C. Eugene Wayne

Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory

AbstractIn this paper the nonlinear wave equation

Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R2

u_u - u_{xx} + v(x)u(x,t) + \varepsilon u^3 (x,t) = 0

Existence and Stability of Traveling Pulses in a Continuous Neuronal Network

is studied. It is shown that for a large class of potentials,v(x), one can use KAM methods to construct periodic and quasi-periodic solutions (in time) for this equation.

On Irwin’s proof of the pseudostable manifold theorem

The Korteweg-de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two-dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long-wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves.

The Non-Linear Stability of Front Solutions for Parabolic Partial Differential Equations

Abstract We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows us to extend those results in a number of ways.

Kawahara dynamics in dispersive media

Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called “Oseen’s vortex”. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we give precise estimates on the rate of convergence toward the vortex.

The KAM Theory of Systems with Short Range Interactions I

We examine the existence and stability of traveling pulse solutions of a set of integro-differential equations that describe activity in a spatially extended population of synaptically connected neurons. These equations have been employed extensively to model wave propagation during normal and epileptic brain activity. Compared to previous studies, we make relatively weak assumptions on the pattern of spatial connectivity, namely, that it is positive, homogeneous, and symmetric and that it decays with distance. A Heaviside step function governs the activation of each neuron. We incorporate a relatively slow local recovery variable within each neuron but make no other assumptions about the recovery rate. Our results are guided by the local behavior of individual neurons. When neurons have a single stable state, we demonstrate the existence of two traveling pulse solutions in a connected network. When the neurons are bistable, we demonstrate the existence of a stationary pulse solution and, in some cases, a...

Periodic Solutions of Nonlinear Schrödinger Equations and the Nash-Moser Method

We simplify and extend Irwins proof of the pseudostable manifold theorem.