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

SYSTEM AUTOMATYCZNEGO ZARZĄDZANIA ZESPOŁEM SAMOCHODÓW Z SILNIKAMI DIESLA NA PRZYKŁADZIE TRANSPORTU SAMOCHODOWEGO W KAMIENIOŁOMACH

The paper investigates the optimal control system of diesel automotive engineering with application of complex criteria, depending on fuel consumption rate and travel time, with adjustable coefficients of physical process mathematical model, considering influence of disturbing effects factors. This control principle allows saving fuel consumption rate, reducing transport influence on environment, and also reducing the importance of human factor for motor transport control.W artykule zostało rozpatrzone optymalne sterowanie procesem przemieszczania pojazdów z zastosowaniem złożonego kryterium, zależnego od zużycia paliwa i czasu przejazdu, ze zmiennymi współczynnikami modelu matematycznego fizycznego procesu, uwzględniając wpływ czynników zakłócających. Otrzymany system sterowania pozwala na oszczędne zużycie paliwa, zmniejszenie wpływu transportu na środowisko i obniżenie wpływ czynnika ludzkiego na zarządzanie transportem samochodowym

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

Analytic expressions of hydrothermal waves

When subjected to a horizontal temperature difference, a fluid layer with a free surface becomes unstable and hydrothermal waves develop in the bulk. Such a system is modelized by two coupled amplitude equations of the one-dimensional, complex, cubic Ginzburg-Landau type. By transposing the method developed for one CGL3 equation, we obtain several new exact solutions expressed by closed form, singlevalued, analytic expressions. Some of them are the analogue of the famous amplitude hole solution of Bekki and Nozaki.Comment: LaTeX, 12 pages, no figure, to appear, Reports on Math. Physic

Nonlinear Analysis of the Eckhaus Instability: Modulated Amplitude Waves and Phase Chaos with Non-zero Average Phase Gradient

We analyze the Eckhaus instability of plane waves in the one-dimensional complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability of plane waves and cease to exist due to saddle-node bifurcations (SN). These MAWs can be characterized by their average phase gradient $\nu$ and by the spatial period P of the periodic amplitude modulation. A numerical bifurcation analysis reveals the existence and stability properties of MAWs with arbitrary $\nu$ and P. MAWs are found to be stable for large enough $\nu$ and intermediate values of P. For different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude. Defects form from perturbed plane waves for parameter values above the SN of the corresponding MAWs. The break-down of phase chaos with average phase gradient $\nu$ > 0 (``wound-up phase chaos'') is thus related to these SNs. A lower bound for the break-down of wound-up phase chaos is given by the necessary presence of SNs and an upper bound by the absence of the splitting instability of MAWs.Comment: 24 pages, 14 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