Partially Integrable Evolution Equations in Physics

ISBN-13: 9789401067546 / Angielski / Miękka / 2011 / 626 str.

R. Conte; N. Boccara

Opis

In the many physical phenomena ruled by partial differential equations, two extreme fields are currently overcrowded due to recent considerable developments: 1) the field of completely integrable equations, whose recent advances are the inverse spectral transform, the recursion operator, underlying Hamiltonian structures, Lax pairs, etc 2) the field of dynamical systems, often built as models of observed physical phenomena: turbulence, intermittency, Poincare sections, transition to chaos, etc. In between there is a very large region where systems are neither integrable nor nonintegrable, but partially integrable, and people working in the latter domain often know methods from either 1) or 2). Due to the growing interest in partially integrable systems, we decided to organize a meeting for physicists active or about to undertake research in this field, and we thought that an appropriate form would be a school. Indeed, some of the above mentioned methods are often adaptable outside their original domain and therefore worth to be taught in an interdisciplinary school. One of the main concerns was to keep a correct balance between physics and mathematics, and this is reflected in the list of courses.

Spis treści

I. Waves in Physical Systems.- Competing interactions and complexity in condensed matter.- Exact solutions of the Boltzmann equation.- Nonlinear interaction between short and long waves.- Nonlinear optics.- The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers I.- Nonlinear evolution equations, quasi-solitons and their experimental manifestation.- The initial-boundary value problem for the Davey-Stewartson 1 equation; how to generate and drive localized coherent structures in multidimensions.- II. Instabilities and Defects.- Ginzburg-Landau models of non-equilibrium.- Nonlinear Ginzburg-Landau equation and its application to fluid-mechanics.- Defects and disorder of nonlinear waves in convection.- III. Concepts Of Integrability and Singularity Analysis.- An introduction to Kowalevski’s exponents.- Singularity analysis and its relation to complete, partial and non-integrability.- A concept of integrability based on the symmetry approach.- Bäcklund transformations and the Painlevé property.- IV. Mathematical Methods.- Differential geometry techniques for sets of nonlinear partial differential equations.- Inertial manifolds and attractors of partial differential equations.- Hirota’s bilinear method and partial integrability.- Generalized symmetries, recursion operators and bihamiltonian systems.- Nonlinear dispersive equations without inverse scattering.- Group theory and exact solutions of partially integrable differential systems.- Contributed Papers.- The nonlinear evolution equation for the order parameter in superfluid Helium-Four.- Quasimonomial transformations and integrability.- Partial integrability of the damped kink equation.- New similarity reductions of Boussinesq-type equations.- The homographic invariance of PDE Painlevé analysis.- The nonlocal amplitude equation.- Pressure waves in fluid-filled nonlinear viscoelastic tubes.- Elliptic function solutions for Landau-Ginzburg equation.- Application of a Macsyma program for the Painlevé test to the Fitzhugh-Nagumo Equation.- The strongly dissipative Toda lattice.- A perturbative approach to Hirota’s bilinear equations of KdV-type.- Construction of two dimensional super potentials for classical super systems.- Applications of nonlinear PDE’s to the modelling of ferromagnetic inhomogeneities.- Author Index.