Chongchun Zeng

Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Let M be a compact C1 manifold which is invariant and normally hyperbolic with respect to a C1 semiflow in a Banach space. Then in an -neighborhood of M there exist local C1 center-stable and center-unstable manifolds W cs( ) and W cu( ), respectively. Here we show that W cs( ) and W cu( ) may each be decomposed into the disjoint union of C1 submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects M in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.

Persistence of Overflowing Manifolds for Semiflow

This paper, which is a sequel to a previous one [4] by the same authors, is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a Banach space. We consider a C1 semiflow defined on a Banach space X; that is, it is continuous on [0,∞)×X , and for each t ≥ 0, T t : X → X is C1, and T t ◦ T (x) = T t+s(x) for all t, s ≥ 0 and x ∈ X . A typical example is the solution operator for a differential equation. In [4] we proved that a compact, normally hyperbolic, invariant manifold M persists under small C1 perturbations in the semiflow. We also showed that in a neighborhood of M , there exist a center-stable manifold and a center-unstable manifold that intersect in the manifold M . In [4] the compactness and invariance of the manifold M were important assumptions. In the present paper, we study the more general case where the manifold M is overflowing (“negatively invariant and the semiflow crosses the boundary transversally”) or inflowing (“positively invariant and the semiflow crosses the boundary transversally”). We do not assume that M is compact or finite-dimensional. Also, M is not necessarily an imbedded manifold, but an immersed manifold. As an example, a local unstable manifold of an equilibrium point is an overflowing manifold. In brief, our main results on the overflowing manifolds may be summarized as follows (the precise statements are given in Section 2). We assume that the immersed manifoldM does not twist very much locally,M is covered by the image under T t of a subset a positive distance away from boundary, DT t contracts along the normal direction and does so more strongly than it does along the tangential direction, and DT t has a certain uniform continuity in a neighborhood of M . If the C1 perturbation T t of T t is sufficiently close to T t, then T t has a unique C1 immersed overflowing manifold M nearM . Furthermore, if T t isCk and a spectral gap condition holds, then M isCk. Similar results for inflowing manifolds are also obtained and given in Section 7.

Periodic solutions for Hamiltonian systems under strong constraining forces

Abstract We consider periodic solutions of Hamiltonian systems in Euclidean spaces whose motion is constrained to a submanifold M . We prove that under some nondegeneracy assumptions, periodic solutions persist when the constraint is replaced by a strong restoring potential.

Orbits homoclinic to centre manifolds of conservative PDEs

In this paper, perturbations to orbits homoclinic to saddle-centres for conservative systems are considered. We prove that if the Hessian of the conserved energy at the saddle-centre is positive definite in the centre directions, then either single bump orbits homoclinic to the saddle-centre persist or its centre-unstable and centre-stable manifolds intersect transversally, where the centre manifold is stable. The return map induced by the homoclinic orbits to the centre manifold and applications to sine-Gordon breathers, homoclinic orbits for nonlinear Schrodinger equations, and periodic travelling waves for Klein–Gordon equations are discussed.

Homoclinic orbits for a perturbed nonlinear Schrödinger equation

In recent years, there have been extensive studies on the existence of homoclinic orbits for nearly integrable Hamiltonian PDEs, which are closely related to chaos. In this work, we consider a perturbed nonlinear Schrodinger equation for u even and periodic in x. The diffusion ieuxx is an unbounded perturbation term. When the diffusion is replaced by its bounded Fourier truncation, Li, McLaughlin, Shatah, and Wiggins [26] proved the existence of homoclinic orbits for the perturbed equation. The method was based on invariant manifolds, foliations, and Melnikov analysis. The unboundedness of the diffusion prevents the equation from being solved for t < 0 for general initial values and destroys some geometric structures, however. We overcome these difficulties and prove the existence of homoclinic orbits for the diffusively perturbed NLS.

Travelling water waves with compactly supported vorticity

In this paper, we prove the existence of two-dimensional, travelling, capillary-gravity, water waves with compactly supported vorticity. Specifically, we consider the cases where the vorticity is a δ-function (a point vortex), or has small compact support (a vortex patch). Using a global bifurcation theoretic argument, we construct a continuum of finite-amplitude, finite-vorticity solutions for the periodic point vortex problem. For the non-periodic case, with either a vortex point or patch, we prove the existence of a continuum of small-amplitude, small-vorticity solutions.

Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D

In this note we propose a new set of coordinates to study the hyperbolic or nonelliptic cubic nonlinear Schrodinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.

On the Wind Generation of Water Waves

AbstractIn this work, we consider the mathematical theory of wind generated water waves. This entails determining the stability properties of the family of laminar flow solutions to the two-phase interface Euler equation. We present a rigorous derivation of the linearized evolution equations about an arbitrary steady solution, and, using this, we give a complete proof of the instability criterion of Miles [16]. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air–sea interface). We are thus able to give a unified equation connecting the Kelvin–Helmholtz and quasi-laminar models of wave generation.

Invariant Manifolds of Traveling Waves of the 3D Gross–Pitaevskii Equation in the Energy Space

We study the local dynamics near general unstable traveling waves of the 3D Gross–Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the center unstable manifold attracts nearby orbits exponentially before they go away from the traveling waves along the center or unstable directions and (ii) if an initial data is not on the center-stable manifolds, then the forward orbit leaves traveling waves exponentially fast. Furthermore, under an additional non-degeneracy assumption, we show the orbital stability of the travelingwaves on the centermanifolds,which also implies the uniqueness of the local invariant manifolds. Our method based on a geometric bundle coordinate system should work for a general class of Hamiltonian PDEs.

Equivariante and Self-similar Standing Waves for a Hamiltonian Hyperbolic-hyperbolic Spin-field System

In this paper we study the existence of special symmetric solutions to a Hamiltonian hyperbolic-hyperbolic coupled spin-field system, where the spins are maps from