A. V. Shapovalov

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.

Separation of variables in the Dirac equation in Stackel spaces

The subspaces of Riemannian space of signature (+---) that admit separation of the Dirac equation have been found in the case of Riemannian space admitting the separation of the Hamilton-Jacobi equation. For the separation of variables in the Hamilton-Jacobi equation it is necessary for the complete set of Killing vectors and tensors to be of a special kind. Every complete set defines its own type of metric of Riemannian space which is called Stackel space. The Dirac equation does not permit the separation of variables in general cases of Stackel space. The main idea of the paper is in the construction, in Stackel space, of a complete set of another kind. This complete set consists of three matrix first-order differential symmetry operators of the Dirac equation. The operators are pairwise commuting and linearly independent. The complete set structure is in agreement with the structure of the Killing vectors and tensors of Stackel space. The separation of variables in the Dirac equation has been carried out in the explicit form in Stackel spaces which admit complete sets of symmetry operators. These operators have been used essentially in the process of separation that differs from Chandrasekhar method.

Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an ex- ternal field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concen- trated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.

A new method of exact solution generation for the one-dimensional Schrödinger equation

Abstract A new exact solution generating method for the one-dimensional Schrodinger equation is found. It gives rise to an infinite set of new exactly solvable potentials and the corresponding wave functions on the basis of a known exact solution for a given potential.

Pattern formation in terms of semiclassically limited distribution on lower dimensional manifolds for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation

We have investigated the pattern formation in systems described by the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation for the cases where the dimension of the pattern concentration domain is lower than that of the domain of independent variables. We have obtained a system of integro-differential equations which describe the dynamics of the concentration domain and the semiclassically limited density distribution for a pattern in the class of trajectory concentrated functions. Also, asymptotic large time solutions have been obtained that describe the semiclassically limited distribution for a quasi-steady-state pattern on the concentration manifold. The approach is illustrated by an example for which the analytical solution is in good agreement with the results of numerical calculations.

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.

Dynamics of the Thermal Instability Evolution in Dielectric Breakdown

A mathematical model of the dynamics of temperature, electric field strength, and charge density in thermal dielectric breakdown is examined. The characteristics of the evolution of a thermal instability initiated by a local temperature disturbance are studied by numerical modeling. The conditions of initiation and growth of an electrothermal structure resulting in the formation of a highly conductive channel and shunting of the dielectrics current are identified.

Separation of variables in the Dirac equation in Stackel spaces. II. External gauge fields

For pt.I see ibid., vol.7, p.517, (1990). This is the second paper in a series of research started by Bagrov et al. The Abelian complete sets of matrix first-order differential symmetry operators for the Dirac equation are used for the separation of variables in the Dirac equation in Stackel spaces with external gauge fields. The permissible space metrics and gauge field potentials are not limited by field equations.