Second order Killing tensors related to symmetric spaces

We discuss the pairs of quadratic integrals of motion belonging to the $n$-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures. In contrast to the Eisenhart theory, additional integrals of motion are polynomials of the fourth, sixth and other orders in momenta. The main focus is on the second-order Killing tensors corresponding to quadratic integrals of motion and relating to the special combinations of rotations and translations in Euclidean space.Comment: 27 pages, LaTeX with amsfont

Stable and chaotic solutions of the complex Ginzburg-Landau equation with periodic boundary conditions

We study, analytically and numerically, the dynamical behavior of the solutions of the complex Ginzburg-Landau equation with diffraction but without diffusion, which governs the spatial evolution of the field in an active nonlinear laser cavity. Accordingly, the solutions are subject to periodic boundary conditions. The analysis reveals regions of stable stationary solutions in the model’s parameter space, and a wide range of oscillatory and chaotic behaviors. Close to the first bifurcation destabilizing the spatially uniform solution, a stationary single-humped solution is found in an asymptotic analytical form, which turns out to be in very good agreement with the numerical results. Simulations reveal a series of stable stationary multi-humped solutionsComment: 9 pages, 15 figure

Instability and stability properties of traveling waves for the double dispersion equation

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation $~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~$ for $~p>1$, $~a\geq b>0$. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms $u_{xxxx}$ and $u_{xxtt}$. We obtain an explicit condition in terms of $a$, $b$ and $p$ on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation ($b=0$), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide both analytical and numerical results on the variation of the stability region of wave velocities with $a$, $b$ and $p$ and then state explicitly the conditions under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure

Strain localization in two-dimensional lattices

Two-dimensional localized strain wave solutions of the nonlinear equation for shear waves in two-dimensional lattices are studied. The corresponding equation does not possess an invariance in one of the spatial direction while its exact plane traveling wave solution does not reflect that. However, the numerical simulation of a two-dimensional localized wave reveals a non-symmetric evolution.DFG, 405631704, Anormaler Energietransfer in kristallinen Materialien vom Standpunkt der diskreten Mechanik und der Kontinuumstheori

Разработка легкого коммерческого электромобиля

The paper describes the process and results of the development of the light commercial electric vehicle. In order to ensure maximum energy efficiency of the developed vehicle the key parameters of the original electric motor. The article also presents the results of power electronic thermal calculation. For the mathematical model of the vehicle, the driving cycle parameters of the electric platform were determined in accordance with UNECE Regulations No 83, 84. The driving cycle was characterized by four successive urban and suburban cycles. The mathematical model also takes into account the time phases of the cycle, which include idling, vehicle idling, acceleration, constant speed movement, deceleration, etc. The model of the electric part of the vehicle was developed using MatLab-Simulink (SimPowerSystems library) in addition to the mechanical part of the electric car. The electric part included the asynchronous electric motor, the motor control system and the inverter. This model at the output allows to obtain such characteristics of the electric motor as currents, flows and voltages of the stator and rotor in a fixed and rotating coordinate systems, electromagnetic moment, angular speed of rotation of the motor shaft. The developed model allowed to calculate and evaluate the performance parameters of the electric vehicle. Technical solutions of the electric vehicle design were verified by conducting strength calculations. In conclusion, the results of field tests of a commercial electric vehicle are presented.В статье описаны процесс и результаты создания легкого коммерческого электромобиля. С целью обеспечения максимальной энергоэффективности разрабатываемого транспортного средства определены основные параметры оригинального электродвигателя. Представлены результаты теплового расчета силовой электроники. Для математической модели транспортного средства определены параметры ездового цикла электрической платформы в соответствии с Правилами № 83, 84 ЕЭК ООН. Цикл движения характеризовался четырьмя последовательными циклами городского и пригородного режимов движения. Математическая модель также учитывает временные фазы цикла, которые включают холостой ход транспортного средства, ускорение, движение с постоянной скоростью, замедление и т. д. Модель электрической части транспортного средства разработана с использованием MatLab-Simulink (библиотека SimPowerSystems) в дополнение к механической части электромобиля. Электрическая часть включала асинхронный электродвигатель, систему управления двигателем и инвертор. Данная модель на выходе позволяет получить такие характеристики электродвигателя, как токи, поведение магнитного поля, напряжения статора и ротора в неподвижной и вращающейся системах координат, электромагнитный момент, угловая скорость вращения вала двигателя. Разработанная модель позволила рассчитать и оценить параметры производительности электромобиля. Технические решения конструкции электромобиля были проверены на прочность путем расчетов. Представлены результаты полевых испытаний коммерческого электромобиля

On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within

In this chapter, we analyze the steady-state microscale fluid--structure interaction (FSI) between a generalized Newtonian fluid and a hyperelastic tube. Physiological flows, especially in hemodynamics, serve as primary examples of such FSI phenomena. The small scale of the physical system renders the flow field, under the power-law rheological model, amenable to a closed-form solution using the lubrication approximation. On the other hand, negligible shear stresses on the walls of a long vessel allow the structure to be treated as a pressure vessel. The constitutive equation for the microtube is prescribed via the strain energy functional for an incompressible, isotropic Mooney--Rivlin material. We employ both the thin- and thick-walled formulations of the pressure vessel theory, and derive the static relation between the pressure load and the deformation of the structure. We harness the latter to determine the flow rate--pressure drop relationship for non-Newtonian flow in thin- and thick-walled soft hyperelastic microtubes. Through illustrative examples, we discuss how a hyperelastic tube supports the same pressure load as a linearly elastic tube with smaller deformation, thus requiring a higher pressure drop across itself to maintain a fixed flow rate.Comment: 19 pages, 3 figures, Springer book class; v2: minor revisions, final form of invited contribution to the Springer volume entitled "Dynamical Processes in Generalized Continua and Structures" (in honour of Academician D.I. Indeitsev), eds. H. Altenbach, A. Belyaev, V. A. Eremeyev, A. Krivtsov and A. V. Porubo

On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation

The Conte-Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of our approach to the quintic complex one-dimensional Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave form. We also find restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5. Using the investigation of the CGLE5 as an example, we demonstrate that to find elliptic solutions the analysis of a system of differential equations is more preferable than the analysis of the equivalent single differential equation.Comment: LaTeX, 21 page