Author | Li, Charles. author |
---|---|

Title | Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrรถdinger Equations [electronic resource] / by Charles Li, Stephen Wiggins |

Imprint | New York, NY : Springer New York : Imprint: Springer, 1997 |

Connect to | http://dx.doi.org/10.1007/978-1-4612-1838-8 |

Descript | VIII, 172 p. online resource |

SUMMARY

This book presents a development of invariant manifold theory for a speยญ cific canonical nonlinear wave system -the perturbed nonlinear Schrooinger equation. The main results fall into two parts. The first part is concerned with the persistence and smoothness of locally invariant manifolds. The secยญ ond part is concerned with fibrations of the stable and unstable manifolds of inflowing and overflowing invariant manifolds. The central technique for proving these results is Hadamard's graph transform method generalized to an infinite-dimensional setting. However, our setting is somewhat different than other approaches to infinite dimensional invariant manifolds since for conservative wave equations many of the interesting invariant manifolds are infinite dimensional and noncom pact. The style of the book is that of providing very detailed proofs of theorems for a specific infinite dimensional dynamical system-the perturbed nonlinear Schrodinger equation. The book is organized as follows. Chapter one gives an introduction which surveys the state of the art of invariant manifold theory for infinite dimensional dynamical systems. Chapter two develops the general setup for the perturbed nonlinear Schrodinger equation. Chapter three gives the proofs of the main results on persistence and smoothness of invariant manยญ ifolds. Chapter four gives the proofs of the main results on persistence and smoothness of fibrations of invariant manifolds. This book is an outgrowth of our work over the past nine years concerning homoclinic chaos in the perturbed nonlinear Schrodinger equation. The theorems in this book provide key building blocks for much of that work

CONTENT

1 Introduction -- 1.1 Invariant Manifolds in Infinite Dimensions -- 1.2 Aims and Scope of This Monograph -- 2 The Perturbed Nonlinear Schrรถdinger Equation -- 2.1 The Setting for the Perturbed Nonlinear Schrรถdinger Equation -- 2.2 Spatially Independent Solutions: An Invariant Plane -- 2.3 Statement of the Persistence and Fiber Theorems -- 2.4 Explicit Representations for Invariant Manifolds and Fibers -- 2.5 Coordinates Centered on the Resonance Circle -- 2.6 (6 = 0) Invariant Manifolds and the Introduction of a Bump Function -- 3 Persistent Invariant Manifolds -- 3.1 Statement of the Persistence Theorem and the Strategy of Proof -- 3.2 Proof of the Persistence Theorems -- 3.3 The Existence of the Invariant Manifolds -- 3.4 Smoothness of the Invariant Manifolds -- 3.5 Completion of the Proof of the Proposition -- 4 Fibrations of the Persistent Invariant Manifolds -- 4.1 Statement of the Fiber Theorem and the Strategy of Proof -- 4.2 Rate Lemmas -- 4.3 The Existence of an Invariant Subbundle E -- 4.4 Smoothness of the Invariant Subbundle E -- 4.5 Existence of Fibers -- 4.6 Smoothness of the Fiber fE(Q) as a Submanifold -- 4.7 Metric Characterization of the Fibers -- 4.8 Smoothness of Fibers with Respect to the Base Point -- References

Mathematics
Mathematical analysis
Analysis (Mathematics)
Geometry
Manifolds (Mathematics)
Complex manifolds
Mathematics
Manifolds and Cell Complexes (incl. Diff.Topology)
Analysis
Geometry