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Enrique G. Reyes

Publications and source records attributed to Enrique G. Reyes.

At least 19 recordsLinked to original sources
arXiv2026

Lund--Regge Geometry and Integrability of a Generalized Konno--Oono System

We extend recent work on the relation between classical surface theory and partial differential equations, focusing on equations of pseudo-spherical type in the sense of Chern--Tenenblat and on a non-trivial generalization motivated by the Lund--Regge system describing surfaces immersed in $S^3$. As our main application, we study a generalized Konno--Oono system with three dependent variables introduced in a previous paper by one of the authors. We construct an associated parameter-dependent overdetermined linear problem and {\em we establish the existence of infinitely many non-trivial local conservation laws}, hence, integrability. The latter is the most technically demanding part of this paper: it requires a refined analysis of a Riccati pseudo-potential expansion, the use of stereographic coordinates at the full equation manifold level, the construction of special representatives, and a direct proof of non-triviality in horizontal cohomology. We also analyse an illustrative class of travelling wave solutions and show that they can be used to generate surfaces immersed in $S^3$ whose Gaussian curvature changes sign periodically, while their mean curvature are non-vanishing periodic functions. In a limit case, we obtain surfaces that are locally congruent to generalized Clifford tori.

nlin.SI↗
arXiv2025

On the geometry of profinite diffeological spaces

We consider the class of profinite diffeological spaces, that is, diffeological spaces which diffeologies are deduced by pull-back of diffeologies on finite-dimensional manifolds through a system of projection mappings. This class includes inductive limits of finite-dimensional manifolds, as well as solution spaces of differential relations and spaces that agree with cylindrical approximations. We analyze tangent and cotangent spaces, differential forms, Riemannian metrics, connections, differential operators, Laplacians, symplectic forms, momentum maps and the relation between de Rham and singular cohomology on them, unifying constructions partially developed in various contexts.

math.DG↗
arXiv2024

Kadomtsev-Petviashvili hierarchies with non-formal pseudo-differential operators, non-formal solutions, and a Yang-Mills--like formulation

We start from the classical Kadomtsev-Petviashvili hierarchy posed on formal pseudo-differential operators, and we produce two hierarchies of non-linear equations posed on non-formal pseudo-differential operators lying in the Kontsevich and Vishik's odd class, one of them with values in formal pseudo-differential operators. We prove that the corresponding Zakharov-Shabat equations hold in this context, and we express one of our hierarchies as the minimization of a class of Yang-Mills action functionals on a space of pseudo-differential connections whose curvature takes values in the Dixmier ideal. We finish by comparing our Kadomtsev-Petviashvili hierarchies in terms of the solutions that they produce to the KP-II equation: existence, uniqueness and formality.

math-ph↗
arXiv2022

Frolicher structures, diffieties, and a formal KP hierarchy

We propose a definition of a diffiety based on the theory of Frolicher structures. As a consequence, we obtain a natural Vinogradov sequence and, under the assumption of the existence of a suitable derivation, we can form on it a Kadomtsev-Petviashvili hierarchy which is well-posed.

nlin.SI↗
arXiv2022

Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

In this paper we discuss integrable higher order equations {\em of Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically integrable", that is, they describe one-parametric families of pseudo-spherical surfaces, in a sense explained in Section 1, and they are integrable in the sense of zero curvature formulation ($\simeq$ Lax pair) with infinitely many local conservation laws. The major focus of the present paper is on a specific fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is, weak bounded solutions with differentiable first derivative and continuous and bounded second derivative, but such that any higher order derivative blows up. Furthermore, we investigate the Cauchy problem of this fifth order CH-type equation on the real line and prove local well-posedness under the initial conditions $u_0 \in H^s(\mathbb{R})$, $s > 7/2$. In addition, we study conditions for global well-posedness in $H^4(\mathbb{R})$ as well as conditions causing local solutions to blow up in a finite time. We conclude our paper with some comments on the geometric content of the high order CH-type equations.

math.AP↗
arXiv2022

On $(t_2,t_3)-$Zakharov-Shabat equations of generalized Kadomtsev-Petviashvili hierarchies

