## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

DCDS

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.

DCDS-B

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.
DCDS-S

This paper deals with a new kind of generalized functions, called
``ultrafunctions", which have been introduced recently in [5] and
developed in [10] and [11]. 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.

DCDS

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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