Doubly periodic solutions of the modified Kawahara equation

Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form. Our approach is to introduce a new auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly periodic solutions of the modified Kawahara equation. When the module m → 1, these solutions degenerate to the exact solitary wave solutions of the equation. Then we reveal the relation of some exact solutions for the modified Kawahara equation obtained by other authors.     

Periodic wave solutions and asymptotic analysis of the Hirota–Satsuma shallow water wave equation

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one‐periodic and two‐periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.     

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation x?x−1, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.     

A Numerical Method for Computing Time‐Periodic Solutions in Dissipative Wave Systems

A numerical method is proposed for computing time‐periodic and relative time‐periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi‐Rayleigh quotients, so that the resulting integrodifferential equation is for the time‐periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton‐conjugate‐gradient iterations. The proposed method applies to both stable and unstable time‐periodic solutions; its numerical accuracy is spectral; it is fast‐converging; its memory use is minimal; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto–Sivashinsky equation and the cubic‐quintic Ginzburg–Landau equation, whose time‐periodic or relative time‐periodic solutions with spatially periodic or spatially localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in the Appendices to illustrate the simple implementation of the proposed method.     

The existence of periodic solutions of higher order nonlinear periodic difference equations

Sufficient conditions for the existence of at least one periodic solution of two classes of nonlinear higher order periodic difference equations are established, respectively. The results show us that sufficient conditions for the existence of T ? periodic solutions of difference equation are different from those ones for the existence of T ? periodic solutions of differential equation. Copyright © 2012 John Wiley & Sons, Ltd.     

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

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.     

On Doubly Periodic Standing Wave Solutions of the Coupled Higgs Field Equation

In this paper, we derive a class of doubly periodic standing wave solutions for a coupled Higgs field equation by employing the Hirota bilinear method and theta function identities. Such solutions are expressed in terms of theta functions with variable separation form. Moreover, it is shown that these solutions can be converted into Jacobi elliptic function representations, and their long‐wave limit yields collision of dark solitons. In comparing with known solutions of the canonical evolution equation, three new aspects will be developed in this paper. First, the periods in the spatial and temporal directions, measured in terms of the theta function parameters τ and τ₁, are independent of each other, quite unlike most similar solutions found earlier in the literature. Second, the doubly periodic wave solutions possess two families of the local maxima, whose locations and types are then examined in detail. Third, we obtain new doubly periodic standing wave solutions for the Davey–Stewartson equation through its similarity transformation to the coupled Higgs field equation.     

On Periodic Solutions of the Periodic

By treating the periodic Riccati equation ${\rm\dot{z}=a(t)z^2+b(t)z+c(t)}$ as a dynamical system on the sphere S, the number and stability of its periodic solutions are determined. Using properties of Moebius transformations, an exact algebraic relation is obtained between any periodic solution and any complex-valued periodic solution. This leads to a new method for constructing the periodic solutions.     

New periodic wave and soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation

The periodic wave solutions for Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

Dynamic behaviors of the periodic predator–prey system with distributed time delays and impulsive effect

In this paper, a periodic predator–prey system with distributed time delays and impulsive effect is investigated. By using the Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We improve some results in Guo and Chen (2009) [1].     

ON A PERIODIC VOLTERRA EQUATION WITH SEVERAL FINITE DELAYS AND AN INFINITE DELAY

The existence of positive periodic solutions for a periodic Volterra equation with several finite delays and an infinite delay is established. Sufficient conditions for the periodic solutions having global attractivity are obtained.     

On Weierstrass elliptic function solutions for a (N+1) dimensional potential KdV equation

This paper is concerned with several aspects of travelling wave solutions for a (N+1) dimensional potential KdV equation. The Weierstrass elliptic function solutions, the Jaccobi elliptic function solutions, solitary wave solutions, periodic wave solutions to the equation are acquired under certain circumstances. It is shown that the coefficients of derivative terms in the equation cause the qualitative changes of physical structures of the solutions.     

On the Relationship between Properties of Solutions of Difference Equations and the Corresponding Differential Equations

We establish conditions under which the existence of a periodic solution of a differential equation is preserved if a solution of the corresponding difference equation possesses the same property. We prove the convergence of periodic solutions of a system of difference equations to a periodic solution of a system of differential equations. Analogous problems are considered for bounded solutions. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 989–996, July, 2005.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

On the periodic solutions of certain class of seventh-order differential equations

The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).     

Limit periodic motions in some systems with aftereffect under resonance condition

Volterra-type integrodifferential equations and their solutions are considered which, when the time increases without limit, exponentially tend to periodic modes. In the critical case of stability, when the characteristic equation has a pair of pure imaginary roots and the remaining roots have negative real parts, the problem of the existence of limit periodic solutions with resonance, caused by coincidence between the periodic part of the limit external periodic perturbation and the natural frequency of the linearized system, is solved. It is shown that, if the right-hand side of the equation is an analytic function and the existence of limit periodic solutions is determined by terms of the (2m + 1)-th order, these solutions are represented by power series in the arbitrary initial values of the non-critical variables and the parameter μ^1/(2m+1), where μ is a small parameter, characterizing the magnitude of the maximum external periodic perturbation. The amplitude equations are presented.     

On a theorem of Favard

We obtain sufficient conditions for the existence of almost periodic solutions of almost periodic linear differential equations thereby extending Favard's classical theorem. These results are meant to complement previous results of the authors who have shown by means of a counterexample that the boundedness of all solutions is not, by itself, sufficient to guarantee the existence of an almost periodic solution to a linear almost periodic differential equation.

     

Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation

Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation are derived with the help of the resulting Riccati equation. As an application, some explicit solutions of the bidirectional Sawada-Kotera equation are obtained, including rational solutions, periodic solutions, and soliton solutions.     

周期系数 Riccati 方程的周期解存在准则

陈省身教授在报告空间曲线的封闭性时,提出了周期系数 Riccati 方程在什么条件下存在实周期解的问题.这个问题的解决,具有重要的理论价值和实践意义.因此很多人不断地从事这方面的研究,但给出的判据大都是充分性的.本文给出几类判定Riccati 方程存在周期解的充要条件.