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

Modulation of Camassa--Holm equation and reciprocal transformations

We derive the modulation equations or Whitham equations for the Camassa--Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot

Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk

Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non--negative part of real Grassmannians (see Refs. [54,43,44,58,7]). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Ref. [5]. We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Refs. [4,6] for the present class of graphs.Comment: 47 pages, many figures; V2 and V3: minor modification

NANDAI PADA ETNIK SERAWAI DI KABUPATEN SELUMA SEBAGAI SUMBER PEMBELAJARAN APRESIASI SASTRA LAMA

This study aims (1) to inventory or collect marks on the Serawai ethnicity in Seluma Regency so that it can be a source of learning for the appreciation of old literature. (2) Finding and explaining the moral values ​​contained in the mark of the Serawai ethnicity in Seluma Regency. The method used in this research is a descriptive qualitative method. The data collection technique was carried out by (1) Observation (2) Defect technique (3) Recording (4) Documentation. Data Analysis Techniques (1) perform translation (2) find and explain moral values ​​(3) conclusions. Research Results (1). Until now, markers are still found in 5 sub-districts in Seluma district, although they are no longer told productively because it is difficult to find sources who can still mark, plus many have had health problems and died. The Nandai that was found was the Nandai ghenai which is a type of legend and fairy tale. (2) There are moral messages and values ​​that can be taken in the markings on the Serawai ethnicity such as individual moral values ​​(honest, patient, disciplined, and responsible) and social moral values ​​(respect for every human being, respect for women, respect for the opinions of others , loyal, polite, and true to promises). (3) Nandai can be used as a source of learning appreciation for old literature in Seluma district at the Junior High School, Senior High School, and General levels because it has two languages ​​(Serawai and Indonesian) and Glossary.Abstrak             Penelitian Ini bertujuan (1) Untuk inventarisasi atau mengumpulan nandai pada etnik Serawai di Kabupaten Seluma sehingga dapat menjadi sumber pembelajaran apresiasi sastra lama. (2) Menemukan dan menjelaskan nilai moral yang terkandung dalam nandai pada etnik Serawai di Kabupaten Seluma. Metode yang digunakan dalam penelitian ini metode deskriptif kualitatif. Teknik pengumpulan data dilakukan dengan teknik (1) Observasi (2) Teknik cacat (3) Perekaman (4) Dokumentasi. Teknik Analisis Data (1) melakukan terjemahan (2) menemukan dan menjelaskan Nilai moral (3) kesimpulan. Hasil Penelitian (1). sampai saat ini nandai masih ditemukan di 5 Kecamatan di kabupaten Seluma meskipun tidak lagi di ceritakan lagi secara produktif dikarekan sulit menemukan narasumber yang masih bisa bernandai ditambah sudah banyak yang terkendala kesehatan dan meninggal. Nandai yang ditemukan adalah nandai ghenai yang berjenis legenda dan dongeng. (2) Terdapat pesan dan nilai moral yang dapat diambil di dalam nandai pada etnik serawai seperti nilai moral individual (jujur, sabar, disiplin, dan tanggung jawab) dan nilai moral sosial (penghargaan setiap manusia, penghargaan terhadap perempuan, penghargaan terhadap pendapat orang lain,setia, sopan, dan tepat janji). (3) Nandai dapat digunakan sebagai sumber pembelajaran apresiasi sastra lama di kabupaten Seluma pada jenjang Sekolah Menengah Pertama, Sekolah Menengah Atas, dan Umum karena memiliki dua bahasa (bahasa Serawai dan bahasa Indonesia) dan glosarium.   Kata kunci : Nandai Etnik Serawai,Nilai Pembelajaran,Sumber Pembelajara

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--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