Belokolos E. D.

For the integrable generalized model of superconductivity, the solution of the Richardson equations is studied for a model spectrum. For the case of a narrow band, the solution is presented in terms of generalized Laguerre or Jacobi polynomials. In the asymptotic limit, when the Richardson equations are transformed into a singular integral equation, the properties of the integration contour are discussed and the spectral density is calculated. The conditions of appearance of gaps in the spectrum are investigated.

A survey of interrelations between kinetic equations and integrable systems is presented. We discuss common origin of special classes of solutions of the Boltzmann kinetic equation for Maxwellian particles and special solutions for integrable evolution equations. The thermodynamic limit and soliton kinetic equation for the integrable Korteweg-de Vries equation are considered. The existence of decaying and degenerate dispersion laws and the appearance of additional integrals of motion for the interacting waves is discussed.