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This article is cited in 7 scientific papers (total in 7 papers)
Hamiltonian PDEs and Frobenius manifolds
B. A. Dubrovinab a Steklov Mathematical Institute, Russian Academy of Sciences
b International School for Advanced Studies (SISSA)
Abstract:
In the first part of this paper the theory of Frobenius manifolds is applied to the problem of classification of Hamiltonian systems of partial differential equations depending on a small parameter. Also developed is a deformation theory of integrable hierarchies including the subclass of integrable hierarchies of topological type. Many well-known examples of integrable hierarchies, such as the Korteweg–de Vries, non-linear Schrödinger, Toda, Boussinesq equations, and so on, belong to this subclass that also contains new integrable hierarchies. Some of these new integrable hierarchies may be important for applications. Properties of the solutions to these equations are studied in the second part. Consideration is given to the comparative study of the local properties of perturbed and unperturbed solutions near a point of gradient catastrophe. A Universality Conjecture is formulated describing the various types of critical behaviour of solutions to perturbed Hamiltonian systems near the point of gradient catastrophe of the unperturbed solution.
Received: 01.09.2008
Citation:
B. A. Dubrovin, “Hamiltonian PDEs and Frobenius manifolds”, Russian Math. Surveys, 63:6 (2008), 999–1010
Linking options:
https://www.mathnet.ru/eng/rm9242https://doi.org/10.1070/RM2008v063n06ABEH004575 https://www.mathnet.ru/eng/rm/v63/i6/p7
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Abstract page: | 1237 | Russian version PDF: | 447 | English version PDF: | 36 | References: | 114 | First page: | 39 |
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