Integrable Abel equations and Vein’s Abel equation
We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations.
Next, these results are applied to Vein’s Abel equation whose solutions are expressed in terms of the third order hyperbolic functions and a phase space analysis of the corresponding nonlinear oscillator is also provided.
Keywords: Abel equation; Appell invariant; normal form; canonical form; third order hyperbolic function.
Math. Meth. Appl. Sci. 39 (2016) 1376-1387
The first order Abel nonlinear equation of the second kind has the form
where , , , , and are all some functions of . The case has been introduced more than two hundred years ago by Abel a1 (). Many solvable equations of this type are collected in polzai () and other ones can be found in more recent works ct1 (); ct2 (); mak1 (); mak2 (); pz (); sh (); mr (). The transformation converts this equation in the form
which is Abel’s equation of the first kind. We see that Abel’s original equation of the second kind is actually a homogeneous case of that of the first kind.
One can also write (2) in the canonical form
where only the case leads to the simple identifications: , , , and .
The integrability features of Abel’s equation in the canonical form are extremely important because of its connection with nonlinear second order differential equations that phenomenologically describe a wide class of nonlinear oscillators which in this way can be treated analytically. In particular, Vein ve () introduced an Abel equation with and rational forms of , , and whose solutions are expressed in terms of third order hyperbolic functions. However, to the best of our knowledge, the paper of Vein went unnoticed for many years and only recently Yamaleev yama1 (); yama2 () provided a generalization in the framework of third order multicomplex algebra. Needless to say, the corresponding Vein’s nonlinear oscillator has not been studied in the literature. This was the main motivation for writing this paper, which is organized as follows. In section II, we present a lemma that encodes the connection of Abel equation with nonlinear oscillator equations and some related simple results. In section III, we reformulate several integrability results for Abel’s equation with the purpose to apply them to Vein’s Abel equation, which is the subject of section IV. The dynamical systems analysis of Vein’s oscillator is developed in section V, and we end up with the conclusions.
Ii Connection with second order nonlinear ODEs
The known importance of Abel’s equation in its canonical form (3) stems from the fact that its integrability leads to closed form solutions to a general nonlinear ODE of the form
where the variables and are some parametric solutions that depend on a generalized coordinate . This can be expressed by the following lemma mr ()
Proof. To show the equivalence, one just needs the chain rule
which turns (4) into the Abel equation of the second kind in canonical form
Via the inverse transformation
The case where is seen from (2) to be the one actually considered by Abel, for in this case the reduced Abel equation
can be always put in the form
where and are reduced functions and are given by the expressions
Using the lemma, the reduced Abel equation corresponds to a linear ODE without higher-order dissipative terms
which is equivalent to
This corresponds to the simple fact that an inverse power of a Riccati solution also satisfies a Riccati equation with redistributed coefficients. Now, we eliminate the linear part to obtain the reduced Riccati equation
which corresponds to a nonlinear ODE with higher-order dissipative terms
where now the coefficients are
Iii Some Abel integrability cases
iii.1 Abel’s equation of constant coefficients
Denote the coefficients of (3), where are constants and so that . It is obvious that the roots of the equation are themselves solutions of (3). More generally, the general solution of (3) is obtained via factorization of the denominator in the right-hand side of
which leads to the following cases:
iii.2 Integrability based on the normal form of Abel’s equation
If the following transformations as given in Kamke’s book kam ()
are applied to equation (3), then one obtains Abel’s equation in normal form
where the invariant is given by
Thus, we conclude that if , then (27) is integrable since it is separable. If one chooses relations between the functions such that the invariant is null and letting , then
is a Riccati equation, which is always integrable because it is obtained from Abel’s equation in normal form, which is integrable. Thus, the solution to (27) is
and explicitly, the solution to (3) with null invariant is
iii.3 Integrability of Abel equation with non-constant coefficients and
In this subsection, we will consider the original Abel equation (2) of the first-kind
which corresponds to (4) without the cubic nonlinearity.
Let us use the transformation
Thus, the original equation (1) is considerably reduced only by having .
where is a constant, then
is integrable, provided that
which leads to . Hence, the necessary condition for the integration of (32) is
iii.4 Canonical form of Abel’s equation and the integrating factor
which lead to the canonical form
is the Appell invariant. Then, the integrating factor can be used to formulate the following interesting result:
Proof. In (34), let , where , which leads to
that is separable, with solution
which after simplification leads back to (41).
Iv Vein’s Abel equation
The following Abel equation:
where , is known from a paper of Vein to be integrable ve (). We first notice that if we define the following determinants,
Using these expressions, the invariant becomes
Therefore, the normal form of Abel’s Equation (45) is
But because , we find
From the last two equations, one might infer that
and consequently using (26) would get
To see how the implicit solutions (55), (57) can be untangled and written explicitly, we will proceed as Vein did. Firstly, we show that there are actually three solutions of the type (58), which are the solutions of (49) and can be put in the form ve ():
with the -functions defined cyclically as
where is an arbitrary constant as it does not appear in the differential equation. The functions are the following triad:
also known as the third-order hyperbolic functions, and are plotted in Fig. 1.
