Nearly Integrable Infinite Dimensional Dynamical System

Name: Nearly Integrable Infinite Dimensional Dynamical System
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Boling Guo Jun Zhang Jing Li Lan Zeng(Author)

EDP sciences (Publisher)

Published on 22. August 2024

582 pages

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Cover-1
Cover-2
Nearly Integrable Infinite Dimensional Dynamical System
Preface
Contents
Chapter 1 Chaos for Nearly Integrable Systems
1.1 Direct methods of perturbation theory for solitons
1.2 Perturbation theory based on the inverse scattering transform
1.3 Motion of a soliton in a driven Sine-Gordon equation
1.3.1 Soliton motion of Sine-Gordon equation
1.3.2 Motion of a SG soliton in the fields of two waves
1.3.3 Stochastic dynamics of a three-dimensional bubble in a driven SG equation
1.3.4 SG soliton similar to the Fermi-Pasta-Ulam problem
1.3.5 Dynamical chaos of a breather under the action of an external field
1.3.6 Dynamical chaos in the SG system with parametric excitation
1.3.7 Stochastization of soliton lattices in the perturbed SG equation
1.4 Motion of the soliton of nonlinear Schrodinger equation with damping under the action of an external field
1.4.1 Nonlinear Schrodinger equation
1.4.2 Stochastic dynamics of NLS solitons in a periodic potential
1.5 Dynamical chaos of the KdV equation and the perturbation equations
1.5.1 Chaotic state of the cnoidal wave in the periodic inhomogeneous medium
1.5.2 Karamoto-Sivashinsky equation
Chapter 2 Some Numerical Results and Their Analysis
2.1 Coherent structure and numerical calculation results
2.2 Fundamental Analysis
2.2.1 Connections between NLS equation and Sine-Gordon equation
2.2.2 Space independent fixed point
2.2.3 Space dependent fixed point
2.2.4 Integrable structure of nonlinear Schrodinger equation
2.2.4.1 Fundamental matrix M(y
?
q
r)
2.2.4.2 Spectral of L(q, r)
2.2.4.3 Floquet discriminant ?(?
q
r)
2.2.4.4 Isospectral level set M(q
r)
2.2.4.5 Dirichlet spectral {µj}
2.2.5 The Whisker ring of focusing nonlinear Schrodinger equation
2.2.5.1 The critical potential of the Floquet discriminant
2.2.5.2 Second-order variation ? 2Fj(qc): A saddle point structure of invariant
2.2.5.3 Morse function of the level set of the Floquet discriminant
2.2.5.4 Theory of linearization and the basis of squared eigenfunctions
2.2.5.5 Homoclinic orbit and Backlund transformation
Chapter 3 Homoclinic Orbits in a Four Dimensional Model of a Perturbed Nonlinear Schrodinger Equation
3.1 Dynamics and geometric structure for the unperturbed systerm
3.1.1 M0 and Ws(M0)nWu(M0)
3.1.2 The dynamics on M0
3.1.3 The unperturbed homoclinic orbits and their relationship to the dynamics on M0 and Ws(M0)nWu(M0)
3.2 Geometric structure of the perturbed systerm
3.2.1 The persistence of M0, Ws(M0) and Wu(M0) under perturbation
3.2.2 The dynamics on Me near resonance
3.3 Fiber representations of stable and unstable manifolds
3.3.1 Representation of Ws(M0) and Wu(M0) through homoclinic orbits
3.3.2 An intuitive introduction to fibrations of stable and unstable manifolds
3.3.3 A second example
3.3.4 Fibers for Ws(M0) and Wu(M0) of the two mode equations
3.3.5 Properties and characteristics of the fibers
3.3.6 Fibers representations for the subset of Wu (qe) and Wsloc(A?Me)
3.4 Homoclinic orbits for qe
3.4.1 Homoclinic coordinates and the hyperplane S
3.4.2 The Melnikov function for Ws(A?Me) nWu(qe)
3.4.3 Explicit expression of the Melnikov function at I = 1
3.4.4 The existence of orbits homoclinic to qe
