30,490 research outputs found
Finite Mechanical Proxies for a Class of Reducible Continuum Systems
We present the exact finite reduction of a class of nonlinearly perturbed
wave equations, based on the Amann-Conley-Zehnder paradigm. By solving an
inverse eigenvalue problem, we establish an equivalence between the spectral
finite description derived from A-C-Z and a discrete mechanical model, a well
definite finite spring-mass system. By doing so, we decrypt the abstract
information encoded in the finite reduction and obtain a physically sound proxy
for the continuous problem.Comment: 15 pages, 3 figure
Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications
We provide a theory to establish the existence of nonzero solutions of
perturbed Hammerstein integral equations with deviated arguments, being our
main ingredient the theory of fixed point index. Our approach is fairly general
and covers a variety of cases. We apply our results to a periodic boundary
value problem with reflections and to a thermostat problem. In the case of
reflections we also discuss the optimality of some constants that occur in our
theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page
Orbital stability of periodic waves in the class of reduced Ostrovsky equations
Periodic travelling waves are considered in the class of reduced Ostrovsky
equations that describe low-frequency internal waves in the presence of
rotation. The reduced Ostrovsky equations with either quadratic or cubic
nonlinearities can be transformed to integrable equations of the Klein--Gordon
type by means of a change of coordinates. By using the conserved momentum and
energy as well as an additional conserved quantity due to integrability, we
prove that small-amplitude periodic waves are orbitally stable with respect to
subharmonic perturbations, with period equal to an integer multiple of the
period of the wave. The proof is based on construction of a Lyapunov
functional, which is convex at the periodic wave and is conserved in the time
evolution. We also show numerically that convexity of the Lyapunov functional
holds for periodic waves of arbitrary amplitudes.Comment: 34 page
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