Topological methods in stability analysis of travelling waves

2001 ◽  
pp. 45-62 ◽  
Author(s):  
Shunsaku Nii
Keyword(s):  
Stability Analysis ◽  
Travelling Waves ◽  
Topological Methods

Linear stability analysis for periodic travelling waves of the Boussinesq equation and the Klein–Gordon–Zakharov system

2014 ◽  
Vol 144 (3) ◽  
pp. 455-489 ◽  
Author(s):  
Sevdzhan Hakkaev ◽  
Milena Stanislavova ◽  
Atanas Stefanov
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Wide Class ◽  
Boussinesq Equation ◽  
Travelling Waves ◽  
Periodic Waves ◽  
Zakharov System ◽  
Klein Gordon ◽  
Spatially Periodic ◽  
Stability Results

The question of the linear stability of spatially periodic waves for the Boussinesq equation (in the cases p = 2, 3) and the Klein–Gordon–Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability) when the perturbations are taken with the same period T. In particular, our results allow us to completely recover the linear stability results, in the limit T → ∞, for the whole-line case.

scholarly journals Multi-dimensional stability analysis of planar travelling waves

1994 ◽  
Vol 7 (5) ◽  
pp. 1-4 ◽  
Author(s):  
C.-M. Brauner ◽  
A. Lunardi ◽  
Cl. Schmidt-Lainé
Keyword(s):  
Stability Analysis ◽  
Dimensional Stability ◽  
Travelling Waves

Stability of plane Couette flow of a granular material

1998 ◽  
Vol 377 ◽  
pp. 99-136 ◽  
Author(s):  
MEHEBOOB ALAM ◽  
PRABHU R. NOTT
Keyword(s):  
Stability Analysis ◽  
Granular Material ◽  
Bulk Density ◽  
Couette Flow ◽  
Flow Direction ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Plane Couette Flow ◽  
Wall Properties ◽  
Wave Instabilities

This paper presents a linear stability analysis of plane Couette flow of a granular material using a kinetic-theory-based model for the rheology of the medium. The stability analysis, restricted to two-dimensional disturbances, is carried out for three illustrative sets of grain and wall properties which correspond to the walls being perfectly adiabatic, and sources and sinks of fluctuational energy. When the walls are not adiabatic and the Couette gap H is sufficiently large, the base state of steady fully developed flow consists of a slowly deforming ‘plug’ layer where the bulk density is close to that of maximum packing and a rapidly shearing layer where the bulk density is considerably lower. The plug is adjacent to the wall when the latter acts as a sink of energy and is centred at the symmetry axis when it acts as a source of energy. For each set of properties, stability is determined for a range of H and the mean solids fraction [barvee ]. For a given value of [barvee ], the flow is stable if H is sufficiently small; as H increases it is susceptible to instabilities in the form of cross-stream layering waves with no variation in the flow direction, and stationary and travelling waves with variation in the flow and gradient directions. The layering instability prevails over a substantial range of H and [barvee ] for all sets of wall properties. However, it grows far slower than the strong stationary and travelling wave instabilities which become active at larger H. When the walls act as energy sinks, the strong travelling wave instability is absent altogether, and instead there are relatively slow growing long-wave instabilities. For the case of adiabatic walls there is another stationary instability for dilute flows when the grain collisions are quasi-elastic; these modes become stable when grain collisions are perfectly elastic or very inelastic. Instability of all modes is driven by the inelasticity of grain collisions.

scholarly journals Topology and Bifurcations in Nonholonomic Mechanics

2015 ◽  
Vol 25 (10) ◽  
pp. 1530028 ◽  
Author(s):  
Ivan A. Bizyaev ◽  
Alexey Bolsinov ◽  
Alexey Borisov ◽  
Ivan Mamaev
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Qualitative Analysis ◽  
Poisson Bracket ◽  
Nonholonomic Mechanics ◽  
Hamiltonian Form ◽  
Topological Methods ◽  
Integral Manifolds ◽  
Nonholonomic Dynamical Systems

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

Instability of the buoyancy layer on an evenly heated vertical wall

2007 ◽  
Vol 587 ◽  
pp. 453-469 ◽  
Author(s):  
G. D. McBAIN ◽  
S. W. ARMFIELD ◽  
GILLES DESRAYAUD
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Prandtl Number ◽  
Travelling Waves ◽  
Vertical Wall ◽  
Reynolds Numbers ◽  
Low Reynolds Numbers ◽  
Linearized Stability ◽  
The Stability ◽  
Volume Method

The stability of the buoyancy layer on a uniformly heated vertical wall in a stratified fluid is investigated using both semi-analytical and direct numerical methods. As in the related problem in which the excess temperature of the wall is specified, the basic laminar flow is steady and one-dimensional. Here flows varying in time and with height are considered, the behaviour being determined by the fluid's Prandtl number and a Reynolds number proportional to the ratio of two temperature gradients: the horizontal one imposed at the wall and the vertical one existing in the far field. For low Reynolds numbers, the flow is stable with variation only in the wall-normal direction. For Reynolds numbers greater than a critical value, depending on the Prandtl number, the flow is unstableand supports two-dimensional travelling waves. The critical Reynolds number and other properties have been obtained via linearized stability analysis and are shown to accuratelypredict the behaviour of the full nonlinear solution obtained numerically for Prandtl number 7. The stability analysis employs a novel Laguerre collocation scheme while the direct numerical simulations use a second-order finite volume method.

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