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Igor Leite Freire

Publications and source records attributed to Igor Leite Freire.

At least 37 records · Page 2Linked to original sources
arXiv2019

Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation

In this paper we consider a four-parameter equation including the Camassa-Holm and the Dulling-Gottwald-Holm equations, among others. We prove the existence and uniqueness of solutions to a Cauchy problem involving the equation using Kato's approach. Conservation laws of the equation up to second order are also investigated. From these conservation laws we establish some properties of the solutions of the equation under consideration and we also find a quadrature for it. The quadrature obtained is of capital importance in a classification of bounded travelling wave solutions of the equation studied. We also find some explicit solutions, given in terms of elliptic integrals. Finally, we classify the members of the equation describing pseudo-spherical surfaces.

math-ph↗
arXiv2018

A family of wave equations with some remarkable properties

We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found to two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair we construct a Miura-type transformation relating the original equation to the KdV equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schrödinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second order nonlinear evolution equation.

math-ph↗
arXiv2017

A generalised multicomponent system of Camassa-Holm-Novikov equations

In this paper we introduce a two-component system, depending on a parameter $b$, which generalises the Camassa-Holm ($b=1$) and Novikov equations ($b=2$). By investigating its Lie algebra of classical and higher symmetries up to order $3$, we found that for $b\neq 2$ the system admits a $3$-dimensional algebra of point symmetries and apparently no higher symmetries, whereas for $b=2$ it has a $6$-dimensional algebra of point symmetries and also higher order symmetries. Also we provide all conservation laws, with first order characteristics, which are admitted by the system for $b=1,2$. In addition, for $b=2$, we show that the system is a particular instance of a more general system which admits an $\mathfrak{sl}(3,\mathbb{R})$-valued zero-curvature representation. Finally, we found that the system admits peakon solutions and, in particular, for $b=2$ there exist 1-peakon solutions with non-constant amplitude.

nlin.SI↗
arXiv2017

Mathematical modelling for the transmission of dengue: symmetry and traveling wave analysis

In this paper we propose some mathematical models for the transmission of dengue using a system of reaction-diffusion equations. The mosquitoes are divided into infected, uninfected and aquatic subpopulations, while the humans, which are divided into susceptible, infected and recovered, are considered homogeneously distributed in space and with a constant total population. We find Lie point symmetries of the models and we study theirs temporal dynamics, which provides us the regions of stability and instability, depending on the values of the basic offspring and the basic reproduction numbers. Also, we calculate the possible values of the wave speed for the mosquitoes invasion and dengue spread and compare them with those found in the literature.

math.AP↗
arXiv2017

Peakon and kink solutions for a system of $0-$Holm-Staley equations

In this paper we consider a two-component system of the Holm-Staley equation with no stretching and a one-parameter nonlinearity in the convection term. Point symmetries are found, conditions for the existence of multipeakon and multikink solutions are determined. Solutions for the 1-peakon and 1-kink are explicitly obtained and they exhibit a behaviour quite different of their analogous in the scalar equation.

math-ph↗
arXiv2016

Study of a fifth order PDE using symmetries

We study a family of PDEs, which was derived as an approximation of an extended Lotka-Volterra system, from the point of view of symmetries. Also, by performing the self adjoint classification on that family we offer special cases possessing non trivial conservation laws. Using both classifications we justify the particular cases studied in the literature, and we give additional cases that may be of importance.

math.AP↗
arXiv2015

A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations

A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced. A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family. These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved $H^1$ norm and it possesses $N$-peakon solutions, when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case $N=2$.

nlin.SI↗
arXiv2015

Symmetry analysis of a class of autonomous even-order ordinary differential equations

A class of autonomous, even-order ordinary differential equations is discussed from the point of view of Lie symmetries. It is shown that for a certain power nonlinearity, the Noether symmetry group coincides with the Lie point symmetry group. First integrals are established and exact solutions are found. Furthermore, this paper complements, for the one-dimensional case, some results in the literature of Lie group analysis of poliharmonic equations and Noether symmetries obtained in the last twenty years. In particular, it is shown that the exceptional negative power discovered in [A. H. Bokhari, F. M. Mahomed and F. D. Zaman, Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation, J. Math. Phys., vol. 51, 053517, (2010)] is a member of an one-parameter family of exceptional powers in which the Lie symmetry group coincides with the Noether symmetry group.

math-ph↗
arXiv2014

Strict self-adjointness and shallow water models

We consider a class of third order equations from the point of view of strict self-adjointness. Necessary and sufficient conditions to the investigated class be strictly self-adjoint are obtained. Then, from a strictly self-adjoint subclass we consider those who admits a suitable scaling transformation. Consequently it is derived a family of equations including the Benjamin-Bona-Mahony, Camassa-Holm and Novikov equation. By a suitable choice of the parameters, we deduce an one-parameter family of equations unifying the last two mentioned equations. Then, using some recent techniques for constructing conserved vectors, we show that from the scale invariance it is obtained as a conserved density the same quantity employed to construct one of the well known Hamiltonians for the cited integrable equations.

math-ph↗
arXiv2012

Group Analysis of the Novikov Equation

We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential equations, we find the conservation law corresponding to the dilations symmetry and show that other symmetries do not provide nontrivial conservation laws. Then we investigat the invariant solutions.

math-ph↗
arXiv2012

New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order

In a recent communication Nail Ibragimov introduced the concept of nonlinearly self-adjoint differential equation [N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., vol. 44, 432002, 8 pp., (2011)]. In the present communication a nonlinear self-adjoint classification of a general class of fifth-order evolution equation with time dependent coefficients is presented. As a result five subclasses of nonlinearly self-adjoint equations of fifth-order and four subclasses of nonlinearly self-adjoint equations of third-order are obtained. From the Ibragimov's theorem on conservation laws [N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., vol. 333, 311--328, (2007)] conservation laws for some of these equations are established.

math-ph↗
arXiv2011

Conservation laws for self-adjoint first order evolution equations

In this work we consider the problem on group classification and conservation laws of the general first order evolution equations. We obtain the subclasses of these general equations which are quasi-self-adjoint and self-adjoint. By using the recent Ibragimov's Theorem on conservation laws, we establish the conservation laws of the equations admiting self-adjoint equations. We illustrate our results applying them to the inviscid Burgers' equation. In particular an infinite number of new symmetries of these equations are found and their corresponding conservation laws are established.

math-ph↗
arXiv2010

On the paper "Symmetry analysis of wave equation on sphere" by H. Azad and M. T. Mustafa

Using the scalar curvature of the product manifold S^{2}X R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl., 333 (2007) 1180--1888] to nonlinear Klein-Gordon equations on the two-dimensional sphere.

math.AP↗
arXiv2009

Special Conformal Groups of a Riemannian Manifold and Lie Point Symmetries of the Nonlinear Poisson Equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on $M$ are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.

math.AP↗
arXiv2008

Symmetry Coefficients of Semilinear Partial Differential Equations

We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial differential equations in Analysis, Geometry and Mathematical - Physics are given in order to enlighten the main result.

math.AP↗