A. Yu. Trifonov

The trajectory-coherent approximation and the system of moments for the Hartree type equation

The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ℏ (ℏ→0), are constructed with a power accuracy of O(ℏ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.

The evolution operator of the Hartree-type equation with a quadratic potential

Based on the ideology of the Maslov complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated functions. The nonlinear evolution operator has been obtained in explicit form in the class of semiclassically concentrated functions. Parametric families of symmetry operators have been found for the Hartree-type equation. With the help of symmetry operators, families of exact solutions of the equation have been constructed. Exact expressions are obtained for the quasi-energies and their respective states. The Aharonov–Anandan geometric phases are found in explicit form for the quasi-energy states.

Semiclassical Solutions of the Nonlinear Schrödinger Equation

Abstract A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schrodinger equation (NLSE) with an external field. These solutions are considered as multidimensional solitary waves. The center of mass of such a solution is shown to move along with the bicharacteristics of the basic symbol of the corresponding linear Schrodinger equation. The leading term of the asymptotic WKBsolution is constructed for the multidimensional NLSE. Special cases are considered for the standard one-dimensional NLSE and for NLSE in cylindrical coordinates.

Berry phases for the nonlocal Gross–Pitaevskii equation with a quadratic potential

A countable set of asymptotic space-localized solutions is constructed by the complex germ method in the adiabatic approximation for the nonstationary Gross–Pitaevskii equation with nonlocal nonlinearity and a quadratic potential. The asymptotic parameter is 1/T, where T 1 is the adiabatic evolution time. A generalization of the Berry phase of the linear Schrodinger equation is formulated for the Gross–Pitaevskii equation. For the solutions constructed, the Berry phases are found in explicit form.

Maslov's complex germ method and Berry's phase

In the adiabatic approximation the connection of the Beny phase with the quasi- classical trajectory-coherent slates of the Schrirdinger-type equation (with the arbitrary scalar %(pseudo) differential operator) and the Dirac equation in the external penodic electromagnetic field is studied.

Quantization of closed orbits in Dirac theory by Maslov's complex germ method

On the basis of Maslovs complex germ method for the Dirac operator in external electromagnetic and torsion fields the quasi-classical spectral series corresponding within the limit h(cross) to 0 to the electron motion along closed stable orbits has been constructed. The quasi-classical energy spectrum is found from the condition of quantization of these orbits, and the quasi-classical asymptotics corresponding to the latter form a complete set of localized quantum state. The method is illustrated in all details by the electron motion in the axial fields as an example.

Quasi-classical spectral series of the Dirac operators corresponding to quantized two-dimensional Lagrangian tori

Based on the Maslov complex germ theory, a method of constructing the quasi-classical spectral series for the Dirac operator is proposed. The case when the corresponding relativistic Hamiltonian system is non-integrable and it admits a family of invariant two-dimensional stable Lagrangian tori containing the focal points is considered. The resulting quantization conditions for the above family generalize the Bohr-Sommerfeld-Maslov conditions and include new additional characteristics. The quasi-classical asymptotics obtained are regular over the full classically allowed domain. They also form an asymptotically complete and orthonormal set. Examples which use the proposed technique of the quasi-classical quantization are analysed.

Theory of spontaneous radiation by electrons in a trajectory-coherent approximation

The first-order quantum correction for the characterization of spontaneous radiation is calculated by means of electron quasi-classical trajectory-coherent states in an arbitrary electromagnetic field. Well known expressions for the characterization of spontaneous radiation are obtained using quasi-classical approximation. The first-order quantum correction is derived as a functional from a classical trajectory (among which is a classical spin vector). Transitions with spin flip and without spin flip are distinguished. Those elements connected with photon kick and quantum motion characteristics are selected for first-order quantum correction. It is shown that, using an ultra-relativistic approximation, the latter may be ignored, but when using a non-relativistic approximation their contributions are approximately equal. A special trajectory-coherent representation that significantly simplifies the investigation of spontaneous radiation is proposed.

Asymptotics semiclassically concentrated on curves for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation

In this paper we construct asymptotic solutions for the nonlocal multidimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation in the class of functions concentrated on a one-dimensional manifold (curve) using a semiclassical approximation technique. We show that the construction of these solutions can be reduced to solving a similar problem for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov in the class of functions concentrated at a point (zero-dimensional manifold) together with an additional operator condition. The general approach is exemplified by constructing a two-dimensional two-parametric solution, which describes quasi-steady-state patterns on a circumference.

Semiclassical trajectory-coherent states of the nonlinear Schrödinger equation with unitary nonlinearity

Semiclassically concentrated states of the nonlinear Schrödinger equation (NLSE) with unitary nonlinearity, representing multidimensional localized wave packets, are constructed on the basis of the Maslov complex germ theory. A system of ordinary differential equations of Hamilton-Ehrenfest (HE) type, describing the motion of the wave packet centroid, is derived. The structure of the HE system is strongly influenced by the initial conditions of the Cauchy problem for the NLSE. Wave packets of Gaussian type are constructed in an explicit form. Possible use of the solutions constructed in the problem of optical pulse propagation in a nonlinear medium with nonstationary dispersion is discussed.