Peierls-Fröhlich problem and potentials with finite number of gaps. II

A generalized Peierls–Fröhlich problem on the formation of a forbidden band in the energy spectrum of electrons due to the deformation of a potential which originally has $n$ bands is formulated. It is shown that the solutions to this problem, which are the extremals of the generalized functional of the Peierls–Fröhlich free energy, form a $(n+1)$-parameter manifold of $(n+1)$-gap potentials. Equations are obtained which the boundaries of the gaps of these potentials satisfy. It is shown that the motions on the manifold of solutions of the considered problem described by Korteweg–de Vries equations are Fröhlieh collective modes. The theory makes it possible to describe phase transitions of a lattice between periodic and quasiperiodie structures.