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This article proposes a set of ideas concerning the introduction of nonlinear analysis, particularly nonlinear PDE, in the theory of particles. The Quantum Mechanics theory is essentially a linear theory, due mainly to the fact that there was a the lack of nonlinear mathematics at the time of the discoveries in particle physics. The main idea is to perturb the Schroedinger equation by a nonlinear term. This nonlinear term has two main parts, a second order quasilinear differential operator responsible for the smoothing of the solutions and a nonlinear 0-order term with a singularity providing topology to the space. By minimizing the energy functional, solutions to the equation are obtained in each topological class. Then the qualitative properties of the soliton is analyzed. By rescaling arguments, the asymptotic behavior of the static solutions is studied. Next the evolution is studied, deriving stability of the soliton and the guidance formula. In this way the equations of Bohmian Mechanics are obtained. Most proofs are omitted, but in all cases a proper reference is given.
We present some new results that relate information to chaotic dynamics. In our approach the quantity of information is measured by the Algorithmic Information Content (Kolmogorov complexity) or by a sort of computable version of it (Computable Information Content) in which the information is measured by using a suitable universal data compression algorithm. We apply these notions to the study of dynamical systems by considering the asymptotic behavior of the quantity of information necessary to describe their orbits. When a system is ergodic, this method provides an indicator that equals the Kolmogorov-Sinai entropy almost everywhere. Moreover, if the entropy is null, our method gives new indicators that measure the unpredictability of the system and allows various kind of weak chaos to be classified. Actually, this is the main motivation of this work. The behavior of a 0-entropy dynamical system is far to be completely predictable except that in particular cases. In fact there are 0-entropy systems that exhibit a sort of weak chaos, where the information necessary to describe the orbit behavior increases with time more than logarithmically (periodic case) even if less than linearly (positive entropy case). Also, we believe that the above method is useful to classify 0-entropy time series. To support this point of view, we show some theoretical and experimental results in specific cases.
This paper deals with a new kind of generalized functions, called ``ultrafunctions", which have been introduced recently in  and developed in  and . Their peculiarity is that they are based on a Non Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions. Some applications of this kind will be presented in the second part of this paper.
This paper is devoted to the study of solitons whose existence is related to the ratio energy/charge. These solitons are called hylomorphic. In the first part of the paper we prove an abstract theorem on the existence of hylomorphic solitons which can be applied to the main situations considered in literature. In the second part, we apply this theorem to the nonlinear Schrödinger and Klein Gordon equations defined on a lattice.
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