Quasi-periodic Solutions of the General Nonlinear Beam Equations

In this paper,one-dimensional (1D) nonlinear beam equations of the form utt - uxx + uxxxx + mu = f(u)with Dirichlet boundary conditions are considered,where the nonlinearity f is an analytic,odd funct...     

KAM Tori for the Derivative Quintic Nonlinear Schrodinger Equation

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].     

KAM tori for higher dimensional beam equation with a fixed constant potential

In this paper, we consider the higher dimensional nonlinear beam equation:utt + △2u + σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.     

Quantum Birkhoff-Gustavson normal form in some completely resonant cases

The goal of this paper is to present an explicit computation of the Birkhoff-Gustavson normal form in the 1:1, 1:2 and 1:3 resonances, by using the concept of Weyl quantization, especially when it is about the first terms. We use in particular the Weyl symbolic calculus and the abstract theorem of the quantum Birkhoff-Gustavson normal form in infinite dimension.     

Diffusion bound and reducibility for discrete Schr?dinger equations with tangent potential

In this paper, we consider the lattice Schr?dinger equations $$idot q_n (t) = tan pi (nalpha + x)q_n (t) + varepsilon left( {q_{n + 1} (t) + q_{n - 1} (t)} right) + delta v_n (t)left| {q_n (t)} right|^{2tau - 2} q_n (t),$$ with ?? satisfying a certain Diophantine condition, x ?? ?/?, and ?? = 1 or 2, where v _n (t) is a spatial localized real bounded potential satisfying |v _n (t)| ? Ce^???|n|. We prove that the growth of H ¹ norm of the solution {q _n (t)}_n??? is at most logarithmic if the initial data {q _n (0)} _n??? ?? H ¹ for ? sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e., $$idot q_n (t) = tan pi (nalpha + x)q_n (t) + varepsilon left( {q_{n + 1} (t) + q_{n - 1} (t)} right) + delta v_n left( {theta ^0 + tomega } right)q_n left( t right).$$ Then the linear equation can be reduced to an autonomous equation for a.e. x and most values of the frequency vectors ?? if ? and ?? are sufficiently small.     

Camassa-Holm方程的拟周期解及其渐近行为

近20年来,浅水波模型Camassa-Holm（CH）方程受到诸多研究者关注。在之前的工作中,通过Hirota双线性方法得到了CH方程的单周期解.基于此,该文将对N=2时CH方程的拟周期解及其渐近行为进行研究.首先,回顾了坐标变换,扩展的双线性形式和Riemann(黎曼)θ-函数等内容,并在此基础上利用Hirota双线性方法构造了在N=2时CH方程的含有多个参数的拟周期解,并且此拟周期解是由Riemannθ-函数表示的。其次,发现了此拟周期解渐近行为的一个特点,即CH方程的此拟周期解可以退化为其二孤子解.     

Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation

In this paper, one-dimensional (1D) nonlinear wave equation utt−uxx+mu+u³=0, subject to Dirichlet boundary conditions is considered. We show that for each given m>0, and each prescribed integer b>1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, which correspond to b-dimensional invariant tori of an associated infinite-dimensional dynamical system. In particular, these Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions

In this paper, one-dimensional (1D) nonlinear Schrödinger equation

     

Making solutions invariant

In this paper a procedure is developed to modify a non-invariant solution in such a way that the resulting solution is invariant. Furthermore it is investigated which properties of the solution are inherited by the modified solution. Received: April 1995     

ON THE LASOTA-WAZEWSKA MODEL WITH PIECEWISE CONSTANT ARGUMENT

In this article, a delay differential equation with piecewise constant argument is considered; the existence and global attractivity condition ofalmost periodic solution and quasi-periodic solution are obtained.     

一类m-KP方程的谱表示及其拟周期解

This paper gives the spectral representation of a class of (2 1)-dimensional mod- ified Kadomtsev-Petviashvili(m-KP) equation with a constant parameter.Its quasi-periodic solution is obtained in terms of Riemann theta functions.     

Positive quasi-periodic solutions to Lotka-Volterra system

In this paper,we prove the existence of positive quasi-periodic solutions for a class of Lotka-Volterra system with quasi-periodic coefficients by KAM technique.     

A self-similar solution to the equation of gas filtration in a spherically symmetric porous medium

We consider the Cauchy problem for the Boussinesq equation which describes filtration of a gas in a spherically symmetric porous medium. For the self-similar solution to this problem we construct a formal in the neighborhood of the point r → ∞ expansion and a convergent near r = 0 one.     

中间逻辑中的良构范式

本文在中间逻辑中引入了良构范式的概念,为该类范式的研究提出了一个通用方法,并应用该方法证明了HT逻辑是存在一般蕴含范式的最弱中间逻辑,经典命题逻辑CPL是存在限制蕴含范式的仅有中间逻辑.     

Projective DNF formulae and their revision

Combinatorial and learnability results are proven for projective disjunctive normal forms, a class of DNF expressions introduced by Valiant.