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Frobenius Manifolds and Hamiltonian Partial Differential Equations

Objective

The basic idea of the project is to apply methods and results of the theory of integrable systems to non-integrable PDEs. We do not promise to solve any PDE; however, in certain strongly nonlinear regimes, solutions to a conservative non-integrable PDE exhibit integrable behaviour. The realization of this idea, supported by some preliminary analytical and numerical results, will consist of three main tasks: 1) classify normal forms of quasilinear Hamiltonian PDEs and their perturbations; 2) reduce the lists of asymptotic solutions to an abridged list of universal forms represented via Painlevé transcendents, theta-functions, etc.; 3) establish matching rules between the universal asymptotic expansions. Differential-geometric methods based on the theory of Frobenius manifolds will be crucial in solving the classification problems; analytic and algebro-geometric techniques applied to the Hurwitz spaces of Riemann surfaces will be instrumental in the description of nonlinear oscillatory regimes; selected solutions to Painlevé equations and their generalizations will be needed for the analytic description of transitions from regular to oscillatory behaviour. The project is aiming at creation of an online library of the main qualitative types of behaviour of solutions to large classes of nonlinear evolutionary PDEs supplied with analytic expressions, numerical codes and visualization tools, as well as with tests of existence of a Hamiltonian structure, integrability or almost integrability. Such a library will both stimulate the research in the field and lead to a high visibility of the project.

Field of science

  • /natural sciences/mathematics/pure mathematics/mathematical analysis/differential equations/partial differential equations

Call for proposal

ERC-2008-AdG
See other projects for this call

Funding Scheme

ERC-AG - ERC Advanced Grant

Host institution

SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Address
Via Bonomea 265
34136 Trieste
Italy
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 864 000
Principal investigator
Boris Dubrovin (Prof.)
Administrative Contact
Gabriele Rizzetto (Mr.)

Beneficiaries (1)

SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE
Italy
EU contribution
€ 864 000
Address
Via Bonomea 265
34136 Trieste
Activity type
Higher or Secondary Education Establishments
Principal investigator
Boris Dubrovin (Prof.)
Administrative Contact
Gabriele Rizzetto (Mr.)