Soliton Equations and their Algebro-Geometric Solutions: Volume 1, (1+1)-Dimensional Continuous Models
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.
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Abel map AKNS hierarchy Alber algebraic algebro-geometric initial value algebro-geometric solutions analogous Appendix associated Assume Hypothesis assume the affine asymptotic auxiliary divisors Baker-Akhiezer vector branch points classical massive Thirring coefficients computes constraints context corresponding Darboux transformation defined denotes derive differential equations differential expression Dirichlet discussed Dubrovin equations eigenvalues elliptic elliptic function equivalent explicit Fn(z genus Green's function hence Herglotz homogeneous hyperelliptic curve implies infers initial value problem integration constants introduce isospectral KdV equation KdV hierarchy KdV potentials KdV solutions Lax pairs Lemma linear matrix meromorphic function Moreover nonlinear nonsingular nonspecial notation Novikov nth stationary open and connected polynomial Proof proves real-valued recursion relation Remark respect Riemann theta function Riemann-Roch theorem satisfies the nth Schrodinger self-adjoint singular soliton suppose symmetric functions Theorem theta function representations time-dependent trace formulas yields zero-curvature zeros
Page 1 - It often happens that the understanding of the mathematical nature of an equation is impossible without a detailed understanding of its solutions. The black hole is a case in point. One could say without exaggeration that Einstein's equations of general relativity were understood only at a very superficial level before the discovery of the black hole.