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Tables of Contents for Algebro-Geometric Approach to Nonlinear Integrable Equations
Chapter/Section Title
Page #
Page Count
Introduction
1
12
The History of the Search for Periodic Solutions of the Korteweg - de Vries Equation
1
4
Outline of the Content of This Book
5
8
Riemann Surfaces and Theta Functions
13
44
Riemann Surfaces
13
5
Coverings
18
1
Elliptic Curves
18
5
Functions, Differentials and Integrals on Riemann Surfaces
23
11
General Properties of Functions on Riemann Surfaces
23
1
Abelian Differentials
24
2
Periods of Abelian Differentials
26
3
Period Matrix. Jacobian Variety
29
2
Divisors
31
3
Abelian Functions and Theta Functions
34
4
Addition Theorems for Theta Functions
38
3
Construction of Functions and Differentials on a Riemann Surface Using Theta Functions
41
5
Hyperelliptic Curves
46
11
Uniformization of Riemann Surfaces
52
5
Finite-Gap Solutions of the Kadomtsev-Petviashvili and the Korteveg - de Vries Equations
57
30
Differential Equations for the Baker-Akhiezer Functions
57
3
Solution of the Kadomtsev-Petviashvili Equation
60
2
Real Non-Singular Solutions of the KP1 and KP2 Equations
62
2
Reduction to Korteweg - de Vries and Boussinesq Equations
64
3
Spectral Properties of the Finite-Gap Solutions
67
9
Spectral Properties of the Schrodinger Operator with a Finite-Gap Periodic Potential
76
11
Monodromy Matrix and Bloch Solutions of the Hill Equation
76
3
Finite-Gap Potentials and the Gelfand-Levitan-Dikii Trace Formulas
79
1
Criterion for the Periodic Potential to be Finite-Gap
80
1
Charles Hermite and the Lame Equation
81
1
Analytical Properties of the Bloch Functions and the Inverse Spectral Problem for Finite-Gap Potentials
82
2
Jules Drach and General Finite-Gap Potentials
84
3
Vector Valued Baker-Akhiezer Functions and Finite-Gap Solutions of the Nonlinear Schrodinger and Sine-Gordon Equations
87
64
Finite-Gap Solutions of the Nonlinear Schrodinger Equation
87
8
Finite-Gap Solutions of the Sine-Gordon Equation
95
11
Reality Conditions. Reduction of the NS System to the Nonlinear Schrodinger Equation
106
11
NS Equation. Reduction y* = y
106
4
NS Equation with Attraction. Reduction y* = -y
110
3
Sine-Gordon Model. Reality Condition v = v
113
4
Degeneracy of Finite-Gap Solutions. Multi-Soliton Solutions
117
14
Partial Degeneracy of the Finite-Gap Formulas. Multi-Phase Modulations of the Cnoidal Wave
131
12
Alternative Approach to the Finite-Gap Integration of Matrix Systems
143
8
Uniformization of Riemann Surfaces and Effectivization of Theta Function Formulas
151
38
The Schottky Uniformization
151
3
Convergence of Poincare Series
154
2
Schottky Uniformization of Riemann Surfaces of the Decomposition Type
156
1
Schottky Uniformization of the M-Curves
157
3
Solutions of the KP2 Equation
160
5
Solutions of the KP1 Equation
165
3
Multi-Soliton Solutions and Small-Amplitude Waves
168
2
Solutions of the KdV Equation
170
4
Qualitative Analysis of Solutions of the KdV Equation
174
3
Solutions of the Sine-Gordon Equation
177
4
Uniformization and Coverings
181
8
Description of the Schottky Space of M-Curves
185
4
Theta Function Formulas for Classical Tops
189
44
The Lax Equation and Analytic Properties of the Baker-Akhiezer Function
189
2
Integrable Systems
191
2
Kowalewski Top
193
13
Kowalewski's Paper
193
1
The Lax Pair for the Kowalewski Top
194
1
The Spectral Curve
195
2
Analyticity Properties of the Baker-Akhiezer Function
197
2
Explicit Formulas for the Baker-Akhiezer Function
199
2
The Prym Condition
201
1
The Poisson Vector
202
2
The Geometry of Liouville Tori
204
1
Reduction to Two-Dimensional Theta Functions
205
1
The Goryachev-Chaplygin Top
206
8
The Lax Pair for the Goryachev-Chaplygin Top
207
1
The Spectral Curve
208
1
Analytic Properties of the Baker-Akhiezer Function
209
2
Construction of the Baker-Akhiezer Function
211
1
Formulas for Dynamical Variables
212
2
Integration of the Lax Representations with the Spectral Parameter on an Elliptic Curve. XYZ Landau-Lifshitz Equation
214
5
Curves of Lower Genera. The Euler Case and the Neumann System
219
3
Manakov and Clebsch Cases
222
2
The Steklov Cases
224
4
Complete Description of Motion in the Rest Frame
228
5
Algebro-Geometric Principles of Superposition of Finite-Gap Solutions of Integrable Nonliner Equations
233
70
The Simplest Reduction Case
234
2
Poincare Theorem
236
4
Coverings of Genus 2 Over a Torus. Humbert Varieties
240
3
Expression of Moduli for a Genus 2 Curve in Terms of Theta Constants
243
5
Reduction Scheme for Genus 2
248
6
Choice of a Basis
248
2
Reduction Condition
250
1
Description of a Covering
251
1
Reduction of Holomorphic Differentials to Elliptic Ones
251
1
Description of Covers
251
3
Reduction of Abelian Differentials of the Second and Third Types
254
1
Example: 2- and 4-Sheeted Covers of Genus 2 over a Torus
254
5
2-Sheeted Covers
254
1
4-Sheeted Coverings
255
4
Elliptic Finite-Gap Potentials
259
7
Elliptic Solutions of the KdV Equation and Particle Dynamics
266
8
Elliptic Solutions of the Sine-Gordon Equation
274
3
On One Periodic Solutions of the Kowalewski Problem
277
4
Normal Coverings
281
2
The Appel Theorem
283
2
Reduction of the Theta Functions of Normal Coverings
285
2
Example: 3-Gap Solution of the Sine-Gordon Equation in Terms of Elliptic Functions
287
16
Relations Between Theta Constants for g = 2
293
3
Addition Formulas for Second-Order Theta Functions at g = 2
296
3
Theta Constants of 2- and 4-Sheeted Coverings Over a Torus
299
4
An Application: The Peierls-Frohlich Problem and Finite-Gap Potentials
303
26
Finite-Gap Potentials
303
10
The Peierls Problem
313
8
Theory of the Frohlich Conductivity
321
6
Conclusion
327
2
References
329