arXiv subjects

Stefan C. Mancas

Publications and source records attributed to Stefan C. Mancas.

29 records · Page 2Linked to original sources

arXiv2014

Pulses and Snakes in Ginzburg--Landau Equation

Using a variational formulation for partial differential equations (PDEs) combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of dissipative solitons, and analyze the dependence of both their shape and stability on the physical parameters of the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. Numerical simulations reveal very interesting bifurcations sequences as the parameters of the CGLE are varied. Our predictions on the variation of the soliton amplitude, width, position, speed and phase of the solutions using the variational formulation agree with simulation results.

nlin.PS↗

arXiv2014

Variable viscosity condition in the modeling of a slider bearing

To reduce tear and wear of machinery lubrication is essential. Lubricants form a layer between two surfaces preventing direct contact and reduce friction between moving parts and hence reduce wear. In this short letter the lubrication of two slider bearings with parallel and nonparallel is studied. First, we show that bearings with parallel plates cannot support any load. For bearings with nonparallel plates we are interested on how constant and temperature dependent viscosity affects the properties of the bearings. Also, a critical temperature for which the bearings would fail due to excess in temperature is found for both latter cases. If the viscosity is constant, the critical temperature is given by an explicit formula, while for the non-constant viscosity the critical temperature can be always found from a closed form formula involving Weber functions

physics.class-ph↗

arXiv2013

Spatiotemporal two-dimensional solitons in the complex Ginzburg-Landau equation

We introduce spatiotemporal solitons of the two-dimensional complex Ginzburg-Landau equation (2D CCQGLE) with cubic and quintic nonlinearities in which asymmetry between space-time variables is included. The 2D CCQGLE is solved by a powerful Fourier spectral method, i.e., a Fourier spatial discretization and an explicit scheme for time differencing. Varying the system's parameters, and using different initial conditions, numerical simulations reveal 2D solitons in the form of stationary, pulsating and exploding solitons which possess very distinctive properties. For certain regions of parameters, we have also found stable coherent structures in the form of spinning (vortex) solitons which exist as a result of a competition between focusing nonlinearities and spreading while propagating through medium.

nlin.PS↗

arXiv2013

Traveling Wavetrains in the Complex Cubic-Quintic Ginzburg-Landau Equation

In this paper we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic-quintic Ginzburg-Landau equation. The primary tools used here are Hopf bifurcation theory and perturbation theory. Explicit results are obtained for the post-bifurcation periodic orbits and their stability. Generalized and degenerate Hopf bifurcations are also briefly considered to track the emergence of global structure such as homoclinic orbits.

nlin.PS↗

arXiv2013

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers-Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second-order nonlinear equations

math-ph↗

arXiv2013

Numerical Simulations of Snake Dissipative Solitons in Complex Cubic-Quintic Ginzburg-Landau Equation

Numerical simulations of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal five entirely novel classes of pulse or solitary waves solutions, viz. pulsating, creeping, snaking, erupting, and chaotical solitons. Here, we develop a theoretical framework for analyzing the full spatio-temporal structure of one class of dissipative solution (snaking soliton) of the CCQGLE using the variational approximation technique and the dynamical systems theory. The qualitative behavior of the snaking soliton is investigated using the numerical simulations of (a) the full nonlinear complex partial differential equation and (b) a system of three ordinary differential equations resulting from the variational approximation.

nlin.PS↗

arXiv2013

Interactions and Focusing of Nonlinear Water Waves

The coupled cubic nonlinear Schrödinger (CNLS) equations are used to study modulational instabilities of a pair of nonlinearly interacting two-dimensional waves in deep water. It has been shown that the full dynamics of these interacting waves gives rise to localized large-amplitude wavepackets (wave focusing). In this short letter we attempt to verify this result numerically using a Fourier spectral method for the CNLS equations.

nlin.CD↗

arXiv2013

Aerodynamic Stability of Satellites in Elliptic Low Earth Orbits

Topical observations of the thermosphere at altitudes below $200 \, km$ are of great benefit in advancing the understanding of the global distribution of mass, composition, and dynamical responses to geomagnetic forcing, and momentum transfer via waves. The perceived risks associated with such low altitude and short duration orbits has prohibited the launch of Discovery-class missions. Miniaturization of instruments such as mass spectrometers and advances in the nano-satellite technology, associated with relatively low cost of nano-satellite manufacturing and operation, open an avenue for performing low altitude missions. The time dependent coefficients of a second order non-homogeneous ODE which describes the motion have a double periodic shape. Hence, they will be approximated using Jacobi elliptic functions. Through a change of variables the original ODE will be converted into Hill's ODE for stability analysis using Floquet theory. We are interested in how changes in the coefficients of the ODE affect the stability of the solution. The expected result will be an allowable range of parameters for which the motion is dynamically stable. A possible extension of the application is a computational tool for the rapid evaluation of the stability of entry or re-entry vehicles in rarefied flow regimes and of satellites flying in relatively low orbits.

physics.geo-ph↗

arXiv2012

Periodic and Chaotic Traveling Wave Patterns in Reaction-Diffusion/ Predator-Prey Models with General Nonlinearities

Traveling wavetrains in generalized two-species predator-prey models and two-component reaction-diffusion equations are considered. The stability of the fixed points of the traveling wave ODEs (in the usual "spatial" variable) is considered. For general functional forms of the nonlinear prey birthrate/prey deathrate or reaction terms, a Hopf bifurcation is shown to occur at two different critical values of the traveling wave speed. The post-bifurcation dynamics is investigated for five different functional forms of the nonlinearities. In cases where the bifurcation is supercritical, the post-bifurcation behaviour yields stable periodic orbits of the traveling-wave ODEs in the spatial variable. These correspond to stable periodic wavetrains of the full PDEs. Subcritical Hopf bifurcations yield more complex post-bifurcation dynamics in the PDE wavetrains. In special cases where the subcritical bifurcation marks the end of the regime of stability, the post-bifurcation behavior in the spatial ODEs is chaotic, corresponding to wavetrains of the original PDEs which are spatially coherent, but have chaotic temporal dynamics. All the models are integrated numerically to investigate the post-bifurcation dynamics and chaotic regimes are characterized by computing power spectra, autocorrelation functions, and fractal dimensions.

math.DS↗

arXiv2012

Solitary waves, periodic and elliptic solutions to the Benjamin, Bona & Mahony (BBM) equation modified by viscosity

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions of \eqref{BBMv} experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results.

physics.gen-ph↗

arXiv2012

Bifurcations and Competing Coherent Structures in the Cubic-Quintic Ginzburg-Landau Equation I: Plane Wave (CW) Solutions

Singularity Theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg-Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcations curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space.

nlin.PS↗