INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS Inverse Problems 17 (2001) 1–26 www.iop.org/Journals/ip PII: S0266-5611(01)16386-0 Processing IP/116386/SPE Printed 3/1/2001 Focal Image CRC data File name IP .TEX First page Date req. Last page Issue no. Total pages Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians Mark S Alber 1,2 and Yuri N Fedorov 3 1 Department of Mathematics, Stanford University, Building 380, MC 2125, Stanford, CA 94305, USA 2 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA 3 Department of Mathematics and Mechanics, Moscow Lomonosov University, Moscow 119 899, Russia E-mail: malber@math.stanford.edu, Mark.S.Alber.1@nd.edu and fedorov@mech.math.msu.su Received 15 August 2000 Abstract Algebraic geometrical solutions of a new shallow-water equation and Dym- type equation are studied in connection with Hamiltonian flows on nonlinear subvarieties of hyperelliptic Jacobians. These equations belong to a class of N -component integrable systems generated by Lax equations with energy- dependent Schr¨ odinger operators having poles in the spectral parameter. The classes of quasi-periodic and soliton-type solutions of these equations are described in terms of theta- and tau-functions by using new parametrizations. A qualitative description of real-valued solutions is provided. 1. Introduction The quasi-periodic and soliton solutions of many well known nonlinear equations such as KdV, sine–Gordon and focusing and defocusing nonlinear Schr ¨ odinger equations, which describe a wide variety of important phenomena in physics, biology and engineering, have been studied by applying the algebraic geometric approach. By using the trace formula, families of solutions of such equations can be associated with Hamiltonian flows on invariant sets in finite-dimensional phase spaces. These flows can be described by using the so-called µ-variable representation, leading to an Abel–Jacobi mapping which includes holomorphic and, in some cases, meromorphic differentials (see, amongst others, Its and Kotlyarov 1976, Alber 1979, Dubrovin 1981, Dubrovin et al 1985, Mumford 1983, Novikov et al 1984). The mapping can be inverted in terms of Riemann theta-functions and their degenerations. Recently special attention was given to the shallow-water (SW) equation derived in Camassa and Holm (1993) in the context of the Hamiltonian structure, U t +3UU x = U xxt +2U x U xx + UU xxx - 2κ U x , (1.1) 0266-5611/01/000001+26$30.00 © 2001 IOP Publishing Ltd Printed in the UK 1