3.5 Numerical results of orbits homoclinic to qe
3.5.1 Numerical computation for periodic solution
3.5.2 Computation for homoclinic manifolds
3.6 The dynamical consequences of orbits homoclinic to qe: the existence and property of chaos
3.6.1 Construction of the domains for the maps
3.6.2 Construction of the map P0 near the origin
3.6.3 Construction of the map along the homoclinic orbits outside a neighborhood of the origin
3.6.4 The full map, P=P0°P1 : ?0 ?0
3.6.5 Verification of the hypotheses of the theorem for the two-mode truncation
Chapter 4 Homoclinic Orbits of a Damped and Forced Sine-Gordon Equation
4.1 Structure of the unperturbed system
4.1.1 The normally hyperbolic invariant manifold M
4.1.2 The dynamics on M
4.1.3 Ws(M), Wu(M) and the homoclinic manifold
4.1.4 The dynamics on G and its relation to the dynamics in M
4.2 Structure of the perturbed system
4.2.1 The persistence of M, Ws(M) and Wu(M) under perturbation
4.2.2 The dynamics on Me
4.2.3 The fibering of Ws(Ae) and Wu(Ae): the singular perturbation nature
4.3 The existence of a homoclinic connection to pe
4.3.1 Wu(pe) ? Ws(Ae): The higher dimensional Melnikov theory
4.3.2 Wu(pe) nWs(pe): a homoclinic orbit to pe
4.4 Chaos: Silnikov's theorem
4.5 An application:model dynamics of the damped, driven, nonlinear Schrodinger equation
4.5.1 The unperturbed integrable structure
4.5.1.1 Analytical expressions for the homoclinic orbits
4.5.1.2 Orbits homoclinic to the circle of fixed points
4.5.2 Dynamics near the resonance on Ae
4.5.3 Calculation of the Melnikov function
4.5.4 The existence of an orbit homoclinic to pe
4.5.5 The geometrical interpretation of chaos in phase space
Chapter 5 Persistent Homoclinic Orbits for a Perturbed Nonlinear Schrodinger Equation
5.1 Introduction
5.2 Analysis of space-independent solutions and motion on the invariant plane
5.2.1 Motion on the invariant plane
5.2.2 The stable manifolds at Q in ?c
5.3 The equations in a neighborhood of the circle of fixed points
5.3.1 Basic equations
5.3.2 Normal forms
5.3.3 Local equations
5.4 Theory of invariant manifolds
5.4.1 Existence of local invariant manifolds
5.4.2 The fibration for invariant manifolds
5.4.3 Stable manifold to Q in Me
5.5 Global integrable theory
5.5.1 Lax pair
5.5.2 Zakharov-Shabat spectral problem
5.5.3 The basic example
5.5.4 Homoclinic orbits and whiskered tori
5.5.5 An important invariant
5.5.6 F'(qh)
5.6 Persistent homoclinic orbit
5.6.1 The first measurement
5.6.2 The second measurement
5.6.3 Existence of a homoclinic orbit
Chapter 6 Homoclinic Orbits and Chaos for the Discrete Disturbed Nonlinear Schrodinger Equation
6.1 Integrable case
6.1.1 Spectral theory of Ln
6.1.2 Hyperbolic structure and homoclinic orbits
6.2 Persistent invariant manifolds
6.2.1 Persistent invariant plane
6.2.2 Persistent invariant manifold theorem
6.2.3 The proof of the local persistent invariant manifold theorem
6.3 Fenichel fibers
6.3.1 An example showing fenichel fibers
6.3.2 Fiber theorem
6.3.3 The unique explicit fenichel fiber for "figure 8 ? A"
6.4 Melnikov measurement: Wu(qe)nWcse
6.4.1 Main argument
6.4.2 Derivation of Melnikov integral
6.4.3 Approximation
6.4.4 Computation for MF1
6.4.5 The intersection between Wu(qe) and Ws(Me) ? Wcse
6.5 Existence of orbits homoclinic to qe: the second measurement
6.6 General theory of symbolic dynamics
6.6.1 General framework
6.6.2 Smooth normal form reduction
6.6.3 Some definitions
6.6.4 Poincare map P01
6.6.5 Poincare map P10
6.6.6 Fixed point of Poincare map P=P01°P10
