometric function, elliptic integral function, rational function, hyperbolic function and Jacobi elliptic functions for nonlinear evolution equation in mathematical physics via the fifth-order modified nonlinear Kawahara equation. In this method, the balance number is not constant as we have shown in other methods, but it is changed by changing the trial equation derivative definition. This method is powerful, reliable, effective and simple for solving more complicated nonlinear partial differential equations in mathematical physics.     

一类非线性波方程尖波解及其动力学性质的分析

应用动力系统分岔理论和定性理论研究了一类非线性DegasperisProcesi方程的行波解及其动力学性质,并结合可积系统的特点,利用哈密尔顿系统的能量特征,通过Maple软件绘出其相轨图,再根据行波与相轨道间的对应关系,揭示了不同类型的行波解间的转变与参数变化的关系,并且给出了不同行波间相互转换的参数分岔值,从根本上解释了Peakon产生的原因.数值模拟验证了该方法的正确性.最后给出了相应行波解的表达式.     

一类捕食-食饵系统的新的行波解

利用改进的[(G/G)-]展开法,借助软件的符号计算功能,求出了一类用来描述捕食-食饵群落时空动力性且食饵的平均生长率具有Allee效应的非线性偏微分方程组的新的行波解,这些解的性质合理地反映了生物入侵问题与参数值之间的相互依赖关系。     

Rational solutions and their interaction solutions of the (2+1)-dimensional modified dispersive water wave equation

A bilinear form for the modified dispersive water wave (mDWW) equation is presented by the truncated Painlevé series, which does not lead to lump solutions. In order to get lump solutions, a pair of quartic–linear forms for the mDWW equation is constructed by selecting a suitable seed solution of the mDWW equation in the truncated Painlevé series. Rational solutions are then computed by searching for positive quadratic function solutions. A regular nonsingular rational solution can describe a lump in this model. By combining quadratic functions with exponential functions, some novel interaction solutions are founded, including interaction solutions between a lump and a one-kink soliton, a bi-lump and a one-stripe soliton, and a bi-lump and a two-stripe soliton. Concrete lumps and their interaction solutions are illustrated by 3d-plots and contour plots.     

非对称NNV方程新的广义孤波解和周期解

指数函数法是求解非线性发展方程的一种简单有效的方法,利用该方法并借助Mathematica软件的符号计算功能求解了非对称NNV方程,得到了新的孤波解和周期解。     

Exact periodic,cusp, solitary and loop wave solutions of the EX-ROE

The extended reduced Ostrovsky equation (EX-ROE) is investigated by using the dynamical system theory. The bifurcation phase portraits are drawn in different regions of parameter plane. The bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakon, solitary loop waves and periodic loop waves are obtained. The dynamic characters of these solutions are investigated.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

New exact travelling wave solutions for some famous nonlinear partial differential equations using the improved tanh-function method

The improved tanh-function method is a powerful tool for obtaining exact travelling wave solutions. The method is used for constructing exact travelling wave solutions and new kinds of solutions for the modified dispersive water wave equation, the Abrahams–Tsuneto reaction diffusion system, and for a class of reaction diffusion models. The solutions obtained are different from those reported in the literature.     

Solitary wave solutions for a higher order nonlinear Schrödinger equation

We consider a higher order nonlinear Schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. The derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. These exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive Schrödinger system.     

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.     

Constant-sign periodic and almost periodic solutions of a system of difference equations

We consider the following system of difference equations, ,where I is a subset of . Our aim is to establish criteria such that the above system has a constantsign periodic and almost periodic solution (u₁, u₂, …, u_n). The above problem is also extended to that on , .     

A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffing's equation

Based on the idea of the previous Obrechkoff's two-step method, a new kind of four-step numerical method with free parameters is developed for the second order initial-value problems with oscillation solutions. By using high-order derivatives and apropos first-order derivative formula, the new method has greatly improved the accuracy of the numerical solution. Although this is a multistep method, it still has a remarkably wide interval of periodicity, . The numerical test to the well known problem, the nonlinear undamped Duffing's equation forced by a harmonic function, shows that the new method gives the solution with four to five orders higher than those by the previous Obrechkoff's two-step method. The ultimate accuracy of the new method can reach about 5×¹⁰⁻¹³, which is much better than the one the previous method could. Furthermore, the new method shows the great superiority in efficiency due to a reasonable arrangement of the structure. To finish the same computational task, the new method can take only about 20% CPU time consumed by the previous method. By using the new method, one can find a better ‘exact’ solution to this problem, reducing the error tolerance of the one widely used method (¹⁰⁻¹¹), to below 10⁻¹⁴.     

