Semiclassical approach in geometric phase theory for the nonlinear Hartree type equation
Study of global symmetry of nonlinear physical systems with non-trivial topology demands appropriate mathematical technique. The needed construction is widely used in quantum mechanics and it is named as the geometric (or topological) phase (GP) of wave function of the system. The GP theory is well developed in quantum mechanics due to linearity of quantum mechanical equations and the GP can be associated with a gauge field that, in its turn, results in the correspondent gauge symmetry. The non-trivial topology of the system can be caused by external fields, so the GP theory in nonlinear systems is elaborated far less than in quantum mechanics since exact integrability is extremely limited for nonlinear partial equations with variable coefficients (external fields). In our recent works [1, 2] a general construction of concentrated solutions, based on the complex WKB-Maslov method, was developed to the multidimensional Schrodinger equation with smooth arbitrary external fields and non-local nonlinearity. This equation is named as the Hartree type equation (HTE). In the present work we obtain the GF for the HTE in explicit form in semiclassical approximation in the class of trajectory consentrated solutions when the Hamiltonian of the HTE is the T-periodical operator in time. The gauge field associated with the GP is discussed. A special case of non-local potential is considered as an illustration.