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

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