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The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2017, том 13
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The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
Інший ID:
2010 Mathematics Subject Classification: 58A05; 58A20; 35A30
DOI:10.3842/SIGMA.2017.003
DOI:10.3842/SIGMA.2017.003
Посилання:
The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs / B. Güneysu, M.J. Pflaum // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ.
Підтримка:
The first named author (B.G.) is indebted to W.M. Seiler for many discussions on jet bundles,
and would also like to thank B. Kruglikov and A.D. Lewis for helpful discussions. B.G. has
been financially supported by the SFB 647: Raum–Zeit–Materie, and would like to thank the
University of Colorado at Boulder for its hospitality. The second named author (M.P.) has been
partially supported by NSF grant DMS 1105670 and by a Simons Foundation collaboration
grant, award nr. 359389. M.P. would also like to thank Humboldt-University, Berlin and the
Max-Planck-Institute for Mathematics of the Sciences, Leipzig for their hospitality. Last but not
least the authors thank the anonymous referees for constructive advice which helped to improve
the paper.
Дата:
2017
Переглядів:
519
Завантажень:
200
Анотація:
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.
