Non-existence of KAM torus

Given an integrable Hamiltonian h ₀ with n-degrees of freedom and a Diophantine frequency ω, then, arbitrarily close to h ₀ in the C ^r topology with r < 2n, there exists an analytical Hamiltonian h _ε with no KAM torus of rotation vector ω. In contrast with it, KAM tori exist if perturbations are small in C ^r topology with r > 2n.     

二维非线性五次Schr(o)dinger方程可约化的KAM环面

在周期边界条件下,本文考虑二维非线性五次Schr(o)dinger方程iut-△u+|u|4u=0(t∈R,x∈T2),证明一个无限维的KAM (Kolmogorov-Arnold-Moser)定理.应用无限维的KAM定理,本文获得这个方程一族Whitney光滑的部分双曲的小振幅拟周期解.     

A class of differential games with state-dependent closed-loop feedback solutions

In this paper, a class ofN-person, nonzero-sum differential games with state-dependent closed-loop feedback solutions is presented. In particular, the system of Hamilton-Jacobi equations for a closed-loop feedback solution turns out to be a set ofN-independent linear equations with viscosity solutions.The helpful comments of an anonymous referee are gratefully acknowledged.     

Construction of quasi-periodic solutions of delay differential equations via KAM techniques

This work focuses on the existence of quasi-periodic solutions for linear autonomous delay differential equation under quasi-periodic time-dependent perturbation near an elliptic-hyperbolic equilibrium point. Using the time-1 map of the solution operator, Newton iteration scheme, space splitting and KAM techniques, it is shown that under appropriate hypothesis, there exist quasi-periodic solutions with the same frequencies as the perturbation for most parameters. We show that if the delay differential equation is analytic, we obtain analytic parameterizations of the solutions.     

Quasi-periodic solutions of completely resonant nonlinear wave equations

It is shown that there are many elliptic invariant tori, and thus quasi-periodic solutions, for the completely resonant nonlinear wave equation subject to periodic boundary conditions via KAM theory.     

Persistence of invariant torus in Hamiltonian systems with two-degree of freedom

In this paper with the KAM iteration we prove a KAM theorem for nearly integrable Hamiltonian systems with two degrees of freedom without any non-degeneracy condition.     

KAM theory and the 3D Euler equation

We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold M   is not mixing in the C^k$C^k$ topology (k>4$k>4$ and non-integer) for any prescribed value of helicity and sufficiently large values of energy. This can be regarded as a 3D version of Nadirashvili's and Shnirelman's theorems showing the existence of wandering solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the mixing under the Euler flow of C^k$C^k$-neighborhoods of divergence-free vectorfields on M  . On the way we construct a family of functionals on the space of divergence-free C¹$C^1$ vectorfields on the manifold, which are integrals of motion of the 3D Euler equation. Given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types. We use the KAM theory to establish some continuity properties of these functionals in the C^k$C^k$-topology. This allows one to get a lower bound for the C^k$C^k$-distance between a divergence-free vectorfield (in particular, a steady solution) and a trajectory of the Euler flow.     

Regularity of boundary quasi-potentials for planar systems

We establish local regularity properties for the value function of a variational problem arising in the study of small random perturbations of planar dynamical systems. The approach is to characterize the extremals as solutions to a Hamiltonian system, using the usual Legendre transformation. The differential of the value function is described by a certain stable manifold associated with the Hamiltonian system. The existence and smoothness of this stable manifold is obtained from standard results.Key arguments of this paper were developed while the author was a visitor at the Division of Applied Mathematics of Brown University.     

(mM)∞算子及Hamilton-Jacobi方程粘性解

Hamilton-Jacobi equations are frequently encountered in applications, e.g. , in control theory, differential games, and theory of economics, construct viscosity solutions of Hamilton-Jacobi equations having a nonconvex flux and a nonconvex initial value. The main idea is. decomposit flux into convex flux plus concave flux, with the help of a newly designed operator (mM)^∞ and Legendre transform, the viscosity solutions of Hamilton-Jacobi equations can be exactly ex-pressed. The (mM)^∞ type Solutions is proved to be the viscosity solutions ofHamilton-Jacobi equations. In fact our ( (mM)^∞ ) formula is a nonconvex generalization of the convex Lax-Oleinik-Hopf’s formula.     

Formula for a solution ofu t +H(u,Du) =g

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equationu _t +H(u,Du) =g in ℝ ⁿ x ℝ₊ withu(x, 0) =u ₀(x). The HamiltonianH(s,p) is assumed to be convex and positively homogeneous of degree one inp for eachs in ℝ. IfH is non increasing ins, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.     

Umbilical torus bifurcations in Hamiltonian systems

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally umbilic invariant tori. These lower dimensional tori do not satisfy the usual non-degeneracy conditions that would yield persistence by an adaption of KAM theory, and there are indeed regions in parameter space with no surviving torus. We assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a (generalised) umbilical catastrophe. Combining techniques of KAM theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large Cantor sets. Applications to gyrostat dynamics are pointed out.     

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.     

On the vanishing viscosity method for first order differential-functional IBVP

We consider the initial-boundary value problem for first order differential-functional equations. We present the ‘vanishing viscosity’ method in order to obtain viscosity solutions. Our formulation includes problems with a retarded and deviated argument and differential-integral equations. Supported by KBN grant 2 PO3A, 01811.     

Regularity for solutions of nonlinear second order evolution equations

This paper deals with the existence of solutions for the class of nonlinear second order evolution equations. The regularity and a variation of solutions of the given equations are also given. As particular cases of our general formulation, some results for Volterra integrodifferential equations of the hyperbolic type are given.