Searcharxiv⌕ Search

arXiv subjects

Rafael Hernández Heredero

Publications and source records attributed to Rafael Hernández Heredero.

5 recordsLinked to original sources
arXiv2025

Computing almost commuting bases of ODOs and Gelfand-Dickey hierarchies

Almost commuting operators were introduced in 1985 by George Wilson to present generalizations of the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dickey (GD) hierarchies. In this paper, we review the formal construction of the vector space of almost commuting operators with a given ordinary differential operator (ODO), with the ultimate goal of obtaining a basis by computational routines, using the language of differential polynomials. We use Wilson's results on weighted ODOs to guarantee the solvability of the triangular system that allows to compute the homogeneous almost commuting operator of a given order in the ring of ODOs. As a consequence, the computation of the equations of the GD hierarchies is achieved without using pseudo-differential operators. A new package in SageMath called \texttt{dalgebra} has been designed to perform symbolic calculations in differential domains. The algorithms to calculate the almost commuting basis and the GD hierarchies in the ring of ODOs are implemented in SageMath, and explicit examples are provided.

math.AC↗
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↗
arXiv2019

The symmetry approach to integrability: recent advances

We provide a concise introduction to the symmetry approach to integrability. Some results on integrable evolution and systems of evolution equations are reviewed. Quasi-local recursion and Hamiltonian operators are discussed. We further describe non-abelian integrable equations, especially matrix (ODE and PDE) systems. Some non-evolutionary integrable equations are studied using a formulation of formal recursion operators that allows to study non-diagonalisable systems of evolution equations.

nlin.SI↗
arXiv2018

Formal recursion operators of integrable nonevolutionary equations and Lagrangian systems

We derive the general structure of the space of formal recursion operators of nonevolutionary equations~$q_{tt}=f(q,q_{x},q_t,q_{xx},q_{xt},q_{xxx},q_{xxxx})$. This allows us to classify integrable Lagrangian systems with a higher order Lagrangian of the form~$\mathscr{L}=\frac12 L_2(q_{xx}, q_x, q)\,q_t^2 + L_1(q_{xx}, q_x, q)\, q_{t} + L_0(q_{xx}, q_x, q)$. The key technique relays on exploiting a homogeneity of the determining equations of formal recursion operators. This technique allows us to extend the main results to more general equations~$q_{tt}=f(q,q_{x},\ldots,q_{n};q_{t},q_{xt},\ldots,q_{mt})$.

nlin.SI↗