We review the integration of the KP hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation; an algebra of functions equipped with a derivation that generalizes the gradient operator; an algebra of quaternion-valued functions; a differential Lie algebra; an algebra of polynomials equipped with the Pincherle differential; a Moyal algebra. In all these cases we can formulate and solve the Cauchy problem of the KP hierarchy. Also, in each of these cases we derive different Zakharov-Shabat $(t_2,t_3)$-equations -- that is, different Kadomtsev-Petviashvili equations -- and we prove existence of solutions arising from solutions to the corresponding KP hierarchy.

nlin.SI↗
arXiv2022

Rigid body equations on spaces of pseudo-differential operators with renormalized trace

We equip the regular Fréchet Lie group of invertible, odd-class, classical pseudodifferential operators $Cl^{0,*}_{odd}(M,E)$ -- in which $M$ is a compact smooth manifold and $E$ a (complex) vector bundle over $M$ -- with pseudo-Riemannian metrics, and we use these metrics to introduce a large class of rigid body equations. We adapt to our infinite-dimensional setting Manakov's classical observation on the integrability of Euler's equations for the rigid body, and we show that our equations can be written in Lax form (with parameter) and that they admit an infinite number of integrals of motion. We also prove the existence of metric connections, we show that our rigid body equations determine geodesics on $Cl^{0,*}_{odd}(M,E)$, and we present rigorous formulas for the corresponding curvature and sectional curvature. Our main tool is the theory of renormalized traces of pseudodifferential operators on compact smooth manifolds without boundary.

math.DG↗
arXiv2020

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator for our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral operators in which this correspondence seems to take place. Also, motivated by Mulase's work on the KP hierarchy, we prove a group factorization theorem for our group of Fourier integral operators.

math-ph↗
arXiv2020

Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients which are series in formal parameters, describe a rigorous derivation of the Hamiltonian interpretation of the KP hierarchy, and discuss how solutions depending on formal parameters can lead to sequences of functions converging to a class of solutions of the standard KP-I equation.

math-ph↗
arXiv2019

Compacton equations and integrability: the Rosenau-Hyman and Cooper-Shepard-Sodano equations

We study integrability --in the sense of admitting recursion operators-- of two nonlinear equations which are known to possess compacton solutions: the $K(m,n)$ equation introduced by Rosenau and Hyman \[ D_t(u) + D_x(u^m) + D_x^3(u^n) = 0 \; , \] and the $CSS$ equation introduced by Coooper, Shepard, and Sodano, \[ D_t(u) + u^{l-2}D_x(u) + αp D_x (u^{p-1} u_x^2) + 2αD_x^2(u^p u_x) = 0 \; . \] We obtain a full classification of {\em integrable $K(m,n)$ and $CSS$ equations}; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of $CSS$.

nlin.SI↗
arXiv2018

Nonlinear pseudo-differential equations defined by elliptic symbols on ${\lp}$ and the fractional Laplacian

We develop an $L^p(\mathbb{R}^n)$-functional calculus appropriated for interpreting "non-classical symbols" of the form $a(-Δ)$, and for proving existence in $L^q(\mathbb{R}^n)$, some $q > p$, of solutions to nonlinear pseudo-differential equations of the form $[1 + a(-Δ)]^{s/2} (u) = V(\cdot, u)$. More precisely, we use the theory of Fourier multipliers for constructing suitable domains on which the formal operator appearing in the above equation can be rigorously defined, and we prove existence of solutions belonging to these domains. We also include applications of the theory to equations of physical interest involving the fractional Laplace operator such as the Allen-Cahn equation.

math.AP↗
arXiv2017

A Laplace transform approach to linear equations with infinitely many derivatives and zeta-nonlocal field equations

