|
Nonlinear Evolution Equations and Hyperelliptic Covers of Elliptic Curves
A. Treibich Investigador PEDECIBA, Centro de Matemática, Universidad de la República, Montevideo, Uruguay
Аннотация:
This paper is a further contribution to the study of exact solutions to
KP, KdV, sine-Gordon, 1D Toda and nonlinear Schrodinger equations. We will
be uniquely concerned with algebro-geometric solutions, doubly periodic in
one variable. According to (so-called) Its-Matveev's formulae, the
Jacobians of the corresponding spectral curves must contain an elliptic
curve X, satisfying suitable geometric properties. It turns out that the
latter curves are in fact contained in a particular algebraic surface
$S\perp$, projecting onto a rational surface $\widetilde S$. Moreover, all
spectral curves project onto a rational curve inside $\widetilde S$. We
are thus led to study all rational curves of $\widetilde S$, having
suitable numerical equivalence classes. At last we obtain
$d\,$-$\,1$-dimensional of spectral curves, of arbitrary high genus,
giving rise to KdV solutions doubly periodic with respect to the $d$-th
KdV flow ($d\geq 1$). Analogous results are presented, without proof, for
the 1D Toda, NL Schrodinger an sine-Gordon equation.
Ключевые слова:
elliptic and hyperelliptic curves, Jacobian variety, ruled and rational surfaces, exceptional curve, elliptic soliton.
Поступила в редакцию: 04.03.2010 Принята в печать: 26.08.2010
Образец цитирования:
A. Treibich, “Nonlinear Evolution Equations and Hyperelliptic Covers of Elliptic Curves”, Regul. Chaotic Dyn., 16:3-4 (2011), 290–310
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd439 https://www.mathnet.ru/rus/rcd/v16/i3/p290
|
Статистика просмотров: |
Страница аннотации: | 69 |
|