Nonlocal deformations of autonomous invariant curves for Liénard equations with quadratic damping

Highlights

• Nonlocal transformations preserve autonomous invariant curves.

• Classification of invariant curves can be extended for non-polynomial equations.

• Two integrable families of Lienard-type equations are constructed.

Abstract

We consider a family of nonlinear oscillators with quadratic damping, that generalizes the Liénard equation. We show that certain nonlocal transformations preserve autonomous invariant curves of equations from this family. Thus, nonlocal transformations can be used for extending known classification of invariant curves to the whole equivalence class of the corresponding equation, which includes non-polynomial equations. Moreover, we demonstrate that an autonomous first integral for one of two non-locally related equations can be constructed in the parametric form from the general solution of the other equation. In order to illustrate our results, we construct two integrable subfamilies of the considered family of equations, that are non-locally equivalent to two equations from the Painlevé–Gambier classification. We also discuss several particular members of these subfamilies, including a traveling wave reduction of a nonlinear diffusion equation, and construct their invariant curves and first integrals.

Introduction

We consider the following family of nonlinear differential equations

$$x_{tt}+h\left(x\right)x_t^2+f\left(x\right)x_t+g\left(x\right)=0,$$

where $f\ne 0,g\ne 0$ and $h$ are arbitrary sufficiently smooth functions. Particular members of (1.1) often appear in various applications that include, but are not limited to, physics, chemistry and biology (see, e.g. [1]). Moreover, traveling wave reductions of various partial differential equations, e.g. reaction-diffusion equations, belong to (1.1) [2].

Consequently, integrability of (1.1) has attracted a lot of attention. For example, equivalence problems for (1.1) and its integrable subcases via point or nonlocal transformations were considered in [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. In particular, linearization problem with respect to nonlocal transformations was studied in [3], [4], [12], [13], while in [9], [10] connections between (1.1) and Painlevé type equations were considered. Authors of [14], [15], [16] used an approach based on $\lambda$–symmetries and classification of certain first integrals for (1.1) was obtained. Another approach to the integrability of (1.1) with polynomial coefficients is connected with the classification of algebraic invariant curves admitted by (1.1) and the classical Darboux integrability theory and its generalization to Liouville and Weierstrass integrability (see, e.g. [17], [18], [19], [20], [21], [22], [23], [24] and references therein).

However, the relations between various aspect of integrability of (1.1) has been considered only in a few works (see, e.g. [25] and references therein). Thus, the main aim of this work is to demonstrate the connection between the approach based on nonlocal equivalence transformations and the Darboux integrability approach. Namely, we show that certain nonlocal transformations preserve autonomous invariant curves of equations from (1.1). Furthermore, if invariant curves are classified for one of two non-locally equivalent equations, then we can map this classification into the other equation. For example, one can classify algebraic invariant curves for polynomial equations [20], [23], however, it seems that there are no criteria for classification of invariant curves for non-polynomial equations. On the other hand, if a non-polynomial equation is connected to a polynomial one, for which algebraic invariant curves are classified, one can extend this classification via the corresponding nonlocal transformations to a non-polynomial equation.

In addition, we demonstrate that if we know the general solution of one of two non-locally equivalent equations, then we can explicitly construct an autonomous first integral in the parametric form for the other equation. What is more, it can be done without completely inverting the corresponding transformations. Let us remark that analogous parametric first integrals for two dimensional polynomial dynamical systems via solutions of linear differential equations were previously constructed in [26], [27].

We illustrate our results by constructing two integrable subfamilies of (1.1) that are connected via nonlocal transformations to two equations of the Painlevé type. We show that all equations from these subfamilies possess certain invariant curves and integrating factors. We also consider several particular examples, including a traveling wave reduction of a nonlinear diffusions equation, and construct their invariant curves, integrating factors and parametric first integrals.

The rest of this work is organized as follows. In the next Section we provide the proof of main results and present two integrable subfamilies of (1.1). In Section 3 we provide concrete examples of integrable equations from these subfamilies and construct their invariant curves and first integrals. In the last Section we briefly summarize and discuss our results.