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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