Objective
The main goal of the present project is to combine efforts of experts in nonlinear integrable equations in studying new algebraic and analytic aspects of integrable models and their physical applications.
The objectives of the project include investigation and classification of new classes of integrable lattice models together with their cut-offs and scaling, which preserve integrability; derivation and examination of similarity solutions for NLS-type systems, in particular optical solitons; extension of the theory of inverse scattering to multi-dimensional potentials not decaying to zero at large distances; application of the technique of the theory of integrable systems to several interesting problems in mathematics (differential geometry, discrete geometry) and physics (topological quantum field theory, string theory, Einstein equations).
The research activity under this project is split into seven tasks, which are closely connected and constructively interact with each other.
Task 1 is devoted to the development of the methods to construct and solve new classes of multidimensional integrable equations. Development of different versions of the inverse spectral transform method will be pursued. Particular attention will be paid to the generalized resolvent approach and to the D-bar dressing method. Their use will give new classes of solutions of some multidimensional equations and will allow better understanding of the general structure and properties of infinite hierarchies of integrable equations.
Task 2 consists of investigation and classification of new classes of integrable lattice models. Integrable equations with one or more discrete spatial variables, such as Toda lattice, play important role in the modern nonlinear science. A number of discrete integrable models have been discovered recently in quantum field theory (matrix models), statistical physics, cellular automata, discrete geometry, mathematical ecology and economics. A strict mathematical study of these models seems to be a hard and stimulating problem since the traditional technique of Inverse Scattering Transform (IST) is not well suited for the discrete space-time case.
Task 3 is connected with the development of symmetry type classification of nonlinear equations. Classification of nonlinear differential equations according to their symmetries is a very important field of research. A basic goal consists in extending the detailed symmetry classification of 1+1-dimensional differential equation to the 2+1-dimensional case. Symmetry approach to certain non-abelian ordinary differential equations and some particular classes of transformations will be studied.
Task 4 is to analyse concrete integrable equations important for physics and mathematics. Nonlinear differential equations important in various fields of classical and quantum physics will be studied.
Task 5 is application of the technique of soliton equations to the study of problems of differential geometry of continuous and discrete surfaces. The principal goal of this task is to reveal and study new promising interrelations between differential geometry and theory of integrable equations. Weierstrass type formulae for surfaces in multidimensional spaces and corresponding integrable deformations will be constructed. Discrete surfaces and nets and their relations with discrete integrable equations will be studied. One of the goals is to apply ideas of Projective and Finsler geometries to an analysis of ordinary differential equations.
Development of the Hamiltonian formalism for integrable equations constitutes Task 6.
Bihamiltonian character of soliton equations has been one of the important discoveries in this field. Development of bi-hamiltonian ideas using different formalisms and their applications to various integrable systems is the principal goal of this task.
Finally, Task 7 suggests the study of integrable structures which arise in quantum field theory, string theory, matrix models. Deeper and wider application of integrability ideas in matrix models associated with two-dimensional quantum gravity, models of quantum field theory, in different versions of string theory and in Einstein equations of classical gravity is a goal of great importance. Different techniques to reach such a goal will be used.
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Programme(s)
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Topic(s)
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Funding Scheme
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Coordinator
73100 Lecce
Italy
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.