# Integrable Ermakov-Pinney equations with nonlinear Chiellini ‘damping’

###### Abstract

For the constant frequency case, we introduce a special type of Ermakov-Pinney equations with nonlinear dissipation based on the corresponding Chiellini integrable Abel equation. General solutions of these equations are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type.
In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini ‘dissipative’ function is actually a dissipation-gain function because it can be negative on some intervals. These are the first examples of integrable Ermakov-Pinney equations with nonlinear ‘damping’.

## I Introduction

The nonlinear Ermakov-Pinney (EP) equation has long been known to have profound connections with the linear equations of identical operatorial form without the inverse cubic nonlinearity and because of this it has been seen as an example of ‘nonlinearity from linearity’ KD . A historical overview has been written by Leach and Andriopoulos LA and the fundamental importance of the EP equation for parametric oscillators, both classical and quantum-mechanical, with their vast application reaches, is well established in the literature.

Already in 1880, Ermakov introduced the following pair of equations E80

(1) |

for which the linear equation is endowed with the so-called Ermakov dynamic invariant

(2) |

We label here this invariant by the tuning parameters of the nonlinear term of the two equations. When one gets , where is the Wronskian of two solutions of the linear equation. Thus, Ermakov’s invariant can be considered as a generalization of the Wronskian. Modern research on this invariant started in the 1960s when Lewis rediscovered it and also provided applications in quantum-mechanics lewis67 . In particular, Ray and Reid rr and also Sarlet s introduced the following generalized Ermakov systems

(3) |

and their associated invariants

(4) |

When and the invariant is recovered. Recent works on the connections of the Ermakov systems with the nonlinear superposition principle belong to Cariñena and collaborators c1 ; c2 where the reader can find more references. The importance of Ermakov equations stems from the fact that they can be used to model the propagation of laser beams in nonlinear optics gon , magneto-gas dynamics rs , the mean field dynamics of pancake-shaped Bose-Einstein condensates herring , and cosmology ro ; hl and the same is expected from Ermakov systems.

In this paper, we determine the general solution of an Ermakov equation with an Abel-integrable nonlinear dissipation and constant ‘frequency’ function :

(5) |

We get the solution using the corresponding integrable Abel equation and also we give a theorem for obtaining the general solution if a particular solution is known. In the latter case, the phase of the solution is of the Milne type Milne and the Ermakov invariant for a pair of nonlinear dissipative Ermakov equations of the type (5) with different nonlinearity parameters and is used in the derivation. We emphasize that Ermakov equations with nonlinear dissipation have never been considered before. For the general case of Ermakov equations with a linear dissipative term one has to resort on numerical methods because there are no Lie symmetries and reductions to simpler forms are useful only in particular cases haas-10 .

The paper is structured as follows. We first discuss briefly the basic properties of the simplest nondissipative Ermakov equation corresponding to the constant ‘frequency’ case. We next discuss the special EP equation with dissipation determined by Chiellini’s integrability condition for the corresponding Abel equation of the first kind prev . Its general solution is obtain through a method that makes usage of the Ermakov invariant of a pair of such dissipative Ermakov equations. Finally, the method is also applied to Chiellini-dissipative Ermakov equations with higher order Ermakov nonlinearities for which we also provide the general solutions. The paper ends up with some conclusions and three appendices in which several related mathematical issues are included for self-consistency reasons.

## Ii The simplest EP (SEP) equation

### ii.1 Solution of SEP

As is well known, if we have two linear independent solutions , and to

(6) |

then a particular solution of the corresponding EP equation

(7) |

is given by P

(8) |

where is the Wronskian of the two linearly independent solutions, while the general solution can be written , with the three constants constrained by the condition .

Let’s take the simplest case, i.e., , a constant.

(9) |

We call the EP equation for this case as SEP

(10) |

which we also write in the form

(11) |

From (8), its solution can be written in the form

(12) |

The case corresponds to the simple harmonic oscillator.

