Generalized Twister Correspondences, d-Bar Problems, and the KP Equations

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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Dispersive Estimates for Full Dispersion KP Equations

Journal of Mathematical Fluid Mechanics ◽

10.1007/s00021-021-00557-3 ◽

2021 ◽

Vol 23 (1) ◽

Author(s):

Didier Pilod ◽

Jean-Claude Saut ◽

Sigmund Selberg ◽

Achenef Tesfahun

Keyword(s):

Stationary Phase ◽

Initial Value Problem ◽

Bessel Functions ◽

Linear Part ◽

Independent Interest ◽

Phase Method ◽

Stationary Phase Method ◽

Initial Value ◽

Kp Equations ◽

Well Posed

AbstractWe prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev–Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev–Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev–Petviashvili is locally well-posed in $$H^s(\mathbb R^2)$$ H s ( R 2 ) , for $$s>\frac{7}{4}$$ s > 7 4 , in the capillary-gravity setting.

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Symmetries of the higher‐order KP equations

Journal of Mathematical Physics ◽

10.1063/1.526516 ◽

1985 ◽

Vol 26 (6) ◽

pp. 1158-1159 ◽

Cited By ~ 14

Author(s):

K. M. Case

Keyword(s):

Higher Order ◽

Kp Equations

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THE EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS IN THE GARDNER AND GARDNER–KP EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741250126x ◽

2012 ◽

Vol 22 (05) ◽

pp. 1250126 ◽

Cited By ~ 3

Author(s):

FANG YAN ◽

CUNCAI HUA ◽

HAIHONG LIU ◽

ZENGRONG LIU

Keyword(s):

Traveling Wave ◽

Function Method ◽

Traveling Wave Solutions ◽

Analytic Solutions ◽

Auxiliary Function ◽

Kp Equation ◽

Gardner Equation ◽

Exact Traveling Wave Solutions ◽

Wave Solutions ◽

Kp Equations

By using the method of dynamical systems, this paper studies the exact traveling wave solutions and their bifurcations in the Gardner equation. Exact parametric representations of all wave solutions as well as the explicit analytic solutions are given. Moreover, several series of exact traveling wave solutions of the Gardner–KP equation are obtained via an auxiliary function method.

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