Quantum mathematics: Backgrounds and some applications to nonlinear dynamical systems

Abstract

UDC 517.9The backgrounds of quantum mathematics, a new discipline in mathematical physics, are discussed and analyzed from both historical and analytical points of view. The magic properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to the theory of nonlinear dynamical systems is considered.

Quantized representation of some nonlinear integrable evolution equations on the soliton sector

“…There are many attempts in the literature at rigorous quantization procedures of nonlinear evolution equations [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. The quantized representation of equations and their solutions discussed here is somewhat different.…”

The order parameter and susceptibility of the 3D Ising-like system in an external field near the phase transition point

“…Thus, the problem needs to be treated very carefully in this case. From this point of view the Yukhnovskii's method of a phase transition description [5] used in this work is quite consistent with Bogolyubov's(Jr.) ideas of using the canonical collective variable transformation approach to the corresponding Bogolyubov's functional equation [6][7][8] for the correlation functions of a simple magnet system Hamiltonian instead of that for the standard Ising model. The related functional equation splitting, compatible with the Bogolyubov's principle of correlations weakening, proves to be equivalent to the suitable mean-field approximation of higher order, giving rise to a closed solution in the thermodynamical limit.…”