Korteweg–de Vries Flows with General Initial Conditions (Mathematical Physics Studies)

✍ Scribed by Shinichi Kotani

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 167
• Category: Library

✦ Synopsis

Large numbers of studies of the KdV equation have appeared since the pioneering paper by Gardner, Greene, Kruskal, and Miura in 1967. Most of those works have employed the inverse spectral method for 1D Schrödinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by Hirota–Sato and Marchenko independently, those have not been fully investigated and analyzed.

The present book offers a new approach to the study of the KdV equation, which treats decaying initial data and oscillating data in a unified manner. The author’s method is to represent the tau functions introduced by Hirota–Sato and developed by Segal–Wilson later in terms of the Weyl–Titchmarsh functions (WT functions, in short) for the underlying Schrödinger operators. The main result is stated by a class of WT functions satisfying some of the asymptotic behavior along a curve approaching the spectrum of the Schrödinger operators at +∞ in an order of -(n-1/2)for the nth KdV equation. This class contains many oscillating potentials (initial data) as well as decaying ones. Especially bounded smooth ergodic potentials are included, and under certain conditions on the potentials, the associated Schrödinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the author’s motivation for extending the class of initial data in this book. A large class of almost periodic potentials is also included in these ergodic potentials. P. Deift has conjectured that any solutions to the KdV equation starting from nearly periodic initial data are almost periodic in time. Therefore, our result yields a foundation for this conjecture.

For the reader’s benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrödinger operators, including the notion of the WT functions; (2)Sato’s Grassmann manifold method revised by Segal–Wilson; and (3) basic results of ergodic Schrödinger operators.

✦ Table of Contents

Preface
Contents
1 Introduction
2 Sato's Theory
2.1 Algebra of Formal Pseudo-differential Operators
2.2 Flows on Grassmann Manifolds
3 KdV Flow I: Reflectionless Case
3.1 Hardy Spaces and Toeplitz Operators
3.2 Characteristic Functions and mm-function
3.3 Group Action on script upper A Superscript italic i n vmathcalAinv and Tau-Function
3.4 Non-degeneracy of Tau-Functions
3.5 Identification of mm-functions with WTK Functions
3.6 KdV Flow on script upper Q Superscript italic r e f lmathcalQrefl
3.7 Boussinesq Equation
4 KdV Flow II: Extension
4.1 Hardy Spaces and Toeplitz Operators
4.2 Group Action on script upper A Subscript upper L Superscript italic i n vmathcalALinv and Derivation of Equations
4.2.1 Estimates of upper T left parenthesis g bold italic a right parenthesisT( ga) and Differentiability
4.2.2 Derivation of Schrödinger Operator
4.2.3 Derivation of KdV Equation
4.3 Tau-Function
4.3.1 Properties of Tau-Function
4.3.2 Continuity of Tau-Functions
4.4 Non-negativity Condition of script upper A Subscript upper L Superscript italic i n vmathcalALinv
4.4.1 Non-degeneracy of Tau-Functions for bold italic a element of script upper A Subscript upper L comma plus Superscript italic i n vainmathcalAL,+inv
4.4.2 mm-function and WTK Function
4.5 KdV Flow
4.5.1 Construction of KdV Flow
4.5.2 Tau-Function Representation of the Flow
5 Applications
5.1 Decaying Potentials
5.1.1 q element of upper L Superscript 1 Baseline left parenthesis double struck upper R right parenthesisqinL1( mathbbR) Case
5.1.2 Wigner–von Neumann Type Potentials
5.2 Oscillating Potentials
5.2.1 Reflection Coefficient
5.2.2 Ergodic Potentials
6 Further Topics
6.1 Extension of Remling's Theorem
6.1.1 Transfer Matrix
6.1.2 Reflectionless Property on ac Spectrum
6.2 Multi-component Systems
6.2.1 Nonlinear Schrödinger Equation
6.2.2 Sine–Gordon Equation
7 Appendix
7.1 Herglotz–Nevanlinna (HN) Functions
7.2 Spectral Theory of 1D Schrödinger Operators
7.2.1 Weyl–Titchmarsh–Kodaira (WTK) Functions
7.2.2 Shift Operation and Its Properties
7.2.3 Expansion of WTK Functions when z right arrow normal infinityzrightarrowinfty
7.3 Conformal Maps
7.4 Ergodic Schrödinger Operators
Appendix Bibliography
Index