53,740 research outputs found
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts
Fronts that start from a local perturbation and propagate into a linearly
unstable state come in two classes: pulled and pushed. ``Pulled'' fronts are
``pulled along'' by the spreading of linear perturbations about the unstable
state, so their asymptotic speed equals the spreading speed of linear
perturbations of the unstable state. The central result of this paper is that
the velocity of pulled fronts converges universally for time like
. The parameters ,
, and are determined through a saddle point analysis from the
equation of motion linearized about the unstable invaded state. The interior of
the front is essentially slaved to the leading edge, and we derive a simple,
explicit and universal expression for its relaxation towards
. Our result, which can be viewed as a general center
manifold result for pulled front propagation, is derived in detail for the well
known nonlinear F-KPP diffusion equation, and extended to much more general
(sets of) equations (p.d.e.'s, difference equations, integro-differential
equations etc.). Our universal result for pulled fronts thus implies
independence (i) of the level curve which is used to track the front position,
(ii) of the precise nonlinearities, (iii) of the precise form of the linear
operators, and (iv) of the precise initial conditions. Our simulations confirm
all our analytical predictions in every detail. A consequence of the slow
algebraic relaxation is the breakdown of various perturbative schemes due to
the absence of adiabatic decoupling.Comment: 76 pages Latex, 15 figures, submitted to Physica D on March 31, 1999
-- revised version from February 25, 200
Discrete Analog of the Burgers Equation
We propose the set of coupled ordinary differential equations
dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers
equation. We focus on traveling waves and triangular waves, and find that these
special solutions of the discrete system capture major features of their
continuous counterpart. In particular, the propagation velocity of a traveling
wave and the shape of a triangular wave match the continuous behavior. However,
there are some subtle differences. For traveling waves, the propagating front
can be extremely sharp as it exhibits double exponential decay. For triangular
waves, there is an unexpected logarithmic shift in the location of the front.
We establish these results using asymptotic analysis, heuristic arguments, and
direct numerical integration.Comment: 6 pages, 5 figure
On the formation of singularities of solutions of nonlinear differential systems in antistokes directions
We determine the position and the type of spontaneous singularities of
solutions of generic analytic nonlinear differential systems in the complex
plane, arising along antistokes directions towards irregular singular points of
the system. Placing the singularity of the system at infinity we look at
equations of the form with
analytic in a neighborhood of , with genericity
assumptions; is then a rank one singular point. We analyze the
singularities of those solutions which tend to zero for in some sectorial region, on the edges of the maximal region (also
described) with this property. After standard normalization of the differential
system, it is shown that singularities occuring in antistokes directions are
grouped in nearly periodical arrays of similar singularities as ,
the location of the array depending on the solution while the (near-) period
and type of singularity are determined by the form of the differential system.Comment: 61
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