6.6.6.1 The Silnikov variables
6.6.6.2 C(s)k = 0 solutions
6.6.6.3 Fixed points
6.6.7 Smale horseshoes
6.6.7.1 Definition of slabs
6.6.7.2 Sl and P10 (Sl)
6.6.7.3 P(Sl)
6.6.7.4 Definition of slices
6.6.7.5 Generic intersection and horseshoes
6.6.8 Symbol dynamics
6.6.8.1 The shift automorphism
6.6.8.2 Conley-Moser condition
6.6.8.3 Topological conjugacy
6.7 Application to discrete NLS systems
6.7.1 Transformation of (6.6.1) to the form (6.1.3)
6.7.2 The Generic assumptions
6.7.3 Smale horseshoes and chaos created by a pair of homoclinic orbitsin the discrete nonlinear Schroinger systems
6.7.3.1 Equivariant smooth normal form reduction
6.7.3.2 Chaos created by the symmetric pair of homoclinic orbits
Chapter 7 Persistent Homoclinic Orbits for the Perturbed Sine-Gordon Equation
7.1 Persistent homoclinic orbits for a kind of Sine-Gordon equation under dissipative perturbation
7.2 Persistent homoclinic orbits for another kind of Sine-Gordon equation under dissipative perturbation
7.3 Persistent homoclinic orbits for a kind of Klein-Gordon equation under small perturbation
Chapter 8 Persistent Homoclinic Orbits of Perturbed High-order Nonlinear Schrodinger Equations
8.1 Persistent homoclinic orbits of a perturbed cubic-quintic NLS equation
8.1.1 Some fundamental results
8.1.1.1 Existence and regularity of solutions
8.1.1.2 The analysis of space-independent solutions
8.1.1.3 Some results from integrable theory
8.1.2 The equations in a neighborhood of Cw
8.1.2.1 Basic equations
8.1.2.2 Normal forms and local equations
8.1.3 Invariant manifolds
8.1.3.1 Existence of local invariant manifolds
8.1.3.2 The fibration
8.1.3.3 Stable manifold to Q in Me
8.1.4 Persistent homoclinic orbit
8.1.4.1 The first measurement (Melnikov)
8.1.4.2 The second measurement
8.1.4.3 Existence of a homoclinic orbit
8.2 Homoclinic orbits in a six dimensional model of derivative nonlinear Schrodinger equation
8.2.1 The Fourier truncation of a perturbed derivative NLS equation
8.2.2 Persistence of the normally hyperbolic invariant manifold
8.2.3 Persistence of the homoclinic orbits
8.3 Persistent homoclinic orbits for a perturbed coupled non-linear Schrodinger system
8.3.1 The preliminary results
8.3.2 An equation in a neighborhood of Sw
8.3.3 Existence of local invariant manifolds
8.3.4 Homoclinic orbit of unperturbed system
8.3.5 Persistent homoclinic orbit
8.4 Persistent homoclinic orbits for a perturbed nonlinear Schrodinger equation with derivation term under a small perturbation
8.4.1 The preliminary results
8.4.2 Analysis of space-independent solutions
8.4.3 Equation in a neighborhood of Cw
8.4.4 Invariant manifolds
8.4.5 Persistent homoclinic orbit
Chapter 9 Homoclinic Orbits of a Perturbed Nonlinear Schrodinger Equation
9.1 Main theorems and establishment of basic equations
9.2 Invariant manifolds and invariant foliations
9.3 Homoclinic orbits
9.3.1 Homoclinic orbits for unperturbed NLS
9.3.2 The first measurement
9.3.3 Second measurement
9.3.4 Existence of a Homoclinic Orbit
Chapter 10 Morse Functions and Floquet Theory
10.1 Morse and Melnikov functions for nonlinear Schrodinger equation
10.1.1 Floquet Spectral Theory
10.1.2 Critical Structure of Fj
10.1.3 The Morse description of the isospectral stratification
10.1.4 A Melnikov vector
10.2 Hill equation
10.3 Topological classification of integrable partial differential equations
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Nearly Integrable Infinite Dimensional Dynamical System

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