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Well-Posedness of the Kadomtsev–Petviashvili Hierarchy, Mulase Factorization, and Frölicher Lie Groups

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Abstract

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 Mulase’s deep algebraic 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 which are series in formal parameters, we describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and we discuss how solutions depending on formal parameters can lead to sequences of functions converging to a class of solutions of the standard KP-II equation.

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Notes

  1. As explained in the previous paragraphs, our favoured setting is the category of Frölicher spaces. This category is a full subcategory of the category of diffeological spaces, and therefore it is natural to begin by considering the latter spaces, as we do in Sect. 2.

  2. Jets are considered in [54], in the classical paper [7] and in the more recent review [47]; we note that in [54] the author introduces jets within the category DS of differential spaces, but it is known that Frölicher spaces form a subcategory of DS, see [32]. Moreover, as explained in Sect. 2.5 (see also [19, 32]) the category of Frölicher spaces is Cartesian closed, complete and cocomplete, and so infinite jets in Frölicher spaces can be defined by adapting the constructions of [7, 47].

  3. The following example was essentially suggested by the referee: let us fix a smooth function \(u_0:S^1 \rightarrow { \! \mathrm \ I\!K}\) and think of it as a (trivial) element of \(R_z = C^\infty (S^1,{ \! \mathrm \ I\!K})[[z]]\). Then, we obtain a unique solution \(u = \sum \sum f_n^q(x,qt_1,q^2y,q^3t, qt_4, \cdots )z^n q^k\) to the q-deformed KP-II Eq. (5.19), with y-dependence of u fixed by integration. On the other hand, let us choose a smooth function f on \(S^1\) with \(f(0)=0\) and such that \({\tilde{u}}_0(x,q^2 y) = q u_0(x)+ q f(q^2 y)\) is different from \(u(x,0,q^2 y,0,\cdots )\). Then, the solution to KP-II with initial data \((1/q){\tilde{u}}_0\) arising via Bourgain’s theorem [9] induces, as explained in Sect. 4.2, a solution \({\tilde{u}}\) to our q-deformed KP-II equation. The function \({\tilde{u}}\) cannot coincide with our solution u, even though they coincide at \(t=y=0\).

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Acknowledgements

Both authors have been partially supported by CONICYT (Chile) via the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) operating grant # 1161691.

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Authors and Affiliations

  1. LAREMA, Université d’Angers, 2 Bd Lavoisier, 49045, Angers Cedex 1, France

    Jean-Pierre Magnot

  2. Lycée Jeanne d’Arc, Avenue de Grande Bretagne, 63000, Clermont-Ferrand, France

    Jean-Pierre Magnot

  3. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile

    Enrique G. Reyes

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Dedicated to the memory of Professor Leonid Aleksandrovich Dickey.

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Magnot, JP., Reyes, E.G. Well-Posedness of the Kadomtsev–Petviashvili Hierarchy, Mulase Factorization, and Frölicher Lie Groups. Ann. Henri Poincaré 21, 1893–1945 (2020). https://doi.org/10.1007/s00023-020-00896-3

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