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## Publications

Publications (39)

In the present paper we study the nonlinear system ut + [ϕ(u)]x + v = 0, vt + ψ(u)vx = 0 as a model for the one-dimensional dynamics of dark matter. We prove that under certain conditions this system, such as the Gurevich-Zybin system, can also explain why the observed rotation speed (relative to the galactic center) of stars near galactic halos do...

A Riemann problem for the conservation law ut+[ϕ(u)]x=kH(x-vt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t}+[\phi (u)]_{x}=kH(x-vt)$$\end{document}, where x, t,...

In the setting of a product of distributions, we define a concept of a solution for the Brio system \(u_{t}+\frac{1}{2}(u^{2}+v^{2})_{x}=0\), \(v_{t} +(uv-v)_{x}=0\), which extends the classical solution concept. New results about that product allow us to establish necessary and sufficient conditions for the propagation of distributional travelling...

We consider the nonlinear 2 × 2 Gurevich–Zybin system as a model for the one-dimensional dynamics of dark matter. In this setting, we evaluate the movement of a particle of mass subjected to a possible jump discontinuity in the hydrodynamic speed. This problem is solved by extending the concept of usual solution to the framework of a product of dis...

In this paper travelling waves with distributional profiles for the Camassa–Holm equation are studied. Using a product of distributions a new solution concept is introduced which extends the classical one. As a consequence, necessary and sufficient conditions for the propagation of a distributional profile are established and we present examples wi...

The present paper studies, strictly within the theory of distributions, the propagation of travelling waves with a distributional profile in one-dimensional models ruled by convection–diffusion–reaction equations. By applying a product of distributions (not defined by approximation processes), a rigorous concept of a solution which extends the clas...

Using a solution concept defined in the setting of a product of distributions, we consider the nonlinear equation f(t)ut + (u2)x = 0, where f is a continuous function. This equation can be regarded a generalization of Burgers inviscid equation and allows to study several kinds of interaction of δ-waves. If f(t) ≠ 0 for all t, collisions of δ-waves...

In this paper, we prove that the model known as the zero pressure gas dynamic system with certain singular measures as initial conditions may develop waves defined by distributions that are not measures. This study is done strictly within the theory of distributions and does not assume the knowledge of any classical result about conservation laws....

The present paper concerns the study of travelling waves for the nonconservative model ut+(12u2)x=σx, σt+uσx=k2ux coming from elastodynamics. For this model, that does not admit nonconstant global smooth solutions, necessary and sufficient conditions for the propagation of distributional profiles are established. As an application we study the prop...

The present paper concerns the system ut + [ ϕ ( u )] x = 0, vt + [ψ( u ) v ] x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ , ψ : ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u , v . For this purpose, a consistent extension of the classical solution...

The present paper concerns a Riemann problem for a conservation law with a nonlinear flux involving a step function. In a convenient space of distributions, we will see the emergence of a delta standing wave which extends the corresponding result, obtained by Sun in 2013 and also by Shen in 2014, for a linear flux. In the nonlinear setting, a new p...

The present paper concerns the study of a Riemann problem for the conservation law ut + [ϕ (u)]x = kδ (x − vt) where x, t, k, v and u = u(x,t) are real numbers. We consider ϕ an entire function taking real values on the real axis and δ stands for the Dirac measure. Within a convenient space of distributions we will explicitly see the possible emerg...

We consider a Riemann problem for the shallow water system (Formula presented.), (Formula presented.) and evaluate all singular solutions of the form (Formula presented.), (Formula presented.), where (Formula presented.) are (Formula presented.)-functions of time t, H is the Heaviside function, and (Formula presented.) stands for the Dirac measure...

Newton’s second law is applied to study the motion of a particle subjected to a time dependent impulsive force containing a Dirac delta distribution. Within this setting, we prove that this problem can be rigorously solved neither by limit processes nor by using the theory of distributions (limited to the classical Schwartz products). However, usin...

We consider the system ut +(u²/2)x =σx, σt +uσx = k²ux, where k is a real number and the unknowns u(x, t) and σ(x, t) belong to convenient spaces of distributions. For this simplified model from elastodynamics, a rigorous solution concept defined in the setting of a distributional product is used. The explicit solution of a Riemann problem and the...

The present paper concerns the study of distributional travelling waves for the model problem ut + (u² − υ)x = 0, υt + (u³/3 − u)x = 0, also called the KeyfitzKranzer system. In the setting of a product of distributions, which is not defined by approximation processes, we are able to define a rigourous concept of a solution which extends the classi...

The present paper concerns the study of a Riemann problem for the system , with a one dimensional space variable. We consider φ an entire function that takes real values on the real axis. Under certain conditions, this system provides solutions to the pressureless gas dynamics and the isentropic fluid dynamics systems. We get all solutions of this...