广义Fitzhugh Nagumo方程的无穷序列新解

利用辅助方程的几种结论,构造了广义Fitzhugh-Nagumo方程的多种无穷序列新解.步骤一,利用函数变换与首次积分,给出了辅助方程的新解、Bcklund变换和解的非线性叠加公式.步骤二,通过函数变换,将广义Fitzhugh-Nagumo方程的求解问题转化为非线性常微分方程的求解问题.步骤三,利用符号计算系统Mathematica与辅助方程的几种结论,构造了广义Fitzhugh-Nagumo方程的多种无穷序列新解.     

Exact travelling wave solutions for a diffusion-convection equation in two and three spatial dimensions

The modified extended tanh-function method were applied to the general class of nonlinear diffusion-convection equations where the concentration-dependent diffusivity, D(u), was taken to be a constant while the concentration-dependent hydraulic conductivity, K(u) were taken to be in a power law. The obtained solutions include rational-type, triangular-type, singular-type, and solitary wave solutions. In fact, the profile of the obtained solitary wave solutions resemble the characteristics of a shock-wave like structure for an arbitrary m (where m>1 is the power of the nonlinear convection term).     

A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation

This paper is concerned with semi-global stabilization of discrete-time linear periodic systems subject to actuator saturation. Provided that the open loop characteristic multipliers are within the closed unit circle, a low gain feedback design approach is proposed to solve the problem by state feedback. Our approach is based on the solution to a parametric discrete-time periodic Lyapunov equation. The proposed approaches not only generalize the corresponding results for time-invariant systems to periodic systems, but also reveal some important intrinsic properties of this class of periodic matrix equations. A numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

New solitary wave solutions for generalized regularized long-wave equation

In this paper, the sine–cosine and the tan h methods have been used to obtain solutions of the generalized regularized long-wave equation. New solitary and periodic solutions are formally derived. The change of parameters, that will drastically change the characteristics of the equation, is examined.     

Boundary stabilisation of the wave equation in the presence of singularities

We study the boundary stabilisation of the wave equation by a nonlinear feedback active on a part of the boundary in geometric situations for which the solutions have singularities. These singularities appear at the interfaces at which the mixed Neumann–Dirichlet boundary conditions meet. Under a simple geometrical condition concerning the orientation of the boundary, we obtain sharp energy decay rates under a general growth assumption on the feedback. We show that the singularities do not affect the energy decay rates and give examples.     

Periodic and solitary wave solutions of the attractive and repulsive nonlinear Schrödinger equation

By means of symbolic computation, the first integral method is presented to obtain novel exact solutions of the nonlinear evolution equation. The obtained results include periodic and solitary wave solutions. The power of this manageable method is confirmed and the availability of symbolic computation is demonstrated.     

基于低频周期参数扰动统一混沌系统的分岔分析

本文以低频周期参数扰动下的统一混沌系统为研究对象,应用动力学基础知识,讨论了系统的平衡点的分布及其稳定性,得到了周期扰动系统的静态分岔和Hopf分岔的条件。根据Melnikov方法,计算得到了系统的同宿轨道,以及得到了系统发生同宿轨道分岔的条件。为了验证理论研究结果的正确性,本文采用数值模拟的方法进行了验证,结果表明,理论研究结果正确。本文的研究结果可以看作是对周期激励的Lorenz类系统和Chen类系统的总结,可以有助于混沌系统在计算机应用领域的推广和应用。     

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

We investigate the periodic nature of the positive solutions of the fuzzy difference equation , where k, m are positive integers, A₀, A₁, are positive fuzzy numbers and the initial values x_i, i = −d, −d + 1, … , −1, d = max{k, m}, are positive fuzzy numbers. In addition, we give conditions so that the solutions of this equation are unbounded.