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A new use of nonlocal symmetries for computing Liouvillian first integrals of rational second order ordinary differential equations

I. Deme, L. G. S. Duarte, L. A. C. P. da Mota

TL;DR

This work develops a linear probabilistic framework that leverages a nonlocal symmetry to compute Darboux integrating factors and Liouvillian first integrals for rational second-order ODEs. It constructs three plane vector fields sharing the same Liouvillian first integral with the original 2ODE and uses associated vector fields to obtain a Darboux factor via a linear, probabilistic procedure (PLDIF). The approach offers a practical alternative to Lie, Darboux, and S-function methods, particularly for 2ODEs with high-degree Darboux polynomials or scarce point/spectral symmetries, and demonstrates strong performance on a suite of challenging equations. It also discusses the potential for generalization to higher-order equations and PDEs and raises open questions about degree bounds and structural aspects of the solution spaces.

Abstract

Here we present an efficient method for finding and using a nonlocal symmetry admitted by a rational second order ordinary differential equation (rational 2ODE) in order to find a Liouvillian first integral (belonging to a vast class of Liouvillian functions). In a first stage, we construct an algorithm (improving the methodde veloped in [1]) that computes a nonlocal symmetry of a rational 2ODE. In ase cond stage, based on the knowledge of this symmetry, it is possible to construct three polynomial vector fields (in R2), which "share" the Liouvillian first integral with the rational 2ODE. These "plane" polynomial vector fields can be used to construct a procedure (based on an idea developed in [2]) to determine an integrating factor for the rational 2ODE with a fast probabilistic algorithm. The main advantages of the proposed method are: the obtaining of the nonlocal symmetry is algorithmic and very efficient and, furthermore, its use to find an integrating factor is a sequence of linear or quasilinear processes.

A new use of nonlocal symmetries for computing Liouvillian first integrals of rational second order ordinary differential equations

TL;DR

This work develops a linear probabilistic framework that leverages a nonlocal symmetry to compute Darboux integrating factors and Liouvillian first integrals for rational second-order ODEs. It constructs three plane vector fields sharing the same Liouvillian first integral with the original 2ODE and uses associated vector fields to obtain a Darboux factor via a linear, probabilistic procedure (PLDIF). The approach offers a practical alternative to Lie, Darboux, and S-function methods, particularly for 2ODEs with high-degree Darboux polynomials or scarce point/spectral symmetries, and demonstrates strong performance on a suite of challenging equations. It also discusses the potential for generalization to higher-order equations and PDEs and raises open questions about degree bounds and structural aspects of the solution spaces.

Abstract

Here we present an efficient method for finding and using a nonlocal symmetry admitted by a rational second order ordinary differential equation (rational 2ODE) in order to find a Liouvillian first integral (belonging to a vast class of Liouvillian functions). In a first stage, we construct an algorithm (improving the methodde veloped in [1]) that computes a nonlocal symmetry of a rational 2ODE. In ase cond stage, based on the knowledge of this symmetry, it is possible to construct three polynomial vector fields (in R2), which "share" the Liouvillian first integral with the rational 2ODE. These "plane" polynomial vector fields can be used to construct a procedure (based on an idea developed in [2]) to determine an integrating factor for the rational 2ODE with a fast probabilistic algorithm. The main advantages of the proposed method are: the obtaining of the nonlocal symmetry is algorithmic and very efficient and, furthermore, its use to find an integrating factor is a sequence of linear or quasilinear processes.
Paper Structure (13 sections, 17 theorems, 38 equations, 8 tables)

This paper contains 13 sections, 17 theorems, 38 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Finding a nonlocal symmetry
  3. Using a nonlocal symmetry to determine a Darboux integrating factor
  4. Vector fields (in $\hbox{R}^2$) associated with the 2ODE
  5. The vector fields associated with $\mathfrak{X}_i$
  6. Some important results
  7. Defining vector fields ${\cal X}_i$ such that ${\cal X}_i(R)=0$
  8. Constructing a linear probabilistic algorithm
  9. Performance of the algorithms
  10. Some 'difficult' 2ODEs
  11. First set:
  12. Second set:
  13. Some final observations and possible developments

Key Result

Theorem 2.1

Let $I \in L_S$ be a first integral of the rational 2ODE (r2ode) as above. Then, 2ODE (r2ode) admits a symmetry given by where $\, \int_x \,\,\hbox{\rm is the inverse operator of}\,\, D_x\,,\,\left(D_x \equiv \frac{\mathfrak{X}}{N}\right)$.

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Definition 2.5
  • Remark 2.2
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • ...and 38 more