The system of conservation laws \({u_t} + {\left( {\frac{{{u^2} + {v^2}}}{2}} \right)_x} = 0\), v
t + (uv − v)x = 0 with the initial conditions u(x, 0) = l
0 + b
0H(x), v(x, 0) = k
0 + c
0H(x), where H is the Heaviside function is studied. This strictly hyperbolic system was introduced by M. Brio in 1988 and provides a simplified model for the magn...

We study the possibility of collision of a δ-wave with a stationary δ-wave in a model ruled by equation f (t)u t + [u 2 − β (x − γ(t))u] x = 0, where f , β and γ are given real functions and u = u(x,t) is the state variable. We adopt a solution concept which is a consistent extension of the classical solution concept. This concept is defined in the...

The Brio system is a 2 × 2 fully nonlinear system of conservation laws which arises as a simplified model in the study of plasmas. The present paper offers explicit solutions to this system subjected to initial conditions containing Dirac masses. The concept of a solution emerges within the framework of a distributional product and represents a con...

In the setting of a distributional product, we consider a Riemann problem for the Hunter-Saxton equation
[
u
t
+
(
(
1
/
2
)
u
2
)
x
]
x
=
(
1
/
2
)
u
x
2
in a convenient space of discontinuous functions. With the help of a consistent extension of the classical solution concept, two classes of discontinuous solutions are obta...

In the setting of a product of distributions which is not defined by approximation processes, we are able to consider a Riemann problem for the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0, with the unknown states u, v in convenient spaces of distributions and ϕ, ψ : ℝ → ℝ continuous. A consistent extension of the classical solution concept will show...

This paper contains a study of propagation of singular travelling waves u(x, t) for conservation laws u
t
+[ϕ(u)]
x
= ψ(u), where ϕ, ψ are entire functions taking real values on the real axis. Conditions for the propagation of wave profiles β + mδ and β + mδ′ are presented (β is a real continuous function, m ≠ 0 is a real number and δ′ is the deri...

In this paper, we restrict our attention to the advection-reaction equation u
t
+ [ϕ(u)]x
= ψ(u), where ϕ and ψ are entire functions. Conditions for the propagation of a distributional wave profile are presented and the wave speed is evaluated. As an example, we prove that, under certain conditions, the propagation of delta-waves in models ruled by...

We define the concept of an α-generalized solution for the ordinary differential equation X ′ = UX + V, where X is the unknown and U, V belong to certain spaces which may contain singular distributions. This concept results a consistent extension of the concept of a classical distributional solution and it is defined in the setting of a distributio...

By using a product of distributions, the existence and collision of soliton delta-waves for a singular perturbation of the Burgers conservative equation are established. We also prove that singular solitons under collision behave as in classical soliton collision (for example, as described by the Korteweg–de Vries equation). The impossibility of tw...

With the help of our theory of multiplication of distributions it is possible to give a meaning in D to the composition φ • T , where φ is an entire function and T belongs to a certain class of strongly singular distributions. As an application we are able to prove that, in our set-ting, the nonlinear conservation law u t +[φ(u)] x = 0 has solution...

The propagation of travelling waves is a relevant physical phenomenon. As usual the understanding of a real propagating wave depends upon a correct formulation of a idealized model. Discontinuous functions, Dirac-δ measures and their distributional derivatives are, respectively, idealizations of sharp jumps, localized high peaks and single sharp lo...

Burgers equation for inviscid fluids is a simplified case of Navier–Stokes equation which corresponds to Euler equation for ideal fluids. Thus, from a variational viewpoint, Burgers equation appears naturally in its nonconservative form. In this form, a consistent concept of a weak solution cannot be formulated because the classical distribution th...

With the help of our distributional product we define four types of new solutions for first order linear systems of ordinary differential equations with distributional coefficients. These solutions are defined within a convenient space of distributions and they are consistent with the classical ones. For example, it is shown that, in a certain sens...

We treat linear partial differential equations of first order with distributional coefficients naturally related to physical conservation laws in the spirit of our preceding papers (which concern ordinary differential equations): the solutions are consistent with the classical ones. Under compatibility conditions we prove uniqueness and existence r...

In [3] we have considered the n th order linear Cauchy problem for a class of differential equations with distributional coefficients. We have extended the concept of solution of this problem and we have proved that these solutions are consistent with the classical solutions. Here we give necessary and sufficient conditions for existence, in this e...

In [1, 2] we introduced a distribution product, invariant under the action of compact Lie groups of linear transformations. This product has application to non-relativistic physics. Here we present a general version of the product, invariant under various groups, including the Lorentz group.

The possibility of solving differential equations with distribution coefficients is considered within our theory of products. We study linear differential equations as well as Burgers’ nonlinear equation of fluid dynamics, obtaining for it pure shock-wave solutions.