Semiclassical approach in geometric phase theory for the nonlinear Hartree type equation
Abstract:
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.
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