5,327 research outputs found
Existence of uncountable many homoclinic solutions to periodic orbits in a center manifolds
No full textConsider a Lagrangian of the form
L(x, \dot{x}, q, \dot{q})=\frac{1}{2}(\dot{x}^2-x^2)+ \frac{1}{2}\dot{q]^2+(1+ \delta (x))V(q), \quad x, q \in \mathbb{R}.
Assuming bounded, periodic in with a strict global minimum at , it is shown for each the existence of a homoclinic solution satisfying
\begin{align*}
& \lim_{t \to - \infty} q(t)=0 \\
& \lim_{t \to + \infty} q(t)=2 \pi \\
& \lim_{t \to - \infty} |x(t)- R \cos t|=0 \\
& \lim_{t \to + \infty} |x(t)- R \cos (t+ \varphi)|=0.
\end{align*
Existence and asymptotics for self-dual periodic vortices of topological-type
No full textWe consider vortices of topological-type for a class of selfdual gauge models, with periodic
boundary conditions and as the ratio of the vortex core size to the separation distance between vortex
points (the scaling parameter) tends to zero. We use a gluing technique (shadowing lemma) for solutions
to the corresponding semilinear elliptic equation on the plane, where the vortex points are periodically
arranged. This approach is particularly convenient and natural for the study of the asymptotics as the
scaling parameter tends to zero. In particular, we prove a factorization ansatz for multivortex solutions,
up to an error which is exponentially small
Uniqueness of topological solutions for a class of self-dual vortex theories
No full textWe study the solutions of topological type for a class of self-dual vortex theories in two dimensions. We consider the regime corresponding to the limit of small vortex core size with respect to the separation distance between vortices, namely as the scaling parameter tends to zero. Using a gluing technique for the corresponding nonlinear elliptic equation on the plane, with any number (finite or countable) of prescribed singular sources, we prove the existence of multi-vortex solutions which behave as a single vortex solution near each vortex point, up to an error exponentially small, as . Moreover, in the physically relevant cases, namely when the vortex points are either finite or periodically arranged in the plane, we prove that the multi-vortex solution satisfying a ‘topological condition' is unique, for sufficiently small
Stationary solutions for the non-linear Hartree equation with a slowly varying potential
No full textWe consider the non-linear Hartree equation with a slowly varying external potential Vε and a short range, attractive two-body interaction W. We prove the existence of stationary solutions which are approximatively given by a superposition of several Hartree solitons with their center of mass positions behaving, at the leading order, as classical particles at rest in the background potential Vε
Existence of homoclinic solutions to periodic orbits in a center manifold
No full textConsider a Lagrangian of the form
where ,. Assuming that is bounded and , periodic in , is such that , we prove existence of infinitely many solutions homoclinic to periodic orbits in the center manifold , of the corresponding system
Multibump solutions homoclinic to periodic orbits of large energy in a center manifold
No full textWe consider the Lagrangian
, -periodic and bounded. The corresponding Euler–Lagrange equations have as the origin a saddle-centre stationary point whose (globally defined) centre manifold is foliated in periodic orbits. We prove that for small enough there exist multi-bump solutions, of large energy and heteroclinic to periodic orbits in the centre manifold
Homoclinic solutions to invariant tori in a center manifold
No full textWe consider the Lagrangian
where is non-negative, periodic in and such that . We prove, using critical point theory, the existence of infinitely many solutions of the corresponding Euler-Lagrange equations which are asymptotic, as t \to \pm \infinity, to invariant tori in the center manifold of the origin, that is, to solutions of the form , ,
Asymptotics for selfdual vortices on the torus and on the plane: a gluing technique
No full textWe consider multivortex solutions for the selfdual Abelian Higgs model,
as the ratio of the vortex core size to the separation distance
between vortex points (the scaling parameter) tends to zero.
To this end, we use a gluing technique (a shadowing lemma) for solutions
to the corresponding semilinear equation on the plane,
allowing any number (finite or countable) of arbitrarily
prescribed singular sources.
Our approach is particularly convenient and natural for the study
of the asymptotics.
In particular, in the physically relevant case where the vortex
points are either finite or periodically arranged in the plane,
we prove that a frequently used factorization ansatz for multivortex
solutions is rigorously satisfied,
up to an error which is exponentially small
Existence and asymptotics for self-dual periodic vortices of topological-type
No full textWe consider vortices of topological-type for a class of selfdual gauge models, with periodic boundary conditions and as the ratio of the vortex core size to the separation distance between vortex points (the scaling parameter) tends to zero. We use a gluing technique (shadowing lemma) for solutions to the corresponding semilinear elliptic equation on the plane, where the vortex points are periodically arranged. This approach is particularly convenient and natural for the study of the asymptotics as the scaling parameter tends to zero. In particular, we prove a factorization ansatz for multivortex solutions, up to an error which is exponentially small
Asymptotics for selfdual vortices on the torus and on the plane: a gluing technique
No full textWe consider multivortex solutions for the selfdual Abelian Higgs model,
as the ratio of the vortex core size to the separation distance
between vortex points (the scaling parameter) tends to zero.
To this end, we use a gluing technique (a shadowing lemma) for solutions
to the corresponding semilinear equation on the plane,
allowing any number (finite or countable) of arbitrarily
prescribed singular sources.
Our approach is particularly convenient and natural for the study
of the asymptotics.
In particular, in the physically relevant case where the vortex
points are either finite or periodically arranged in the plane,
we prove that a frequently used factorization ansatz for multivortex
solutions is rigorously satisfied,
up to an error which is exponentially small
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