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. Fields of science natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations Keywords Egoroff metrics Frobenius manifolds Hamiltonian PDEs Painleve transcendents Whitham equations integrable systems theta-functions of Riemann surfaces Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Topic(s) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations 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 EU contribution € 864 000,00 Address VIA BONOMEA 265 34136 Trieste Italy See on map Region Nord-Est Friuli-Venezia Giulia Trieste Activity type Higher or Secondary Education Establishments Principal investigator Boris Dubrovin (Prof.) Administrative Contact Gabriele Rizzetto (Mr.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data Beneficiaries (1) Sort alphabetically Sort by EU Contribution Expand all Collapse all SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE Italy EU contribution € 864 000,00 Address VIA BONOMEA 265 34136 Trieste See on map Region Nord-Est Friuli-Venezia Giulia Trieste Activity type Higher or Secondary Education Establishments Principal investigator Boris Dubrovin (Prof.) Administrative Contact Gabriele Rizzetto (Mr.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data