Vertex operators of the KP hierarchy and singular algebraic curves
The paper relates two classes of solutions to the KP hierarchy:\N\begin{itemize}\N\item[1.] Soliton solutions, those regular in the plane and localised along rays, and\N\item[2.] Quasi-periodic solutions, those expressed in terms of theta functions an a compact connected Riemann surface of genus \(g > 0\).\N\end{itemize}\N\NTo any such Riemann surface \(C\) there is a solution to KP hierarchy \(\tau_0(t)\) (defined on page 8), known to exist since the 1980s. Vertex operators (defined on page 8) create new solutions to the KP hierarchy from old (Theorem 3.1); combining these operators and \(\tau_0\) the authors construct a new solution to the KP hierarchy \(\tau(t)\) (Equation 3.3). In Section 4 it is explained that \(\tau(t)\) corresponds to a soliton solution, and this is verified by direct numerical computation and plotting in the case \(g=1\).\N\NThere is a well known correspondence between points of the Sato Grassmannian and solutions of the KP hierarchy, a brief background of which is given in Section 2. The key result of the paper (Theorem 3.8) is to identify the point \(U_e\) in the Sato Grassmannian which corresponds to \(\tau(-t)\) with a vector space built from the zeroeth cohomology of a certain line bundle over \(C\) (defined on page 11).\N\NSection 5 constructs a singular algebraic curve \(C^\prime\) (defined on page 18) from the data of \(U_e\) via abstract algebro-geometric methods. It is shown that \(C\) is the normalisation of \(C^\prime\) (Proposition 5.8), and that conversely \(C^\prime\) can be constructed from \(C\) by identifying together certain points (Proposition 5.8). \(C^\prime\) is thus interpreted as some singular limit of \(C\).\N\NSection 1 of the paper outlines the core ideas, proof strategies, and provides a historical context within which to situate the results. Most proofs are relegated to appendices in aid of readability.
scientific article; zbMATH DE number 7880229
scientific article