An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.Comment: 28 pages, version accepted for publication in Int. J. Geom. Methods Mod. Phy

The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation

The general construction of quasi-classically 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 \h (\h\to0), are constructed with a power accuracy of O(\h^{N/2}), where N is any natural number. In constructing the quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for middle or centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of quasi-classically concentrated solutions of the Hartree-type equations. The results obtained are exemplified by the one-dimensional equation Hartree-type with a Gaussian potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class: Accelerator PhysicsComment: 36 pages, LaTeX-2

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained

Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity

The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Сравнительная оценка традиционной и эндоскопической аппендэктомии по результатам выполнения первой 1000 лапароскопических аппендэктомий

The aim of the study is improvement of treatment results of patients with acute appendicitis by application of laparoscopic technique in diagnosis and treatment of the disease. The laparoscopic diagnosis of acute appendicitis allows to avoid “unnecessary” appendectomy inevitable in traditional clinical and laboratory diagnosis. Performance of laparoscopic appendectomies for acute appendicitis is possible in 95.9% of patients. Intracorporal laparoscopic appendectomy was performed in 704 patients. We withdrew patients from the study if conversion to open appendectomy was necessary (28 patients — 4.0%). The appendix stump closure method was assigned in accordance with appendix base inflammatory changes. The patients were divided into 4 groups according to stump securing method. The appendix stump was controlled by using two or three titanic clips in 356 (52.6%) patients, two separate ligatures — in 252 (37.3%) patients, using a linear stapler in 56 (8.3%), and immersion into the ceacum cupola by a purse-string suture was performed in 12 (1.8%) patients. Operation time and complications were analyzed. Duration of laparoscopic appendectomy — (53.4±7.6) min does not differ from open surgery — (49.2±8.7) min. Duration of in-hospital treatment after laparoscopic appendectomy — (3.4±0.9) days is shorter than after open surgery — (6.2±1.2) days. The rate of postoperative complications after laparoscopic appendectomy is lower than those after traditional open surgery — 3.5 and 6.1% accordingly.Цель исследования — улучшение результатов лечения больных острым аппендицитом путем применения лапароскопической техники в диагностике и лечении заболевания. При применении лапароскопии ошибочно удалено 0,4 % неизмененных червеобразных отростков. При традиционном методе — 6,5 %. Катаральный аппендицит диагностирован у 36,4 % больных в группе открытых аппендэктомий и лишь у 10,5 % — в группе лапароскопических аппендэктомий, что позволяет говорить о неоправданно выполненной в некоторых случаях аппендэктомии. Выполнение лапароскопической аппендэктомии по поводу острого аппендицита возможно у 95,9 % больных. Продолжительность выполнения эндоскопической аппендэктомии — (53,4± ±7,6) мин достоверно не отличается от таковой открытой операции — (49,2±8,7) мин. Продолжительность лечения больных в стационаре после выполнения лапароскопической аппендэктомии составляет (3,4±0,9) дня, что меньше, чем после открытой операции, — (6,2±1,2) дня. Частота послеоперационных осложнений после выполнения лапароскопической аппендэктомии меньше, чем после открытой операции, — соответственно 3,5 и 6,1 %

Symmetry operators of the two-component Gross–Pitaevskii equation with a Manakov-type nonlocal nonlinearity

We consider an integro-differential 2-component multidimensional Gross-Pitaevskii equation with a Manakov-type cubic nonlocal nonlinearity. In the framework of the WKB-Maslov semiclassical formalism, we obtain a semiclassically reduced 2-component nonlocal Gross- Pitaevskii equation determining the leading term of the semiclassical asymptotic solution. For the reduced Gross-Pitaevskii equation we construct symmetry operators which transform arbitrary solution of the equation into another solution. Constructing the symmetry operator is based on the Cauchy problem solution technique and uses an intertwining operator which connects two solutions of the reduced Gross-Pitaevskii equation. General structure of the symmetry operator is illustrated with a 1D case for which a family of symmetry operators is found explicitly and a set of exact solutions is generated

Thermal interaction of biological tissue with nanoparticles heated by laser radiation

We explore the problem of thermal interaction of nanoparticles heated by laser radiation with a biological tissue after particle flow entering the cell. The solution of the model equations is obtained numerically under the following assumptions: a single particle is located in a neighborhood exceeding the particle size; the environment surrounding the particle is water with the conventional thermal characteristics. The model equations are deduced from the particle and the environment energy conditions taking into account the heat transfer in the particle and in its environment by conduction. We also assume that at the boundary between the particle and the surrounding water the perfect thermal contact takes place

Semiclassical Solutions of the Nonlinear Schrödinger Equation

Abstract A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schrödinger 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 Schrödinger 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