194 research outputs found
Existence of solution for a class of fractional Hamiltonian systems
In this work we want to prove the existence of solution for a class of
fractional Hamiltonian systems given by
{eqnarray*}_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = &
\nabla W(t,u(t)) u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{N}) {eqnarray*
Spatial Hamiltonian identities for nonlocally coupled systems
We consider a broad class of systems of nonlinear integro-differential
equations posed on the real line that arise as Euler-Lagrange equations to
energies involving nonlinear nonlocal interactions. Although these equations
are not readily cast as dynamical systems, we develop a calculus that yields a
natural Hamiltonian formalism. In particular, we formulate Noether's theorem in
this context, identify a degenerate symplectic structure, and derive
Hamiltonian differential equations on finite-dimensional center manifolds when
those exist. Our formalism yields new natural conserved quantities. For
Euler-Lagrange equations arising as traveling-wave equations in gradient flows,
we identify Lyapunov functions. We provide several applications to
pattern-forming systems including neural field and phase separation problems.Comment: 39 pages, 1 figur
Existence of solution for perturbed fractional Hamiltonian systems
In this work we prove the existence of solution for a class of perturbed
fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00}
-{_{t}}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) - L(t)u(t) + \nabla
W(t,u(t)) = f(t), \end{eqnarray} where , , , is
a symmetric and positive definite matrix for all , and is the
gradient of at . The novelty of this paper is that, assuming is
coercive at infinity we show that (\ref{eq00}) at least has one nontrivial
solution.Comment: arXiv admin note: substantial text overlap with arXiv:1212.581
Interaction of sine-Gordon kinks with defects: The two-bounce resonance
A model of soliton-defect interactions in the sine-Gordon equations is
studied using singular perturbation theory. Melnikov theory is used to derive a
critical velocity for strong interactions, which is shown to be exponentially
small for weak defects. Matched asymptotic expansions for nearly heteroclinic
orbits are constructed for the initial value problem, which are then used to
derive analytical formulas for the locations of the well known two- and
three-bounce resonance windows, as well as several other phenomena seen in
numerical simulations.Comment: 26 pages, 17 figure
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