Reducible Mcurves for Lenetworks in the totallynonnegative Grassmannian and KPII multiline solitons
Abstract
We associate real and regular algebraicgeometric data to each multiline soliton solution of KadomtsevPetviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally nonnegative part of real Grassmannians $Gr^{TNN}(k,n)$. In Ref.[3] we were able to construct real algebraicgeometric data for soliton data in the main cell $Gr^{TP} (k,n)$ only. Here we do not just extend that construction to all points in $Gr^{TNN}(k,n)$, but we also considerably simplify it, since both the reducible rational $M$curve $\Gamma$ and the real regular KP divisor on $\Gamma$ are directly related to the parametrization of positroid cells in $Gr^{TNN}(k,n)$ via the Lenetworks introduced by A. Postnikov in Ref [62]. In particular, the direct relation of our construction to the Lenetworks guarantees that the genus of the underlying smooth $M$curve is minimal and it coincides with the dimension of the positroid cell in $Gr^{TNN}(k,n)$ to which the soliton data belong to. Finally, we apply our construction to soliton data in $Gr^{TP}(2,4)$ and we compare it with that in Ref [3].
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.05641
 Bibcode:
 2018arXiv180505641A
 Keywords:

 Mathematical Physics;
 37K40;
 37K20;
 14H50;
 14H70
 EPrint:
 72 pages