Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a loca... more Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localised packet and which preserves this localisation in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behaviour. In this paper we show a new mechanism which might produce solitary waves and solitons for a large class of equations, such as the nonlinear Klein-Gordon equation. We show that the existence of these kind of solitons, that we have called hylomorphic solitons, depends on a suitable energy/charge ratio. We show a variational method that allows to prove the existence of hylomorphic solitons and that turns out to be very useful for numerical applications. Moreover we introduce some classes of nonlinearities which admit hylomorphic solitons of different shapes and with different relations between charge, energy and frequency. Contents 1. Introduction 311 2. The abstract theory 312 3. The Nonlinear Klein-Gordon equation 316 4. Q-balls 319 5. Acknowledgments 331 References 331
In many situations, the notion of function is not sucient and it needs to be extended. A classica... more In many situations, the notion of function is not sucient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to dene generalized solutions of stationary problems in [4, 7, 9, 11, 12]. In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDEs, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDEs, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level.
Several different versions of the theory of numerosities have been introduced in the literature. ... more Several different versions of the theory of numerosities have been introduced in the literature. Here, we unify these approaches in a consistent frame through the notion of set of labels, relating numerosities with the Kiesler field of Euclidean numbers. This approach allows to easily introduce, by means of numerosities, ordinals and their natural operations, as well as the Lebesgue measure as a counting measure on the reals.
Ultrafunctions are a particular class of generalized functions defined on a hyperreal field {\mat... more Ultrafunctions are a particular class of generalized functions defined on a hyperreal field {\mathbb{R}^{*}\supset\mathbb{R}} that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.
Ultrafunctions are a particular class of functions defined on a hyperreal field {\mathbb{R}^{\ast... more Ultrafunctions are a particular class of functions defined on a hyperreal field {\mathbb{R}^{\ast}\supset\mathbb{R}} . They have been introduced and studied in some previous works [2, 6, 7]. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.
The British Journal for the Philosophy of Science, 2016
Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We ... more Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.
In this paper we study existence, multiplicity and concentration of solutions for the following n... more In this paper we study existence, multiplicity and concentration of solutions for the following nonlinear field equation
Progress in Nonlinear Differential Equations and Their Applications, 2006
We study the existence of a solution of the problem $ \left\{ \begin{gathered} - \Delta u + V\lef... more We study the existence of a solution of the problem $ \left\{ \begin{gathered} - \Delta u + V\left( x \right)u = f'\left( u \right)x \in \mathbb{R}^N , \hfill \\ u\left( x \right) > 0, \hfill \\ \end{gathered} \right. $ \left\{ \begin{gathered} - \Delta u + V\left( x \right)u = f'\left( u \right)x \in \mathbb{R}^N , \hfill \\ u\left( x \right)
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a loca... more Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localised packet and which preserves this localisation in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behaviour. In this paper we show a new mechanism which might produce solitary waves and solitons for a large class of equations, such as the nonlinear Klein-Gordon equation. We show that the existence of these kind of solitons, that we have called hylomorphic solitons, depends on a suitable energy/charge ratio. We show a variational method that allows to prove the existence of hylomorphic solitons and that turns out to be very useful for numerical applications. Moreover we introduce some classes of nonlinearities which admit hylomorphic solitons of different shapes and with different relations between charge, energy and frequency. Contents 1. Introduction 311 2. The abstract theory 312 3. The Nonlinear Klein-Gordon equation 316 4. Q-balls 319 5. Acknowledgments 331 References 331
In many situations, the notion of function is not sucient and it needs to be extended. A classica... more In many situations, the notion of function is not sucient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to dene generalized solutions of stationary problems in [4, 7, 9, 11, 12]. In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDEs, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDEs, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level.
Several different versions of the theory of numerosities have been introduced in the literature. ... more Several different versions of the theory of numerosities have been introduced in the literature. Here, we unify these approaches in a consistent frame through the notion of set of labels, relating numerosities with the Kiesler field of Euclidean numbers. This approach allows to easily introduce, by means of numerosities, ordinals and their natural operations, as well as the Lebesgue measure as a counting measure on the reals.
Ultrafunctions are a particular class of generalized functions defined on a hyperreal field {\mat... more Ultrafunctions are a particular class of generalized functions defined on a hyperreal field {\mathbb{R}^{*}\supset\mathbb{R}} that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.
Ultrafunctions are a particular class of functions defined on a hyperreal field {\mathbb{R}^{\ast... more Ultrafunctions are a particular class of functions defined on a hyperreal field {\mathbb{R}^{\ast}\supset\mathbb{R}} . They have been introduced and studied in some previous works [2, 6, 7]. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.
The British Journal for the Philosophy of Science, 2016
Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We ... more Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.
In this paper we study existence, multiplicity and concentration of solutions for the following n... more In this paper we study existence, multiplicity and concentration of solutions for the following nonlinear field equation
Progress in Nonlinear Differential Equations and Their Applications, 2006
We study the existence of a solution of the problem $ \left\{ \begin{gathered} - \Delta u + V\lef... more We study the existence of a solution of the problem $ \left\{ \begin{gathered} - \Delta u + V\left( x \right)u = f'\left( u \right)x \in \mathbb{R}^N , \hfill \\ u\left( x \right) > 0, \hfill \\ \end{gathered} \right. $ \left\{ \begin{gathered} - \Delta u + V\left( x \right)u = f'\left( u \right)x \in \mathbb{R}^N , \hfill \\ u\left( x \right)
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