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In this paper we study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the ratio energy/charge of a function, which turns out to be a useful approach for many field equations.

The adoption of the Kolmogorov–Sinai entropy is becoming a popular research tool among physicists, especially when applied to a dynamical system fitting the conditions of validity of the Pesin theorem. The study of time series that are a manifestation of system dynamics whose rules are either unknown or too complex for a mathematical treatment, is still a challenge since the KS entropy is not computable, in general, in that case. Here we present a plan of action based on the joint action of two procedures, both related to the KS entropy, but compatible with computer implementation through fast and efficient programs. The former procedure, called compression algorithm sensitive to regularity (CASToRE), establishes the amount of order by the numerical evaluation of algorithmic compressibility. The latter, called complex analysis of sequences via scaling and randomness assessment (CASSANDRA), establishes the complexity degree through the numerical evaluation of the strength of an anomalous effect. This is the departure, of the diffusion process generated by the observed fluctuations, from ordinary Brownian motion. The CASSANDRA algorithm shares with CASToRE a connection with the Kolmogorov complexity. This makes both algorithms especially suitable to study the transition from dynamics to thermodynamics, and the case of non-stationary time series as well. The benefit of the joint action of these two methods is proven by the analysis of artificial sequences with the same main properties as the real time series to which the joint use of these two methods will be applied in future research work. [Copyright &y& Elsevier]

Abstract:: In this paper we study a class of Lorentz invariant nonlinear field equations in several space dimensions. The main purpose is to obtain soliton-like solutions. These equations were essentially proposed by C. H. Derrick in a celebrated paper in 1964 as a model for elementary particles. However, an existence theory was not developed. The fields are characterized by a topological invariant, the charge. We prove the existence of a static solution which minimizes the energy among the configurations with nontrivial charge. Moreover, under some symmetry assumptions, we prove the existence of infinitely many solutions, which are constrained minima of the energy. More precisely, for every n∈:N there exists a solution of charge n.

We obtain a discreteness condition for the spectrum of the Schrödinger operator $ - \Delta + V(x)$V is not bounded from below.

We prove that the Hamiltonian system{p˙=−∂V∂qq˙=pp,q∈Rn;V∈C2(Rn)

Let a, U ∈ C2(Ω) where Ω is a bounded set in ℝnand let(*)L(x,ξ)=12a(x)|ξ|2−U(x),x∈Ω;ξ∈ℝn.

In this paper we study the existence of a periodic trajectory with prescribed period, which bounoes against the boundary of an open subset of ℝN, in presence of a potential field. We prove the existence of periodic solutions with at most N + 1 bounce points.

In this paper we study the problem of the existence of geodesies in static space-times which are a particular case of Lorentz manifolds. We prove multiplicity results about geodesies joining two given events and about periodic trajectories having a prescribed period.

Let Ω={(x,y)∈ℝ2|x2+y2<1},Sn={v∈ℝn+1||v|=1}(n⩾ 2), and let γ∈ C2,δ(∂Ω; Sn). We study the following problem(*){u∈C2(Ω;Sn)∩C0(Ω¯;Sn)−Δu=u|∇u|2u=γon∂Ω.