On the periodic solutions of a forced second-order equation

Summary We study the equations for two-dimensional irrotational motion induced in a rectangular tank of water which is forced to oscillate horizontally in a periodic fashion (shallow water sloshing). The problem reduces to a second-order non-autonomous ordinary differential equation. We rigorously show the existence of many solutions obtained previously by numerical computations, and in addition we show that there are many other bounded solutions. We use simple shooting to obtain irregular solutions of the kind obtained by the methods of dynamical systems. In addition, we obtain a kind of kneading theory whereby we can order the initial conditions according to the pattern of spikes exhibited by the solutions.     

Stable periodic solutions in the forced pendulum equation

Consider the pendulum equation with an external periodic force and an appropriate condition on the length parameter. It is proved that there exists at least one stable periodic solution for almost every external force with zero average. The stability is understood in the Lyapunov sense.     

Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities

In this paper we consider the following Schrödinger equation:

     

Zhiber-Shabat方程的孤立波解与周期波解

结合齐次平衡法原理并利用F展开法,再次研究了Zhiber-Shabat方程的各种椭圆函数周期解.当椭圆函数的模m分别趋于1或0时,利用这些椭圆函数周期解,得到了Zhiber-Shabat方程的各种孤子解和三角函数周期解,从而丰富了相关文献中关于Zhiber-Shabat波方程的解的类型.     

Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Bifurcation of periodic solutions for a semilinear wave equation

Bifurcation of time periodic solutions and their regularity are proved for a semilinear wave equation, u_tt?u_xx?λu=f(λ,x,u),x?(0,π), t?R, together with Dirichlet or Neumann boundary conditions at x = 0 and x = π. The set of values of the real parameter λ where bifurcation from the trivial solution u = 0 occurs is dense in $\text{R}$.     

Stationary solutions for a superlinear Dirac equation

K. Guerlebeck In this paper, we consider the following nonlinear Dirac equation By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we prove the existence of nontrivial and ground state solutions for the aforementioned system under conditions weaker than those in Zhang et al. (Journal of Mathematical Physics, 2013). John Wiley & Sons, Ltd.     

Multiple periodic solutions for a nonlinear suspension bridge equation

We investigate nonlinear oscillations in a fourth-order partialdifferential equation which models a suspension bridge. Previouswork establishes multiple periodic solutions when a parameterexceeds a certain eigenvalue. In this paper, we use Leray-Schauderdegree theory to prove that if the parameter is increased further,beyond a second eigenvalue, then additional solutions are created.     

一个组合方程的单孤子解和周期尖波解

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.     

New solitary wave solutions and periodic wave solutions for the compound KdV equation

In this paper, for compound KdV equation, four new solitary wave solutions in the form of hyperbolic secant function and six periodic wave solutions in the form of cosine function are obtained by using undetermined coefficient method. On three different layers, the velocity interval which ensures that bell-shaped solitary wave solutions and periodic wave solutions exist synchronously is obtained, respectively. The length of the interval is related to coefficients of the two nonlinear terms.     

Linear behaviour of solutions of a superlinear heat equation

We give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data. This improves a previous result of Lee and Ni by showing that this behaviour holds for a significantly larger, and rather tightly defined class of solutions.     

Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces

The existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper. Our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of Leray-Schauder type and on a fixed point theorem in cones.     

Multiple periodic solutions for a class of nonlinear nonautonomous wave equations

In this paper, by means of ourZ _p index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations. All previous known results rely either on the oddness of nonlinear terms, or on autonomous systems, and the best result for the general case is the existence of two nontrivial periodic solutions (Amann, Zehnder, K. C. Chang, S. P. Wu, S. J. Li, Z. Q. Wang). In this paper, under the assumption that the nonlinear term isT/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.This research was supported in part by the National Postdoctoral Science Fund.     

Multiple periodic solutions for system of first-order differential equation

Sufficient conditions have been obtained for the existence of at least two non-negative periodic solutions to a system of first-order nonlinear functional differential equations. Applications to some ecological models are given.