707 research outputs found
Dynamics of Patterns
This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects
Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations
The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization
dynamics in the presence of an applied field and a spin polarized current. In
the case of axial symmetry and with focus on one space dimension, we
investigate the emergence of space-time patterns in the form of wavetrains and
coherent structures, whose local wavenumber varies in space. A major part of
this study concerns existence and stability of wavetrains and of front- and
domain wall-type coherent structures whose profiles asymptote to wavetrains or
the constant up-/down-magnetizations. For certain polarization the Slonczewski
term can be removed which allows for a more complete charaterization, including
soliton-type solutions. Decisive for the solution structure is the polarization
parameter as well as size of anisotropy compared with the difference of field
intensity and current intensity normalized by the damping
Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves
Turing patterns on unbounded domains have been widely studied in systems of
reaction-diffusion equations. However, up to now, they have not been studied
for systems of conservation laws. Here, we (i) derive conditions for Turing
instability in conservation laws and (ii) use these conditions to find families
of periodic solutions bifurcating from uniform states, numerically continuing
these families into the large-amplitude regime. For the examples studied,
numerical stability analysis suggests that stable periodic waves can emerge
either from supercritical Turing bifurcations or, via secondary bifurcation as
amplitude is increased, from sub-critical Turing bifurcations. This answers in
the affirmative a question of Oh-Zumbrun whether stable periodic solutions of
conservation laws can occur. Determination of a full small-amplitude stability
diagram-- specifically, determination of rigorous Eckhaus-type stability
conditions-- remains an interesting open problem.Comment: 12 pages, 20 figure
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