The project FroM-PDE is funded by the European Research Council (ERC) through the Advanced Investigator Grant Scheme awarded to Boris Dubrovin (SISSA, Italy).

The basic idea of the project is to apply the methods and results of the theory of integrable systems to non-integrable Hamiltonian PDEs. In many cases solutions to a conservative PDE in certain strongly nonlinear regimes 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 PDEs and their Hamiltonian perturbations;

2) reduce the lists of asymptotic solutions to an abridged list of universal forms represented via Painleve 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 Painleve equations and their generalizations will be needed for analytic description of transitions from regular to oscillatory behaviour. The project is aiming at creation of an online library of main qualitative types 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, of integrability or almost integrability. Such a library will be orientated first of all towards applied mathematicians and physicists, but it will also be useful for researches in pure mathematics.

Principal Investigator

Boris Dubrovin
International School for Advanced Studies - SISSA

via Beirut 2-4, 34014 Trieste, Italy

Team Members

Tom Claeys (K.U.Leuven, Belgium)

Eugene Ferapontov (Loughborough Univ., UK)

Tamara Grava (SISSA, Italy)

Christian Klein (Univ. Dijon, France)

External collaborators

Prof. Marco Bertola (Concordia University, Montreal, Canada)

Prof. Youjin Zhang (Tshingua University, Beijing, China)

Hired researchers

Par Joel Ekstrand (SISSA, Italy)

Robert Jenkins (SISSA, Italy)

Andrey Kapaev (SISSA, Italy)

Antonio Moro (SISSA, Italy)

Kristelle Roidot (SISSA, Italy)

Daniele Valeri (SISSA, Italy)

PhD Students

Matteo Casati (SISSA, Italy)

Dario Merzi (SISSA, Italy)

Duration

The programme has a duration of five years, from January 2009 until December 2013.

Planned Activities

The programme includes the following planned activities:

• The organization of three workshops in 2009, 2010 and 2013 respectively.
• The hiring of two post-docs for 3 years each and one research associate for 5 years.