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A. J. Pan-Collantes

Publications and source records attributed to A. J. Pan-Collantes.

7 recordsLinked to original sources
arXiv2026

From curvature to Kovacic: a geometric approach to integrability of scalar ODEs

We study first-order ordinary differential equations such that the intrinsic Gauss curvature of the associated surface depends only on the independent variable: $\mathcal{K}(x,u)=κ(x)$, showing that this geometrically motivated class of equations admits a threefold connection to the second-order linear operator $L=d^2/dx^2+κ(x)$: the divergence along every solution satisfies a Riccati equation that linearizes to $L(y)=0$; every solution of the first-order equation satisfies the non-homogeneous equation $L(u)=c(x)$; and solutions of $L(y)=0$ give rise to integrating factors for the original nonlinear equation. By means of differential Galois theory, we prove that the nonlinear equation is integrable by quadratures if and only if $L$ admits a non-zero Liouvillian solution; when $κ$ is rational, Kovacic's algorithm provides a complete decision procedure.

math.CA↗
arXiv2026

Exact integration of Hamiltonian dynamics via Jacobi and Poisson Cinf-structures

We develop a geometric framework for the exact integration of Hamiltonian systems based on triangular closure relations among a finite family of functions. Unlike Liouville-Arnold integrability and its noncommutative generalizations, the functions involved in these relations need not be first integrals of the system. Instead, their Hamiltonian vector fields generate a $C^\infty$-structure on phase space that provides an algorithmic procedure for integrating the dynamics. Within this framework, the equations of motion can be reduced to a finite sequence of completely integrable Pfaffian equations, yielding an explicit integration scheme even when a complete set of conserved quantities is unavailable. The resulting geometric structure is called a Poisson $C^\infty$-structure. We further extend the construction to Jacobi Hamiltonian systems, showing that the same mechanism applies naturally to important subclasses of Jacobi geometry, including Poisson, locally conformally symplectic, and contact manifolds. The method is illustrated on two systems of physical interest: the two-particle non-periodic Toda lattice and the multi-waterbag reduction of the Vlasov equation. We also discuss extensions of the theory to time-dependent Hamiltonian systems.

math-ph↗
arXiv2024

Integration of first-order ODEs by Jacobi fields

A new class of vector fields enabling the integration of first-order ordinary differential equations (ODEs) is introduced. These vector fields are not, in general, Lie point symmetries. The results are based on a relation between 2-dimensional Riemannian manifolds and the integrability of first-order ODEs, which was established in a previous work of the authors. An integration procedure is provided, together with several examples to illustrate it. A connection between integrating factors of first-order ODEs and Schrödinger-type equations is highlighted.

math.CA↗
arXiv2023

Integration of differential equations by $\mathcal{C}^{\infty}$-structures

Several integrability problems of differential equations are addressed by using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization of the notion of solvable structure. Specifically, the integration procedure associated with $\mathcal{C}^{\infty}$-structures is used to integrate to a Lotka-Volterra model and several differential equations that lack sufficient Lie point symmetries and cannot be solved using conventional methods.

nlin.SI↗
arXiv2023

$\mathcal{C}^{\infty}$-symmetries of distributions and integrability

An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. The new structures are based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the extension of Lie point symmetries to $\mathcal{C}^{\infty}$-symmetries for ODEs developed in the recent years. These new objects, named $\mathcal{C}^{\infty}$-structures, play a fundamental role in the integrability of the distribution: the knowledge of a $\mathcal{C}^{\infty}$-structure for a corank $k$ involutive distribution permits to find its integral manifolds by solving $k$ successive completely integrable Pfaffian equations. These results have important consequences for the integrability of differential equations. In particular, we derive a new procedure to integrate an $m$th-order ordinary differential equation by splitting the problem into $m$ completely integrable Pfaffian equations. This step-by-step integration procedure is applied to integrate completely several equations that cannot be solved by standard procedures.

nlin.SI↗
arXiv2023

$\mathcal{C}^{\infty}$-structures in the integration of involutive distributions

For a system of ordinary differential equations (ODEs) or, more generally, an involutive distribution of vector fields, the problem of its integration is considered. Among the many approaches to this problem, solvable structures provide a systematic procedure of integration via Pfaffian equations that are integrable by quadratures. In this paper structures more general than solvable structures (named cinf-structures) are considered. The symmetry condition in the concept of solvable structure is weakened for cinf-structures by requiring their vector fields be just cinf-symmetries. For cinf-structures there is also an integration procedure, but the corresponding Pfaffian equations, although completely integrable, are not necessarily integrable by quadratures. The well-known result on the relationship between integrating factors and Lie point symmetries for first-order ODEs is generalized for cinf-structures and involutive distributions of arbitrary corank by introducing symmetrizing factors. The role of these symmetrizing factors on the integrability by quadratures of the Pfaffian equations associated with the \cinf-structure is also established. Some examples that show how these objects and results can be applied in practice are also presented.

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