They are independent because their Wronskian , and they satisfy the following relationships
They also fulfill the following relationships
which are cyclic. That is, the fourth-order derivatives have the same property as the first ones, the fifth-order derivatives as the second ones, and so forth. In particular, they are independent solutions of the differential equation , but also of and of any , with .
where is a constant. Then, fulfills
with . Suppose is defined implicitly as a multi-valued function of as follows
and denote the inverse relation by the explicit formula
Then satisfies Abel’s equation (49).
The quotient can be eliminated by means of (64), which leads to
Differentiating again, this time with respect to we have,
Substituting and dividing by lead to Abel’s equation (49). Thus, the solution can be written explicitly as
with an arbitrary constant.
The other two solutions can be obtained by cyclically replacing the ’s in the function
V Vein’s nonlinear oscillator
To see what kind of oscillator corresponds to Vein’s Abel equation, we will proceed as follows. First, we will eliminate the linear term from (49) using
Then using lemma 1 with , we obtain
Writing equation (72) as a dynamical system
and because (73) cannot be put in the form of (40), because , we will be using instead the standard methods of phase-plane analysis and use the linear approximation at the equilibrium points of (73) to classify them instead of solving the equation by finding the potential .
Also, notice that the system is non-Hamiltonian because there is no potential such that
The Jacobian matrix of (73) is
We have three equilibrium points of the system (73) from which one is real , while the other two are a pair of complex conjugates, , with and . The particular case which Vein considered, will reduce the three fixed points to the case of one real fixed point . The characteristic polynomial of the Jacobian matrix is
while the discriminant is . By evaluating the Jacobian at the real fixed point , where is either real or complex, we obtain
1. Real fixed point: In this case we have
then, the real fixed point living on the axis is a stable spiral. The eigenvalues of the Jacobian matrix are complex conjugate pairs . By calculating the eigenvectors, the linearized solution around the fixed point is
where and are arbitrary constants.
Because both coefficients and are not real nothing can be said about these complex fixed points.
This paper recalls several integrability properties of Abel equations together with some simple consequences, followed by a discussion of Vein’s Abel equation related to third-order hyperbolic functions. The corresponding nonlinear oscillator system is introduced here, and its dynamical systems analysis is provided. Finally, it is worth noticing that the nonlinear oscillators corresponding to Abel equations are in fact a large class of generalized Liénard equations of the form , where , and are arbitrary functions and , where are constants hl () and, as such, have many applications in physics, biology, and engineering m10 (); nay95 ().
The first author wishes to acknowledge support from Hochschule München while on leave from Embry-Riddle Aeronautical University in Daytona Beach, Florida. We also wish to thank the referees for their suggestions that led to significant improvements of this work.
- (1) Abel NH. Précis d’une théorie des fonctions elliptiques. J. Reine Angew. Math. 1829; 4:309-348.
- (2) Polyanin AD, Zaitsev VF. Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, Boca Raton, 1995.
- (3) Cheb-Terrab ES, Roche AD. Abel ODEs: Equivalence and integrable classes. Comp. Phys. Commun. 2000; 130:204-231.
- (4) Cheb-Terrab ES, Roche AD. An Abel ordinary differential equation class generalizing known integrable classes. Eur. J. Appl. Math. 2003; 14:217-229.
- (5) Mak MK, Chan HW, Harko T. Solutions generating technique for Abel-type nonlinear ordinary differential equations. Comp. Math. Appl. 2001; 41:1395-1401.
- (6) Mak MK, Harko T. New method for generating general solution of Abel differential equation. Comp. Math. Appl. 2002; 43:91-94.
- (7) Panayotounakos DE, Zarmpoutis TI. Construction of exact parametric or closed form solutions of some unsolvable classes of nonlinear ODEs (Abel’s nonlinear ODEs of the first kind and relative degenerate equations). Int. J. Math. Math. Sci. 2011; 2011: Article 387429, 13 pages.
- (8) Salinas-Hernández E, Martínez-Castro J, Muñoz R. New general solutions to the Abel equation of the second kind using functional transformations. Appl. Math. Comp. 2012; 218:8359-8362.
- (9) Mancas SC, Rosu HC. Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations. Phys. Lett. A 2013; 377:1434-1438.
- (10) Vein PR. Functions which satisfy Abel’s differential equation. SIAM J. Appl. Math. 1967; 15:618-623.
- (11) Yamaleev RM. Solutions of Riccati-Abel equation in terms of third order trigonometric functions. Indian J. Pure Appl. Math. 2014; 45:165-184.
- (12) Yamaleev RM. Representation of solutions of -order Riccati equation via generalized trigonometric functions. J. Math. Anal. Appl. 2014; 420:334-347.
- (13) Kamke E. Differentialgleichungen: Lösungsmethoden und Lösungen, Chelsea, New York, 1959.
- (14) Davis HT. Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962.
- (15) Harko T, Liang S-D. Exact solutions of the Liénard and generalized Liénard type ordinary non-linear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator. arXiv:1505.02364v3, J. Eng. Math. 2016; to appear.
- (16) Mickens RE. Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods, World Scientific, Singapore, 2010.
- (17) Nayfeh AH, Mook DT. Nonlinear Oscillations, John Wiley & Sons, New York, Chichester, 1995.