Peyresq Lectures on Nonlinear Phenomena

The following sections are included:

• Preface

• Addresses of Contributors

• Contents

We outline the general principles of thin plate elasticity, by emphasizing their connection with the classical results of differential geometry. The relevant FvK equation, can be solved in some specific cases, even though they are strongly and definitely nonlinear. We present two types of solutions. The first one concerns the contact of a spherical shell on a flat plane at increasing pressing forces, the second one is about the buckling of a thin film under pressure on a flat substrate, where we explain the observed “telephone-cord” pattern of delamination.

Chaos is a well defined concept for classical systems. In these lectures, I study the manifestations of chaos for microscopic objects, for which a quantum description must be used. Various examples, mainly but not exclusively coming from atomic physics, are used to illustrate our current understanding of the problem.

The mathematical study of 2D travelling waves in the potential flow of one or several layers of perfect fluid, with free surface and interfaces can be set as an ill-posed evolution problem, where the horizontal space variable plays the role of a “time”. In the finite depth case, the study of near equilibria waves reduces to a low dimensional reversible ordinary differential equation. In most cases, it appears that the problem is a perturbation of an integrable system, where all types of solutions are known. We shall describe the method of study and typical results in these lectures. In addition, we shall give indications on what happens when passing to the limit of infinite depth, which is indeed the case of physical interest. In such a case, the reduction technique fails because the linearized operator possesses a continuous spectrum crossing the imaginary axis, and we are lead to use tools adapted to the type of solution we are looking for. We shall develop such tools in the lectures.

In this article we use a classical description of laser cooling of atoms. In a second part we describe the use of cold atoms for multiple scattering experiments and discuss some effects which appear for dense atomic media.

These lectures give a (hopefully!) pedagogical introduction to the theory of nonlinear interactions of waves and weak turbulence. After having derived the pertubative expansion of the Hamiltonian of surface waves, we write down the kinetic equations for the number density of waves. Then stationary solutions to these kinetic equations allow us to find the weak turbulence spectrum.

These lectures are devoted to phenomena beyond all orders in dynamical systems. We explain how these phenomena are connected with oscillatory integrals and how to obtain an exponentially small equivalent of an oscillatory integral when it involves solutions of nonlinear differential equations. The method proposed herein enables us to study the problem of existence of homoclinic connections to 0 for vector fields admitting a 0²iω or a (iω₀)²iω₁ resonance. These problems could not be solved by a direct application of the Melnikov method since in these cases the Melnikov function is given by an exponentially small oscillatory integral.

The following sections are included:

• Introduction

• Models for pattern formation and morphogenesis
  • Chemical pre-pattern models
  • Cell movement models
  • Cell rearrangement models
  • Applications
  • Coupling pattern generators
  • Domain Growth
  • Discussion

• Models for wound healing
  • Corneal Wound Healing
  • Dermal healing
  • Discussion

• Conclusions

• References

The following sections are included:

• Introduction
  • Hamilton's equations
  • Examples
  • Symmetry
  • Central force problem
  • Lie group actions

• Noether's Theorem and the Momentum Map
  • Noether's theorem
  • Equivariance of the momentum map
  • Reduction
  • Singular reduction
  • Symplectic slice and the reduced space

• Relative Equilibria

• Bifurcations of (relative) equilibria
  • One degree of freedom
  • Higher degrees of freedom

• Geometric Bifurcations

• Examples
  • Point vortices on the sphere
  • Point vortices in the plane
  • Molecules

• References