We study existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives. We develop a natural framework based on Laplace transform as a correspondence between appropriate $L^p$ and Hardy spaces: this point of view allows us to interpret rigorously operators of the form $f(\partial_t)$ where $f$ is an analytic function such as (the analytic continuation of) the Riemann zeta function. We find the most general solution to the equation \begin{equation*} f(\partial_t) ϕ= J(t) \; , \; \; \; t \geq 0 \; , \end{equation*} in a convenient class of functions, we define and solve its corresponding initial value problem, and we state conditions under which the solution is of class $C^k,\, k \geq 0$. More specifically, we prove that if some a priori information is specified, then the initial value problem is well-posed and it can be solved using only a {\em finite number} of local initial data. Also, motivated by some intriguing work by Dragovich and Aref'eva-Volovich on cosmology, we solve explicitly field equations of the form \begin{equation*} ζ(\partial_t + h) ϕ= J(t) \; , \; \; \; t \geq 0 \; , \end{equation*} in which $ζ$ is the Riemann zeta function and $h > 1$. Finally, we remark that the $L^2$ case of our general theory allows us to give a precise meaning to the often-used interpretation of $f(\partial_t)$ as an operator defined by a power series in the differential operator $\partial_t$.

math-ph↗
arXiv2016

From Hopf fibrations to exotic causal replacements

Topological solitons are relevant in several areas of physics [1]. Recently, these configurations have been investigated in contexts as diverse as hydrodynamics [2], Bose-Einstein condensates [3], ferromagnetism [4], knotted light [5] and non-abelian gauge theories [6]. In this paper we address the issue of wave propagation about a static Hopf soliton in the context of the Nicole model. Working within the geometrical optics limit we show that several nontrivial lensing effects emerge due to nonlinear interactions as long as the theory remains hyperbolic. We conclude that similar effects are very likely to occur in effective field theories characterized by a topological invariant such as the Skyrme model of pions.

math-ph↗
arXiv2014

On linear differential equations with infinitely many derivatives

Differential equations with infinitely many derivatives, sometimes also referred to as ``nonlocal'' differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. The goal of this paper is to show how to properly interpret and solve such equations, with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous approaches (N. Barnaby and N. Kamran, Dynamics with infinitely many derivatives: the initial value problem. {\em J. High Energy Physics} 2008 no. 02, Paper 008, 40 pp.; P. Górka, H. Prado and E.G. Reyes, Functional calculus via Laplace transform and equations with infinitely many derivatives. {\em Journal of Mathematical Physics} 51 (2010), 103512; P. Górka, H. Prado and E.G. Reyes, The initial value problem for ordinary equations with infinitely many derivatives. {\em Classical and Quantum Gravity} 29 (2012), 065017). In particular we reconsider generalized initial value problems and disprove various conjectures found in the modern literature. We illustrate various phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory.

math-ph↗
arXiv2012

The initial value problem for ordinary differential equations with infinitely many derivatives

We study existence, uniqueness and regularity of solutions for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal cosmology and string theory. We develop an appropriate Lorentzian functional calculus via Laplace transform which allows us to interpret rigorously an operator of the form $f(\partial_t)$ on the half line, in which $f$ is an analytic function. We find the most general solution to the equation $f(\partial_t) ϕ= J(t)$ (t greater or equal to 0) in the space of exponentially bounded functions, and we also analyze in full detail the delicate issue of the initial value problem. In particular, we state conditions under which the solution $ϕ$ admits a finite number of derivatives, and we prove rigorously that if an a priori data directly connected with our Lorentzian calculus is specified, then the initial value problem is well-posed and it requires only a finite number of initial conditions.

math-ph↗
arXiv2012

Sobolev spaces on locally compact abelian groups

Motivated by a class of nonlinear equations of interest for string theory, we introduce Sobolev spaces on arbitrary locally compact abelian groups and we examine some of their properties. Specifically, we focus on analogs of the Sobolev embedding and Rellich- Kondrachov compactness theorems. As an application, we prove the existence of continuous solutions to a generalized euclidean bosonic string equation possed on an arbitrary compact abelian group.

math-ph↗
arXiv2002

Correspondence theorems for hierarchies of equations of pseudo-spherical type

Hierarchies of evolution equations of pseudo-spherical type are introduced, generalizing the notion of a single equation describing pseudo-spherical surfaces due to S.S. Chern and K. Tenenblat, and providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of $sl(2,{\bf R})$--valued linear problems. As an application, it is shown that there exists a local correspondence between any two (suitably generic) solutions of arbitrary hierarchies of equations of pseudo-spherical type.

nlin.SI↗