### ii.2 SEP equation with Chiellini dissipation

We build now the Ermakov-Pinney equation with Chiellini-type dissipation as an equation of the following format

(13) |

where is as given in (11). The dissipation term is obtained from using Chiellini’s integrability condition prev

(14) |

for Abel’s equation of the first kind corresponding to (13). From (14) one easily gets

(15) |

Before proceeding further, we want to mention that one can not get rid of the Chiellini dissipation term because this term corresponds to the quadratic nonlinearity of the Abel’s equation

(16) |

see Appendix A.

The inverse of the Abel solution, prev , is obtained from , see

(17) |

which gives

(18) |

Depending on the sign of we have the following cases which give the general solution of (13)

(19) |

It is worth noting that if we take , the reduced equation

(20) |

has the harmonic solutions and as if the nonlinear dissipation is not acting at all and the equation (20) is linear. Thus, one can also construct solutions of (13) in the standard Pinney form. The only feature is that the amplitudes of the harmonic modes are inverse proportional to the frequency.

It is also possible to construct a different form of the general solution for a SEP with Chiellini dissipation in terms of a particular solution as follows. We first get a particular solution from (19) by fixing and

(21) |

Theorem. For the Ermakov equation (22), with the particular solution as given in (21), and given by (15), the general solution is obtained by

(23) |

where the phase is of the Milne type Milne , and is the following quantity

(24) |

which is the Ermakov invariant for an Ermakov pair of equations of type (22) of nonlinear parameters and , respectively.

A version of this theorem for the non-dissipative case can be found in a paper by Qin and Davidson qin .

Proof:

First, we construct the (invariant) quantity , using (24).

Then, using , and in (24) we obtain the following separable equation

(25) |

But from , one gets which leads to (23).

Two more lemmas on are given in the Appendix B, whereas the factorization of the dissipative Ermakov equations on which we focus here can be found in Appendix C.

### ii.3 Abel-dissipative SEP equations with negative power nonlinearity

In this subsection, we show that the method of obtaining the general solution of the previous section can be also applied to equations with higher order Ermakov nonlinearties and associated Abel dissipation.

Reid has shown in reid that the general solution to

(26) |

is

(27) |

provided that and are two independent solutions of (6), and

(28) |

Notice that (27) is a direct generalization of (8). We are interested in a general solution to

(29) |

via the machinery of invariants.

First, let us choose , then (26) has solutions

(30) |

(32) |

where , , and by integrating and leads to

(33) |

Using the same , together with , then the solution is the following

(34) |

where

(35) |

Note that for the case which is the standard EP case the solution simplifies to

(36) |

Plots of the solutions for different values of and all the other parameters set to the value of unity are given in Figs. (1)-(5). Solutions are real, while are periodically pure real and pure imaginary. As for the solution displayed for the case in Fig. (6), it is pure imaginary on a symmetric interval around the origin and real in the rest. Notice also the diminishing of the amplitudes with increased order of the negative-power nonlinearity. But the most interesting feature of the ‘dissipative’ solutions is that they may have larger amplitudes than the nondissipative ones on some time intervals. This reveals the presence of gain effects. Indeed, from the plots of the function in Figs. (7)-(9), one can infer that this function is not always dissipative but it also has gain intervals.

## Iii Conclusion

A class of dissipative Ermakov-Pinney equations, either with standard or higher-order nonlinearities, and with nonlinear dissipation of the Abel-Chiellini-integrable type has been introduced. The general solutions are obtained directly by the Abel equation route and also using the dynamic invariant of Ermakov pairs of equations of this type and the particular solution of one member of the pair, both in the standard case and in the case of any odd Ermakov nonlinearity. The technique based on Abel’s equation we used here is unique to the constant frequency systems and cannot be directly generalized to the case of time-dependent oscillators. This is due to the fact that the Chiellini integrability condition which plays a central role in obtaining the results do not apply to the cases when there exist an explicit dependence on the independent variable. Another remarkable aspect is that the Chiellini nonlinear dissipative function is in fact a dissipation-gain function which could have interesting applications in the propagation of pulses.

Appendix A: The Abel equation

The importance of Abel’s equation in its canonical forms stems from the fact that its integrability cases lead to closed form solutions to nonlinear ODEs of the form

(37) |

This can be expressed by the following Lemma.

