1,721,092 research outputs found
Solitons in gauge theories: existence and dependence on the charge
No full textIn this paper we review recent results on the existence of non-topological solitons in classical relativistic nonlinear field theories. We follow the Coleman approach, which is based on the existence of two conservation laws, energy and charge. In particular we show that under mild assumptions on the nonlinear term it is possible to prove the existence of solitons for a set of admissible charges. This set has been studied for the nonlinear Klein–Gordon equation, and in this paper we state new results in this direction for the Klein–Gordon–Maxwell system
Entropy and complexity in dynamical systems and PDEs
No full textIn this chapter we aim at presenting applications of notions from Information Theory to the study of the statistical properties of dynamical systems. In particular we review the notion of Algorithmic Information Content, or Kolmogorov Complexity, and recall the definition of complexity of an orbit of a dynamical system. The main result is that for ergodic dynamical systems, the complexity of an orbit is almost everywhere constant and coincides with the Kolmogorov-Sinai entropy of the system. We remark that the interest in these results has at least two motivations: applications of this approach to time series of e.g. physical or biomedical origin; investigations on statistical properties of dynamical systems which present ``critical'' behaviours with respect to other classical indicators or for which it may be difficult to compute them
A Complexity approach to the soliton resolution conjecture
Full text linkThe soliton resolution conjecture is one of the most interesting open problems in the theory of nonlinear dispersive equations. Roughly speaking it asserts that a solution with generic initial condition converges to a finite number of solitons plus a radiative term. In this paper we use the complexity of a finite object, a notion introduced in Algorithmic Information Theory, to show that the soliton resolution conjecture is equivalent to the analogous of the second law of thermodynamics for the complexity of a solution of a dispersive equation
Long time dynamics of highly concentrated solitary waves for the nonlinear Schroedinger equation
Full text linkIn this paper we study the behavior of solutions of a nonlinear Schroedinger equation in presence of an external potential, which is allowed to be singular at one point. We show that the solution behaves like a solitary wave for long time even if we start from a unstable solitary wave, and its dynamics coincide with that of a classical particle evolving according to a natural effective Hamiltonian
An analytical formulation for the MOID and its consequences
No full textWe give an analytical formulation for the approximation of the Minimum Orbital Intersection Distance (MOID) between two elliptical orbits, and we apply it to the case of the Earth and of an asteroid. With this formulation we are able to find an algorithm to compute the variance of the MOID for a given asteroid and we make the computation in the case of the asteroids whose orbit has been determined, even if with a great uncertainty, to find cases of asteroids that have a good probabil- ity of having a small MOID even if their nominal orbit gives a relatively large MOID
Lattice determination of the topological susceptibility slope χ′ of 2d CPN−1 models at large N
No full textWe compute the topological susceptibility slope χ′, related to the second moment of the two-point correlator of the topological charge density, of 2d CPN-1 models for N=5, 11, 21 and 31 from lattice Monte Carlo simulations. Our strategy consists in performing a double limit: first, we take the continuum limit of χ′ at fixed smoothing radius in physical units; then, we take the zero-smoothing-radius limit. Since the same strategy can also be applied to 4d gauge theories and full QCD, where χ′ plays an intriguing theoretical and phenomenological role, this work constitutes a step toward the lattice investigation of this quantity in such models
Existence and multiplicity of stable bound states for the nonlinear Klein–Gordon equation
Full text linkWe are interested in the problem of existence of soliton-like solutions for the nonlinear Klein-Gordon equation. In particular we study some necessary and sufficient conditions on the nonlinear term to obtain solitons of a given charge. We remark that the conditions we consider can be easily verified. Moreover we show that multiplicity of solitons of the same charge is guaranteed by the ``shape'' of the nonlinear term for equations on , hence without appealing to topological or geometrical properties of the domain
On the generalised transfer operators of the Farey map with complex temperature
Full text linkWe consider the problem of showing that 1 is an eigenvalue for a family of
generalised transfer operators of the Farey map. This problem is related to the
spectral theory of the modular surface via the Selberg Zeta function and the
theory of dynamical zeta functions of maps. After briefly recalling these
connections, we show that the problem can be formulated for operators on an
appropriate Hilbert space and translated into a linear algebra problem for
infinite matrices.Comment: 12 page
Escape rates for the Farey map with approximated holes
Full text linkWe study the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. Due to the ergodic properties of the map, the standard theoretical approaches to this problem cannot be applied. It has been recently shown in cite{KM} how to apply the standard analytical methods to a piecewise linear version of the Farey map with holes depending on the associated partition, but their results cannot be obtained in the general case we consider here. To overcome these difficulties we propose here to study approximations of the hole by means of real analytic functions. We introduce a particular family of approximations and study numerically the behavior of the escape rate for approximated holes with vanishing measure. The results suggest that the scaling of the escape rate depends on the ``shape'' of the approximation, and we show that this is a typical feature of systems with an indifferent fixed point, not an artifact of the particular family we consider
Nonlinear Schroedinger equations with strongly singular potentials
Full text linkIn this paper we look for standing waves for nonlinear
Schr\"odinger equations
with cylindrically symmetric potentials vanishing at infinity and non-increasing,
and a nonlinear term satisfying weak assumptions. In
particular we show the existence of standing waves with
non-vanishing angular momentum with prescribed norm. The
solutions are obtained via a minimization argument, and the proof
is given for an abstract functional which presents lack of
compactness. As a particular case we prove the existence of standing waves with
non-vanishing angular momentum for the nonlinear hydrogen atom equation
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