一类Hamilton系统的同宿解存在性

We study the existence of homoclinic orbits for some Hamiltonian system.A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a sequence of systems of differential equations.     

Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results.     

Homoclinic Orbits for a Perturbed Lattice Modified KdV Equation

We establish the splitting of homoclinic orbits for a near-integrable lattice modified KdV (mKdV) equation with periodic boundary conditions. We use the Bäcklund transformation to construct homoclinic orbits of the lattice mKdV equation. We build the Melnikov function with the gradient of the invariant defined through the discrete Floquet discriminant evaluated at critical points. The criteria for the persistence of homoclinic solutions of the perturbed lattice mKdV equation are established.     

Existence of homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary-value problems which are obtained by the minimax methods.     

Homoclinic solutions for Davey-Stewartson equation

In this paper, we firstly prove the existence of homoclinic solutions for Davey-Stewartson I equation (DSI) with the periodic boundary condition. Then we obtain a set of exact homoclinic solutions by the novel method-Hirota’s method. Moreover, the structure of homoclinic solutions has been investigated. At the same time, we give some numerical simulations which validate these theoretical results.     

Transverse homoclinic orbit bifurcated from a homoclinic manifold by the higher order melnikov integrals

Consider an autonomous ordinary differential equation in $\mathbb{R}^n$ that has a $d$ dimensional homoclinic solution manifold $W^H$. Suppose the homoclinic manifold can be locally parametrized by $(\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R}$. We study the bifurcation of the homoclinic solution manifold $W^H$ under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed $(\alpha_0,\theta_0)$ on $W^H$, if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near $W^H$.     

(3+1)维变系数Kudryashov-Sinelshchikov(K-S)方程的同宿呼吸波解和高阶怪波解

基于Hirota双线性方法,利用拓展的同宿呼吸检验法得到了(3+1)维变系数Kudryashov-Sinelshchikov(K-S)方程的同宿呼吸波解,对该解的参数选取合适的数值,可得到不同结构的同宿呼吸波.通过对同宿呼吸波解的周期取极限,推导出方程的怪波解.最后,构造出一个特殊的高阶多项式作为测试函数,求得该方程的一阶怪波解和二阶怪波解.     

Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system

In this paper, we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulsive conditions via variational methods. However, without impulses, there is no homoclinic or periodic solution for the system considered in this paper. Moreover, our results can be used to study the existence of periodic and homoclinic solutions of difference equations.     

一类二阶微分方程同宿解的存在性

讨论下列二阶微分方程(y|¨)+ay+U_y(t,y)=0.的同宿解的存在性,其中t∈R,y∈Rⁿ,n∈N,a>0是一个常数,U(t,y)∈Cn,n∈N,a>0是一个常数,U(t,y)∈C¹(R×R1(R×Rⁿ,R),U_y(t,y)表示U(t,y)关于y的梯度.引入快同宿解的概念并给出方程存在快同宿解的判定准则.     

Homoclinic Solutions for a Forced Liénard Type System

In this paper, we find a special class of homoclinic solutions which tend to 0 as t → ±∞, for a Liénard type system with a time-dependent force. Since it is not a small perturbation of a Hamiltonian system, we cannot employ the well-known Melnikov method to determine the existence of homoclinic solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. We prove that the sequence of those periodic solutions has an accumulation which gives an homoclinic solution of the forced Liénard type system.     

A global condition for quasi-random behavior in a class of conservative systems

Devaney has shown that an autonomous Hamiltonian system in dimension 4, with an orbit homoclinic to a saddle-focus equilibrium, admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds. In this paper we deal with two classes of saddle-focus systems: Lagrangian systems defined on a two-manifold in the presence of a gyroscopic force, and fourth-order systems arising in water-wave theory. We first establish, by a standard variational method, the existence of a homoclinic orbit. Then, under a weak nondegeneracy condition, we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic. Our condition is much easier to check than transversality. For example, it is automatically satisfied for gyroscopic systems on a two-torus, for topological reasons. © 1996 John Wiley & Sons, Inc.     

Homoclinics for a Hamiltonian with wells at different levels

We consider a Hamiltonian equation of the form (HS) for which V has two distinct non-degenerate maxima at different levels: 0 is a local maximum and is an absolute maximum. Under standard non-degeneracy conditions on V, our main result is that there is a solution of (HS) homoclinic to 0. Then, supposing that another geometric condition holds, we show the existence of infinitely many solutions of (HS) homoclinic to 0 that are distinguished from one another by the number of times and regions where the solutions stay away from 0. As a corollary, we show that if there is a solution of (HS) homoclinic to , then there are infinitely many solutions of (HS) homoclinic to 0, distinguished by the number and position of intersections with 1/2.

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The EHTA for nonlinear evolution equations

Two types of important nonlinear evolution equations are investigated by using the extended homoclinic test approach (EHTA). Some exact soliton solutions including breather type of soliton, periodic type of soliton and two soliton solutions are obtained. These results show that the extended homoclinic test technique together with the bilinear method is a simple and effective method to seek exact solutions for nonlinear evolution equations.     

Infinitely many homoclinic solutions for some second order Hamiltonian systems

In this paper, we investigate the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems. By using fountain theorem due to Zou, we obtain two new criteria for guaranteeing that second order Hamiltonian systems have infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.     

Multibump solutions for discrete periodic nonlinear Schrödinger equations

In this paper, we study the existence of multibump solutions for discrete nonlinear Schrödinger equations with periodic potentials. We first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group. Then, we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear Schrödinger equations, and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case. As an application, we get, without any symmetry assumptions, infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity.     

Homoclinic solutions for fourth order differential equations with superlinear nonlinearities

In this paper we investigate the existence of homoclinic solutions for a class of fourth order differential equations with superlinear nonlinearities. Under some superlinear conditions weaker than the well-known (AR) condition, by using the variant fountain theorem, we establish one new criterion to guarantee the existence of infinitely many homoclinic solutions.     

The local linking theorem with an application to a class of second-order systems

In the present paper, some existence theorems are obtained concerning periodic and homoclinic solutions for a class of second-order systems by means of a local linking theorem.     

Exponential trichotomy and homoclinic bifurcation with saddle-center equilibrium

In this paper the bifurcation of a homoclinic orbit is studied for an ordinary differential equation with periodic perturbation. Exponential trichotomy theory with the method of Lyapunov–Schmidt is used to obtain some sufficient conditions to guarantee the existence of homoclinic solutions and periodic solutions for this problem. Some known results are extended.     

HOMOCLINIC SOLUTIONS FOR AUTONOMOUS DIFFERENTIAL EQUATIONS

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Homoclinic solutions of discrete nonlinear systems via variational method

Homoclinic solutions arise in various discrete models with variational structure, from discrete nonlinear Schr\"{o}dinger equations to discrete Hamiltonian systems. In recent years, a lot of interesting results on the homoclinic solutions of difference equations have been obtained. In this paper, we review some recent progress by using critical point theory to study the existence and multiplicity results of homoclinic solutions in some discrete nonlinear systems with variational structure.