Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Abstract

We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold-Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.

Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

“…In this paper we have extended the previous studies of the Lax formulation of non-autonomous equations of Liénard type by Sinelshchikov et al [1] to non-autonomous equations including an additional term depending quadratically on the velocity, namely, the Levinson -Smith equation. The first integrals resulting from the trace formula of the Lax pair for such equations are invariably time-dependent.…”

“…Recently Sinelshchikov et al [1] have extended the Lax formalism to a family of nonautonomous second-order differential equations generalizing the Liénard equation [6]. This stands in contrast to the earlier works which dealt mostly with completely integrable systems.…”

Lie group approach for constructing all reciprocal transformations: The two‐dimensional stationary gas dynamics equations

“…In particular, the Sundman type transformation is effective for solving nonlinear ODEs. For example, it has been applied for linearization problems of ordinary differential equations 2–5 and for finding families of analytical solutions 6,7 …”