Geodesic
On the local coordinates on the manifold of finite-gap solutions of the KdV equation
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 23, Tome 210 (1994), pp. 47-56 
R. F. Bikbaev. On the local coordinates on the manifold of finite-gap solutions of the KdV equation. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 23, Tome 210 (1994), pp. 47-56. http://geodesic.mathdoc.fr/item/ZNSL_1994_210_a4/
@article{ZNSL_1994_210_a4,
     author = {R. F. Bikbaev},
     title = {On the local coordinates on the manifold of finite-gap solutions of the {KdV} equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {47--56},
     year = {1994},
     volume = {210},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1994_210_a4/}
}
TY  - JOUR
AU  - R. F. Bikbaev
TI  - On the local coordinates on the manifold of finite-gap solutions of the KdV equation
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 1994
SP  - 47
EP  - 56
VL  - 210
UR  - http://geodesic.mathdoc.fr/item/ZNSL_1994_210_a4/
LA  - ru
ID  - ZNSL_1994_210_a4
ER  - 
%0 Journal Article
%A R. F. Bikbaev
%T On the local coordinates on the manifold of finite-gap solutions of the KdV equation
%J Zapiski Nauchnykh Seminarov POMI
%D 1994
%P 47-56
%V 210
%U http://geodesic.mathdoc.fr/item/ZNSL_1994_210_a4/
%G ru
%F ZNSL_1994_210_a4

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A theorem is proved about nondegeneracy of the map $$ (E_1<E_2<\dots<E_{2g+1})\to(V,W,c), $$ where $E_i$ are the branching points of the hyperelliptic curve $\Gamma$, which corresponds to the finite-gap solution of KdV equation $u_g(x,t)$. Here $V,W$ are frequency vectors and $c$ is the “mean value” of the potential $u_g(x,t)$. The bijectivity of this map for $g=1$ is proved. Complex generalization of the nondegeneracy result is proved. Bibliography: 11 titles.