14,969 research outputs found
Periodic solutions for a generalized p-Laplacian equation
AbstractThe existence and uniqueness of T-periodic solutions for the following boundary value problems with p-Laplacian: (ϕp(x′))′+f(t,x′)+g(t,x)=e(t),x(0)=x(T),x′(0)=x′(T) are investigated, where ϕp(u)=∣u∣p−2u with p>1 and f,g,e are continuous and are T-periodic in t with f(t,0)=0. Using coincidence degree theory, some existence and uniqueness results are presented
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
Generalized elliptic functions and their application to a nonlinear eigenvalue problem with -Laplacian
The Jacobian elliptic functions are generalized and applied to a nonlinear
eigenvalue problem with -Laplacian. The eigenvalue and the corresponding
eigenfunction are represented in terms of common parameters, and a complete
description of the spectra and a closed form representation of the
corresponding eigenfunctions are obtained. As a by-product of the
representation, it turns out that a kind of solution is also a solution of
another eigenvalue problem with -Laplacian.Comment: 17 page
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