The focus of this book is on algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exhaustive bibliography enhance the presentation offered in the main text.
Introduction; 1. The KdV hierarchy; 2. The sGmKdV hierarchy; 3. The AKNS hierarchy; 4. The classical massive Thirring system; 5. The Camassa-Holm hierarchy; Appendix A. Algebraic curves and their theta functions; Appendix B. KdV-type curves; Appendix C. AKNS-type curves; Appendix D. Asymptotic spectral parameter expansions; Appendix E. Lagrange interpolation; Appendix F. Symmetric functions; Appendix G. KdV and AKNS Darboux-type transformations; Appendix H. Elliptic functions; Appendix I. Herglotz functions; Appendix J. Weyl-Titchmarsh theory; List of symbols; Bibliography; Index.