Contact Integrable Extensions of Symmetry Pseudo-Group and Coverings of the r-th Modified Dispersionless Kadomtsev -- Petviashvili Equation

Contact Integrable Extensions of Symmetry Pseudo-Group and Coverings for the r-th Double Modified Dispersionless Kadomtsev--Petviashvili Equation

We find contact integrable extensions and coverings for the r-th double modified dispersionless Kadomtsev--Petviashvili equation.

Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients wh…

具有非形式拟微分算子的 Kadomtsev-Petviashvili 层级、非形式解，以及一种类似 Yang-Mills 的表述

Kadomtsev-Petviashvili hierarchies with non-formal pseudo-differential operators, non-formal solutions, and a Yang-Mills--like formulation

本文基于非形式拟微分算子构造了两个 Kadomtsev-Petviashvili 非线性方程层级，证明了相关 Zakharov-Shabat 方程成立，将其表示为 Yang-Mills 作用泛函的最小化，并比较了它们产生的 KP-II 方程解的存在性、唯一性与形式性。

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator for our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral oper…

Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and II

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are constant on a central quadric. We show how the solutions to the dKP equations can be used to construct some three-dimensional Einstein--Weyl structures, and four--dimensional anti-self-dual null-Kähler metr…

Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian

We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian.…