Lemma 1: Solutions to a general second order ODE of type (37) may be obtained via the solutions to Abel’s equation (42), and vice versa using the following relationship

(38)

Proof: To show the equivalence, one just need the simple chain rule

(39) |

which turns (37) into the second order Abel equation in canonical form

(40) |

Moreover, via transformation of the dependent variable

(41) |

(40) becomes

(42) |

and letting , one gets

(43) |

which is an Abel equation without the linear term.

Appendix B: Two lemmas for

Here we formulate two lemmas related to .

Lemma 1: The invariant is related to the factoring functions of the Ermakov equations of the corresponding system.

Let , and be the first factoring functions of two nonlinear Ermakov ODEs as in the previous appendix. Then, according to (52), , and . Then, using (24) we have

(44) |

where .

Lemma 2: .

Appendix C: Factorization of dissipative Ermakov ODEs

The Ermakov ODEs with nonlinear dissipation-like coefficients, i.e.,

(46) |

can be factored in the form

(47) |

This gives

(48) |

Furthermore, employing the grouping of terms used by Rosu and Cornejo-Pérez rcp1 ; rcp2 ; rcp3 , one gets

(49) |

Then, we have

(50) |

From (47), we also have

(51) |

which leads to

(52) |

This is identical to a well-known result in the case of linear equations, especially in supersymmetric quantum mechanics, where it gives the ground state wavefunction in terms of the superpotential. Since and , and additionally we know that , then we get

which is exactly the dissipative term for which the equation is integrable.

## References

- (1) P.G. Kevrekidis and Y. Drossinos, Math. Comp. Sim. 74, 196 (2007).
- (2) P.G.L. Leach and K. Andriopoulos, Appl. Anal. Discr. Math. 2, 146 (2008).
- (3) V.P. Ermakov, Univ. Izv. Kiev Ser. III 9, 1 (1880) and Appl. Anal. Discr. Math. 2, 123 (2008), translation by A.O. Harin.
- (4) H.R. Lewis, Phys. Rev. Lett. 18, 510 (1967).
- (5) J.R. Ray, J.L. Reid, Phys. Lett. A 71, 317 (1979). J.L. Reid, J.R. Ray, Lett. Math. Phys. 4, 235 (1980).
- (6) W. Sarlet, Phys. Lett. A 82, 161 (1981).
- (7) J.F. Cariñena, J. de Lucas, and M.F. Rañada, Nonlinear superpositions and Ermakov systems, in: F. Cantrijn, M. Crampin, B. Langerock (Eds.), Differential Geometric Methods in Mechanics and Field Theory, Academic Press, 2007.
- (8) J.F. Cariñena, J. de Lucas, Phys. Lett. A 372, 5385 (2008).
- (9) A.M. Goncharenko, Yu. A. Logvin, A.M. Samson, P.S. Shapovalov, Opt. Commun. 81, 225 (1991).
- (10) C. Rogers, W.K. Schief, J. Math. Phys. 52, 083701 (2011).
- (11) G. Herring, P.G. Kevrekidis, F. Williams, T. Christodoulakis, D.J. Frantzeskakis, Phys. Lett. A 367, 140 (2007).
- (12) H. Rosu, P. Espinoza, M. Reyes, Nuovo Cim. B 114, 1439 (1999).
- (13) R.M. Hawkins, J.E. Lidsey, Phys. Rev. D 66, 023523 (2002).
- (14) W.E. Milne, Phys. Rev. 35, 863 (1930).
- (15) F. Haas, Physica Scripta 81, 025004 (2010).
- (16) S.C. Mancas and H.C. Rosu, Phys. Lett. A 377, 1434 (2013).
- (17) E. Pinney, Proc. Am. Math. Soc. 1, 681 (1950).
- (18) H. Qin and R.C. Davidson, Phys. Rev. ST Accel. Beams 9, 054001 (2006).
- (19) J.L. Reid, Proc. Am. Math. Soc. 27, 601 (1971).
- (20) H.C. Rosu, O. Cornejo-Pérez, Phys. Rev. E 71, 046607 (2005).
- (21) O. Cornejo-Pérez, H.C. Rosu, Prog. Theor. Phys. 114, 533 (2005).
- (22) O. Cornejo-Pérez, J. Negro, L.M. Nieto, H.C. Rosu, Found. Phys. 36, 1587 (2006).