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We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemanns theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve.
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We investigate from a mathematical perspective how Feynman amplitudes appear in the low-energy limit of string amplitudes. In this paper, we prove the convergence of the integrands. We derive this from results describing the asymptotic behavior of the height pairing between degree-zero divisors, as a family of Riemann surfaces degenerates. These are obtained by means of the nilpotent orbit theorem in Hodge theory.
Using the determinant representation of gauge transformation operator, we have shown that the general form of $tau$ function of the $q$-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP ($q$-cKP) hierarchy, i.e. $l$-constraints of $q$-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of $q$-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear $q$-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of $q$-deformation ($q$-effects) in single $q$-soliton from the simplest $tau$ function of the $q$-KP hierarchy and in multi-$q$-soliton from one-component $q$-cKP hierarchy, and their dependence of $x$ and $q$, were also presented. Finally, we observe that $q$-soliton tends to the usual soliton of the KP equation when $xto 0$ and $qto 1$, simultaneously.
In the recent paper (R. Willox and M. Hattori, arXiv:1406.5828), an integrable discretization of the nonlinear Schrodinger (NLS) equation is studied, which, they think, was discovered by Date, Jimbo and Miwa in 1983 and has been completely forgotten over the years. In fact, this discrete NLS hierarchy can be directly obtained from an elementary auto-Backlund transformation for the continuous NLS hierarchy and has been known since 1982. Nevertheless, it has been rediscovered again and again in the literature without attribution, so we consider it meaningful to mention overlooked original references on this discrete NLS hierarchy.
We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Malcev-Newmann division ring. The action is constructed from the non-Abelian Hirota-Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.
We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic geometry. We verify this methods via numerical experiments with curves up to genus 7.
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