**Memoirs of the American Mathematical Society**

1998;
216 pp;
Softcover

MSC: Primary 58;

Print ISBN: 978-0-8218-0691-3

Product Code: MEMO/131/624

List Price: $63.00

Individual Member Price: $37.80

**Electronic ISBN: 978-1-4704-0213-6
Product Code: MEMO/131/624.E**

List Price: $63.00

Individual Member Price: $37.80

# A Continuum Limit of the Toda Lattice

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*P. Deift; K. T-R McLaughlin*

In this book, the authors describe a continuum limit of the Toda ODE system, obtained by taking as initial data for the finite lattice successively finer discretizations of two smooth functions. Using the integrability of the finite Toda lattice, the authors adapt the method introduced by Lax and Levermore for the study of the small dispersion limit of the Korteweg de Vries equations to the case of the Toda lattice. A general class of initial data is considered which permits, in particular, the formation of shocks. A novel feature of the analysis in this book is an extensive use of techniques from the theory of Riemann-Hilbert problems.

#### Readership

Graduate students and research mathematicians working in completely integrable systems.

#### Table of Contents

# Table of Contents

## A Continuum Limit of the Toda Lattice

- Contents vii8 free
- Acknowledgments ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Analysis of Log formula 1122 free
- Chapter 3. An Example 1829
- Chapter 4. Monotone Initial Data 3647
- Chapter 5. Shock 1 5263
- Chapter 6. Shock 2 7990
- Chapter 7. Shock 3 93104
- Chapter 8. Shock 4 110121
- Chapter 9. Symmetric data 136147
- Chapter 10. Global Description 146157
- Chapter 11. Large time calculations 162173
- Chapter 12. Appendix I - WKB 170181
- 1. Introduction 170181
- 2. A Theorem of Geronimo and Smith 173184
- 3. Exponential Behavior 175186
- 4. Exponential Behavior: Turning Points 179190
- 5. Oscillatory Behavior 185196
- 6. Oscillatory Behavior - Turning Points 187198
- 7. Turning points - special functions 192203
- 8. Matching and Eigenvalues 202213

- Chapter 13. Appendix II 213224
- Bibliography 216227