Contents

XI

11.6. The double commutation method 198

11.7. Iteration of the double commutation method 204

11.8. The Dirichlet deformation method 207

Notes on literature 215

Part 2. Completely Integrable Nonlinear Lattices

Chapter 12. The Toda system 221

12.1. The Toda lattice 221

12.2. Lax pairs, the Toda hierarchy, and hyperelliptic curves 224

12.3. Stationary solutions 231

12.4. Time evolution of associated quantities 234

Chapter 13. The initial value problem for the Toda system 239

13.1. Finite-gap solutions of the Toda hierarchy 239

13.2. Quasi-periodic finite-gap solutions and the time-dependent Baker-

Akhiezer function 245

13.3. A simple example - continued 248

13.4. Inverse scattering transform 249

13.5. Some additions in case of the Toda lattice 252

13.6. The elliptic case - continued 254

Chapter 14. The Kac-van Moerbeke system 255

14.1. The Kac-van Moerbeke hierarchy and its relation to the Toda

hierarchy 255

14.2. Kac and van Moerbeke's original equations 261

14.3. Spectral theory for supersymmetric Dirac-type difference operators 262

14.4. Associated solutions 263

14.5. iV-soliton solutions on arbitrary background 265

Notes on literature 271

Appendix A. Compact Riemann surfaces - a review 273

A.l. Basic notation 273

A.2. Abelian differentials 275

A.3. Divisors and the Riemann-Roch theorem 277

A.4. Jacobian variety and Abel's map 281

A.5. Riemann's theta function 284

A.6. The zeros of the Riemann theta function 286

A.7. Hyperelliptic Riemann surfaces 289

Appendix B. Herglotz functions 297

Appendix C. Jacobi Difference Equations with Mathematica 307