Rational degeneration of M-curves, totally positive Grassmannians and KP2-solitons

We establish a new connection between the theory of totally positive Grassmannians and the theory of $\mathtt M$-curves using the finite--gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev-Petviashvili 2 equation, which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian $Gr^{TP} (N,M)$ a reducible curve which is a rational degeneration of an $\mathtt M$--curve of minimal genus $g=N(M-N)$, and we reconstruct the real algebraic-geometric data \'a la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth $M$-curves. In our approach we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection $Gr^{TP} (r+1,M-N+r+1)\mapsto Gr^{TP} (r,M-N+r)$.Comment: 49 pages, 10 figures. Minor revision

Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

We associate real and regular algebraic--geometric data to each multi--line soliton solution of Kadomtsev-Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non--negative part of real Grassmannians $Gr^{TNN}(k,n)$. In Ref.[3] we were able to construct real algebraic-geometric 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 Le-networks introduced by A. Postnikov in Ref [62]. In particular, the direct relation of our construction to the Le--networks 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].Comment: 72 pages; several figures. We have decided to split our paper in Arxiv:1801.00208v1 into two parts. This preprint is the fully revised version of the first part of it. In the next version Arxiv:1801.00208 this part will be removed V2: Minor modifications, proof of Theorem 3.1 improve

Real regular KP divisors on M-curves and totally non-negative Grassmannians

In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell STNN M in the totally non-negative Grassmannian GrTNN(k, n) to the spectral data for the relevant class of real regu lar Kadomtsev–Petviashvili II (KP-II) solutions, thus completing the search of real algebraic-geometric data for the KP-II equation started in Abenda and Grinevich (Commun Math Phys 361(3):1029–1081, 2018; Sel Math New Ser 25(3):43, 2019). The spectral curve is modeled on the Krichever construction for degenerate finite-gap solutions and is a rationally degenerate M-curve, , dual to the graph. The divisors are real regular KP-II divisors in the ovals of , i.e. they fulfill the conditions for selecting real regular finite-gap KP-II solutions in Dubrovin and Natanzon (Izv Akad Nauk SSSR Ser Mat 52:267–286, 1988). Since the soliton data are described by points in STNN M , we establish a bridge between real regular finite-gap KP-II solutions (Dubrovin and Natanzon, 1988) and real regular multi-line KP-II solitons which are known to be parameterized by points in GrTNN(k, n) (Chakravarty and Kodama in Stud Appl Math 123:83–151, 2009; Kodama and Williams in Invent Math 198:637–699, 2014). We use the geometric characterization of spaces of relations on plabic networks intro duced in Abenda and Grinevich (Adv Math 406:108523, 2022; Int Math Res Not 2022:rnac162, 2022. https://doi.org/10.1093/imrn/rnac162) to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in Lam (in: Current developments in mathematics, International Press, Somerville, 2014) for the computation of scattering amplitudes for on-shell diagrams N = 4 SYM (Arkani-Hamed et al. in Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, 2016) and govern the totally non-negative amalgamation of the little positive Grassmannians, GrTP(1, 3) and GrTP(2, 3), into any given positroid cell STNN M ⊂ GrTNN(k, n). In our set ting they control the reality and regularity properties of the KP-II divisor. Finally, we explain the transformation of both the curve and the divisor both under Postnikov’s moves and reductions and under amalgamation of positroid cells, and apply our con struction to some examples

Periodic billiard orbits on nn--dimensional ellipsoids with impacts on confocal quadrics and isoperiodic deformations

In our paper we study periodic geodesic motion on multidimensional ellipsoids with elastic impacts along confocal quadrics. We show that the method of isoperiodic deformation is applicable.Comment: Latex, 28 pages, 3 figure

Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

We associate real and regular algebraic\u2013geometric data to each multi-line soliton solution of Kadomtsev\u2013Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non-negative part of real Grassmannians GrTNN(k, n). In ref. 3 (Abenda, Grinevich, CMP 2018) we were able to construct real algebraic\u2013geometric data for soliton data in the main cell GrTP(k, n) only. Here we do not just extend that construction to all points in GrTNN(k, n), but we also considerably simplify it, since both the reducible rational M-curve and the real regular KP divisor on are directly related to the parametrization of positroid cells in GrTNN(k, n) via the Le-networks introduced in ref. 63 (A. Postnikov). In particular, the direct relation of our construction to the Le-networks guarantees that the genus of the underlying smooth M-curve is minimal and it coincides with the dimension of the positroid cell in GrTNN(k, n) to which the soliton data belong to. Finally, we apply our construction to soliton data in GrTP(2, 4) and we compare it with that in Ref. 3

Rational Degenerations of M-Curves, Totally Positive Grassmannians and KP2-Solitons

We establish a new connection between the theory of totally positive Grassmannians and the theory of M-curves using the finite-gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev\u2013Petviashvili 2 equation [see (1)], which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian GrTP(N,M) a reducible curve which is a rational degeneration of an M-curve of minimal genus g=N(M 12N), and we reconstruct the real algebraic-geometric data \ue1 la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction, it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth M-curves. In our approach, we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection GrTP(r+1,M 12N+r+1)\u21a6GrTP(r,M 12N+r

Real Soliton Lattices of the Kadomtsev Petviashvili II Equation and Desingularization of Spectral Curves: The GrTP(2, 4) Case

We apply the general construction developed in our previous papers to the first nontrivial case of GrTP(2, 4). In particular, we construct finite-gap real quasi-periodic solutions of the KP-II equation in the form of a soliton lattice corresponding to a smooth M-curve of genus 4 which is a desingularization of a reducible rational M-curve for soliton data in GrTP(2, 4