On the integrability and Riemann theta functions periodic wave solutions of the Benjamin Ono equation

In this paper, the complete integrability of the Benjamin Ono equation is systematically studied. Its bilinear equation, soliton solutions, bilinear Bäcklund transformation and Lax pair are successfully obtained, by virtue of generalized Bell’s polynomials scheme. Moreover, by using multidimensional Riemann theta functions, the periodic wave solutions of the Benjamin Ono equation are constructed. Further, the asymptotic behaviors of the periodic wave solutions are presented with a limiting procedure, which shows the relations between the periodic wave solutions and soliton solutions.     

ON PERIODIC SOLUTIONS OF SEVERAL CLASSES 0F RICCATI'S EQAUTION AND SECOND一ORDER DIFFERENTIAL EQUATIONS

In this paper, the problem on periodic solutions of several classes of Riccati's equation with periodic coefficients is discussed, and the conditions, under which several classes of secondorder equations with periodic coefficients have periodic solutions, are given.     

Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations

The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic, it is almost periodic in the sense that its wave profile and wave speed are almost periodic.     

Bifurcation of the solution to the problem of steady traveling waves in a layer of viscous liquid on an inclined plane

Traveling waves in a viscous liquid flowing down an inclined plane can be described at small and moderate Reynolds numbers by an ordinary differential equation in the thickness of the layer [1, 2]. As the Reynolds number tends to zero, this equation goes over into an equation of third order with quadratic nonlinearity [3]. Periodic solutions of this last equation bifurcating from the plane-parallel solution have been investigated by Nepomnyashchii and Tsvelodub [3–6]. In the present paper, a study is made of the bifurcation of periodic solutions from periodic solutions, namely, an investigation is made of the values of the wave number for which a periodic solution is not unique; a bifurcation equation is derived, the number of bifurcating solutions is found, and their behavior near a bifurcation point is considered; and the bifurcating solutions are continued numerically with respect to a parameter (the wave number) from the neighborhoods of the bifurcation points.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

Existence of periodic traveling wave solutions for a class of generalized BBM equation

In this paper, a new kind of generalized BBM equation is introduced and discussed. Some existence theorems of periodic traveling wave solutions for this kind generalized BBM equation are given.     

EXISTENCE OF PERIODIC TRAVELNG WAVE SOLUTIONS FOR A CLASS OF GENERALIZED BBM EQUATION

I.Intr9ducti0nInpaperI5]globalexistenceanduniqueness0fsolutionswereestablishedfortheBenjamin-Bona-Mahony(BBM)equation,Il.l)u, uu,-u,,I=o(l'l)wherexERandt>O.TheBBMequationmodelslongwavesinanonlineardispersivesystemisasubstitutemodelfortheKortweg-deVriesequ…     

A stability analysis of periodic solutions to the steady forced Korteweg–de Vries–Burgers equation

The stability of periodic solutions to the steady forced Korteweg–de Vries–Burgers (fKdVB) equation is investigated here. This family of periodic solutions was identified by Hattam and Clarke (2015) using a multi-scale perturbation technique. Here, Floquet theory is applied to the governing equation. Consequently, two criteria are found that determine when the periodic solutions are stable. This analysis is then confirmed by a numerical study of the steady fKdVB equation.     

On the Floquet Multipliers of Periodic Solutions to Non-linear Functional Differential Equations

For periodic solutions to the autonomous delay differential equation

Existence and stability analysis of bifurcating periodic solutions in a delayed five-neuron BAM neural network model

In this paper, a bidirectional associative memory (BAM) neural network model, which consists of two neurons in the X-layer and three neurons in the Y-layer, with two time delays, will be considered. By analyzing the associated characteristic equation, we obtain that Hopf bifurcation occurs and a family of periodic solutions appears. Moreover, the stability and the period of the bifurcating periodic solutions are studied. To illustrate our theoretical results, numerical simulations are presented.     

Symmetric Periodic Solutions of Parabolic Problems With Discontinuous Hysteresis

We develop a framework for treating the long-term behavior of solutions for parabolic equations in multidimensional domains with discontinuous hysteresis. Bearing in mind the thermostat model, we concentrate in this paper on the prototype heat equation with hysteresis in the boundary condition. We provide an algorithm for constructing all periodic solutions with exactly two switchings on the period and study their stability. Coexistence of several periodic solutions with different stability properties is proved to be possible. A mechanism of appearance and disappearance of periodic solutions is investigated.     

Cavity formation and singular periodic oscillations in isotropic incompressible hyperelastic materials

In this paper, a dynamical problem is considered for an incompressible hyperelastic solid sphere composed of the classical isotropic neo-Hookean material, where the sphere is subjected to a class of periodic step radial tensile loads on its surface. A second-order non-linear ordinary differential equation that describes cavity formation and motion is proposed. The qualitative properties of the solutions of the equation are examined. Correspondingly, under a prescribed constant dead-load, it is proved that a cavity forms in the sphere as the dead-load exceeds a certain critical value and the motion of the formed cavity presents a class of singular periodic oscillations. Under periodic step loads, the existence conditions for periodic oscillation of the formed cavity are determined by using the phase diagrams of the motion equation of cavity. In each section, numerical examples are also carried out.     

The twist coefficient of periodic solutions of a time-dependent Newton's equation

This paper gives sufficient conditions for the existence of periodic solutions of twist type of a time-dependent differential equation of the second order. The concept of periodic solution of twist type is defined in terms of the corresponding Birkhoff normal form and, in particular, implies that the solution is Lyapunov stable. Some applications to nonlocal problems are given.     

Une application de diagramme de Newton a la recherche des solutions periodiques d'un oscillateur quasiharmonique et l'etude de la stabilite de ces solutions

Periodic solutions of autonomous quasiharmonic systems are studied in the resonant case if the branching equation has multiple roots. In order to find all the real solutions of this equation, we use Newton's diagram. This problem may have no real periodic solution. This depends upon the configuration of the descending section of Newton's diagram and upon the roots of the appropriate defining equations. The stability of these periodic solutions is also considered. Sufficient conditions of the solutions depend upon the defining equation.     

Convergence of periodic wavetrains in the limit of large wavelength

The Korteweg-de Vries equation was originally derived as a model for unidirectional propagation of water waves. This equation possesses a special class of traveling-wave solutions corresponding to surface solitary waves. It also has permanent-wave solutions which are periodic in space, the so-called cnoidal waves. A classical observation of Korteweg and de Vries was that the solitary wave is obtained as a certain limit of cnoidal wavetrains.This result is extended here, in the context of the Korteweg-de Vries equation. It is demonstrated that a general class of solutions of the Korteweg-de Vries equation is obtained as limiting forms of periodic solutions, as the period becomes large.     

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.     

New exact solutions for a generalized KdV equation

In this paper, we establish a triple-order complete discrimination system to derive the traveling wave solutions of the generalized KdV equation with high power nonlinearities, which consist of solitary patterns solutions, compactons solutions, periodic solutions and Jacobi elliptic functions solutions.     

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.