8,801 research outputs found
Spectral stability of nonlinear waves in KdV-type evolution equations
This paper concerns spectral stability of nonlinear waves in KdV-type
evolution equations. The relevant eigenvalue problem is defined by the
composition of an unbounded self-adjoint operator with a finite number of
negative eigenvalues and an unbounded non-invertible symplectic operator
. The instability index theorem is proven under a generic
assumption on the self-adjoint operator both in the case of solitary waves and
periodic waves. This result is reviewed in the context of other recent results
on spectral stability of nonlinear waves in KdV-type evolution equations.Comment: 15 pages, no figure
On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis
We consider the phenomenon of collapse in the critical Keller-Segel equation
(KS) which models chemotactic aggregation of micro-organisms underlying many
social activities, e.g. fruiting body development and biofilm formation. Also
KS describes the collapse of a gas of self-gravitating Brownian particles. We
find the fluctuation spectrum around the collapsing family of steady states for
these equations, which is instrumental in derivation of the critical collapse
law. To this end we develop a rigorous version of the method of matched
asymptotics for the spectral analysis of a class of second order differential
operators containing the linearized Keller-Segel operators (and as we argue
linearized operators appearing in nonlinear evolution problems). We explain how
the results we obtain are used to derive the critical collapse law, as well as
for proving its stability.Comment: 22 pages, 1 figur
Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the
simplest microscopic description of the thermodynamic transition from a fluid
state to a crystalline state. The resulting phase field crystal model describes
a variety of spatially localized structures, in addition to different spatially
extended periodic structures. The location of these structures in the
temperature versus mean order parameter plane is determined using a combination
of numerical continuation in one dimension and direct numerical simulation in
two and three dimensions. Localized states are found in the region of
thermodynamic coexistence between the homogeneous and structured phases, and
may lie outside of the binodal for these states. The results are related to the
phenomenon of slanted snaking but take the form of standard homoclinic snaking
when the mean order parameter is plotted as a function of the chemical
potential, and are expected to carry over to related models with a conserved
order parameter.Comment: 40 pages, 13 figure
- …