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

Based on the ideology of the Maslov's 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.Comment: 23 pege

Berry phases for 3D Hartree type equations with a quadratic potential and a uniform magnetic field

A countable set of asymptotic space -- localized solutions is constructed by the complex germ method in the adiabatic approximation for 3D Hartree type equations with a quadratic potential. The asymptotic parameter is 1/T, where $T\gg1$ is the adiabatic evolution time. A generalization of the Berry phase of the linear Schr\"odinger equation is formulated for the Hartree type equation. For the solutions constructed, the Berry phases are found in explicit form.Comment: 15 pages, no figure

Complete group classification of a class of nonlinear wave equations

Preliminary group classification became prominent as an approach to symmetry analysis of differential equations due to the paper by Ibragimov, Torrisi and Valenti [J. Math. Phys. 32, 2988-2995] in which partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the infinite-dimensional equivalence algebra of the class under consideration. In the present paper we implement, up to both usual and general point equivalence, the complete group classification of the same class using the algebraic method of group classification. This includes the complete preliminary group classification of the class and finding Lie symmetry extensions which are not associated with subalgebras of the equivalence algebra. The complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel algebra. The set of admissible point transformations of the class is exhaustively described in terms of the partition of the class into normalized subclasses. A version of the algebraic method for finding the complete equivalence groups of a general class of differential equations is proposed.Comment: 39 page

Group classification of heat conductivity equations with a nonlinear source

We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are three, seven, twenty eight and twelve inequivalent classes of partial differential equations of the considered type that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any partial differential equation belonging to the class under study and admitting symmetry group of the dimension higher than four is locally equivalent to a linear equation. This classification is compared to existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page

THE LEFT VENTRICLE DIASTOLIC FUNCTION IN PATIENTS WITH HYPERTENSION UNDER THE USE OF DIFFERENT DRUG GROUPS

The objective: To assess the prevalence of diastolic dysfunction in patients with hypertension and preserved left ventricular ejection fraction under pharmacological correction (monotherapy) with angiotensin converting enzyme inhibitors, angiotensin II receptor blockers and β-blockers. Materials and methods: 82 patients (58 women and 24 men) with stage 2 hypertension were examined. The diastolic function was assessed via echocardiography in accordance with the European Association of Cardiovascular Imaging guidelines (2017). Echocardiography was performed before the onset of the treatment and 6 months after its onset. The treatment onset was considered to start after a 2-week period of elimination of previously used pharmacological substance and 2 weeks of assessing tolerability, dose and regimen adjustment. Results: For all selected drugs, target values of blood pressure were achieved, and no adverse effects were identified. The average values of the left atrial volume index before and after the treatment course did not show significant differences. In the majority of the examined patients, this parameter did not exceed the threshold value of 34 ml/m2 . Values exceeding the specified threshold were observed in Group 1 in 4 patients, in Group 2 in 3 patients and in Group 3 in 8 patients. According to the Tissue Doppler echocardiography results on the velocity of myocardial motion at the early diastolic filling, which was measured at the level of the lateral segments of mitral valve and the interventricular septum, positive, but unreliable changes were observed in the Groups of bisoprolol and valsartan, and no changes — in the Group of perindopril. According to the traditional criteria, diastolic dysfunction was observed in 80 % of patients, while according to the criteria of the European Association of Cardiovascular Imaging (2017) — in 21 % of patients. Conclusion: The same efficacy of all three drugs is observed in terms of achieving target blood pressure values. The most pronounced effect on the morphometric parameters of the left atrium and intracardiac hemodynamics is shown in the Groups of bisoprolol and valsartan