The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and dym type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.Comment: 31 pages, no figures, to appear in Commun. Math. Phy

On Soliton-type Solutions of Equations Associated with N-component Systems

The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.Comment: 43 pages, 16 figure

On Non Commutative Calabi-Yau Hypersurfaces

Using the algebraic geometry method of Berenstein et al (hep-th/0005087), we reconsider the derivation of the non commutative quintic algebra ${\mathcal{A}}_{nc}(5)$ and derive new representations by choosing different sets of Calabi-Yau charges ${C_{i}^{a}}$. Next we extend these results to higher $d$ complex dimension non commutative Calabi-Yau hypersurface algebras ${\mathcal{A}}_{nc}(d+2)$. We derive and solve the set of constraint eqs carrying the non commutative structure in terms of Calabi-Yau charges and discrete torsion. Finally we construct the representations of ${\mathcal{A}}_{nc}(d+2)$ preserving manifestly the Calabi-Yau condition $\sum_{i}C_{i}^{a}=0$ and give comments on the non commutative subalgebras.Comment: 16 pages, Latex. One more subsection on fractional branes, one reference and minor changes are added. To appear in Phy. Let.

Classical Functional Bethe Ansatz for SL(N)SL(N): separation of variables for the magnetic chain

The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which gives separation variables for systems for which an $R$-matrix is known. Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated) $R$-matrices. In this paper I advance Sklyanin's program by giving the FBA for certain systems with $SL(N)$ $R$-matrices. This is achieved by constructing rational functions \A(u) and \B(u) of the matrix elements of $T(u)$, so that, in the generic case, the zeros $x_i$ of \B(u) are the separation coordinates and the P_i=\A(x_i) provide their conjugate momenta. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed.Comment: 14pp LaTex,DAMTP 94-1