Verdier elliptic solitons and the Weierstrass theory of reduction

1989 ◽  
Vol 23 (1) ◽  
pp. 46-47 ◽  
Author(s):  
E. D. Belokolos ◽  
V. Z. �nol'skii
Keyword(s):  
Elliptic Solitons ◽  
Weierstrass Theory

scholarly journals Elliptic solitons and Gröbner bases

2004 ◽  
Vol 45 (2) ◽  
pp. 696-712 ◽  
Author(s):  
Yurii V. Brezhnev
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Elliptic Solitons

scholarly journals Elliptic solitons in optical fiber media

2015 ◽  
Vol 92 (5) ◽  
Author(s):  
Défi Jr. Fandio Jubgang ◽  
Alain M. Dikandé ◽  
A. Sunda-Meya
Keyword(s):  
Optical Fiber ◽  
Elliptic Solitons

Hyperelliptic d-Tangential Covers and d× d-Matrix KdV Elliptic Solitons

2018 ◽  
Vol 2020 (23) ◽  
pp. 9539-9558
Author(s):  
Armando Treibich
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Kdv Equation ◽  
Rational Surface ◽  
Geometric Condition ◽  
Kp Equation ◽  
Nodal Curves ◽  
Severi Varieties ◽  
The Kdv Equation ◽  
Elliptic Solitons

Abstract More than $40$ years ago I. Krichever developed the Theory of (vector) Baker–Akhiezer functions and devised a criterion for a $d$-marked compact Riemann surface to provide $d\times d$-matrix solutions to the KdV equation. Later on he also found a criterion for a $d$-marked curve to provide $d\times d$-matrix solutions to the Kadomtsev-Petviashvili (KP) equation, doubly periodic with respect to $x$, the 1st KP flow. In particular, when both criteria apply, one should obtain $d\times d$-matrix KdV elliptic solitons. It seems, however, that the latter issue has been completely neglected until very recently (cf. [10] where the $d=2$ case is treated). In this article we fix a complex elliptic curve $X=\mathbb{C}/\Lambda$, corresponding to a lattice $\Lambda \subset \mathbb{C}$, and define so-called hyperelliptic $d$-tangential covers as $d$-marked covers of $X$ satisfying a geometric condition inside their Jacobians. They satisfy Krichever’s criteria and give rise, therefore, to families of $d\times d$-matrix KdV elliptic solitons. We also construct an anticanonical rational surface ${\mathcal S}$ naturally attached to $X$, with a Picard group of rank $10$. It turns out that the former covers of $X$ correspond to rational irreducible curves in suitable divisor classes of ${\mathcal S}$. We thus reduce their construction to proving that the associated Severi Varieties (of rational irreducible nodal curves) are not empty. The final key to the problem consists in finding rational reducible nodal curves in the latter divisor classes that can be deformed to irreducible ones, according to A. Tannenbaum’s criterion (see [5]). At last we deduce, for any $d\geq 2$, infinite families (of arbitrary high genus and degree) of hyperelliptic $d$-tangential covers, giving rise to $d\times d$-matrix KdV elliptic solitons.

Sign in / Sign up

Export Citation Format

Share Document