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1.Introduction
The exact solutions of nonlinear ordinary differential equations (ODEs) describe the behavior of a great variety of physical, chemical, biological, and engineering systems. Widespread systems in these vast areas of research can be described by homogeneous Liénard equations, which have been intensively studied over the years, see, e.g., 1 and the recent review 2. On the other hand, the same type of inhomogeneous equations received relatively less attention despite the remarkable leap forward brought by the discovery of an irregular noise, later termed deterministic chaos, in the case of sinusoidally driven triode circuits by van der Pol and van der Mark in 1927 3. Our focus in this short paper is on inhomogeneous Liénard type equations of the form
where the dot denotes the time derivative, d/dt, G(u) and F(u) are arbitrary, but usually polynomial, functions of u, and the forcing term I(t) is an arbitrary continuous function of time.
The main goal of the present paper is to show how the factorization method developed in 4-6 and the factorization conditions thereof can be used to obtain some integrable inhomogeneous Liénard equations for specific forcing terms. The key point is that the factorization method helps to reduce the inhomogeneous Liénard equations to first-order nonlinear equations, such as Abel and Riccati equations, which are presumably easier to solve in some cases. We recall that the reduction to Riccati equations of the linear Schrödinger equations has been extensively used in supersymmetric quantum mechanics and older factorization methods as reviewed in 7,8.
2.The nonlinear factorization
As in 4-6, we consider the factorization of (1)
under the conditions
adding the scheme proposed in 9, where one assumes [ddt-f1(u)]u=Ω(t). This yields the following coupled ODEs for (2),
which we further simplify by taking the second factorizing function as a constant, f2=a2≡const.,
Besides, using the constant function f 2, conditions (3) and (4) imply a relationship between functions F and G given by
where c 2 stands for an integration constant, or equivalently
Denoting I(t)=∫t0e-a2tI(t)dt, the solution to (7) is
where c1 is an integration constant given by c1=Ω(0). This allows to rewrite (8) in the form
whose general solution is also the solution of the Liénard Eq. (1), while further particular solutions can be obtained by setting c1 = 0.
Viceversa, one can say that (12) is a first-order nonlinear reduction of forced Liénard equations of the form
Thus, integrable cases of (12) can provide Liénard solutions in closed form. Since among the most encountered forced Liénard equations are these having F(u) in the form of cubic and quadratic polynomials, in the rest of the paper, we address the applications of this solution method to some cases of these types.
3.The inhomogeneous Duffing-van der Pol oscillator
We choose the particular cubic case F(u)=Au+Cu3 because it corresponds to the forced Duffing-van der Pol oscillator 10
This equation admits the factorization
where α=A/a2 and γ=C/a2.
The corresponding first-order equation is the Abel equation
The change of variables
turns (16) into the normal form
with invariant
Unfortunately, this formula shows that inhomogeneous Abel equations in this category are not integrable by the separation of variables because N(x) cannot be made constant as required by this type of integrability. Only in the force-free particular case I(t) = 0, the invariant can be reduced to the constant
By separation of variables, the solution is given by the implicit relation
This solution has been obtained previously in 10.
4.Quadratic inhomogeneous Liénard equations
If we set F(u)=Au+Bu2, then the first order equivalent equation is the Riccati equation
Equation (22) can be transformed into the normal form 11
where
For integrable cases of separable type, one should have N(t) as an arbitrary real constant that we choose p2/4, implying Ω(t)=(p2+α2)/4β also a constant, as well as a constant driving force
In this simple case, we obtain a Liénard solution of (13) of the form
4.1.Linear polynomial source term
After the constant driving case, it is orderly to consider the source term as the linear polynomial I(t)=t+δ, where δ is an arbitrary constant. We set a2=1 and c1 = 0, and we obtain the Riccati equation
with solution given by
where ̃t=β1/3[α2/4β+(t+̃δ], the prime denotes the ̃t derivative, and k2 is an integration constant. However, the presence of the rational term in Airy functions turns singular such as Liénard solutions.
4.2.Quadratic polynomial source term
Let the source term be the quadratic polynomial of type I(t)=a2βt2+(a2α-2β)t-(a2+α). According to Eqs. (11) and (22), and by setting c1 = 0, we have the Riccati equation
This equation has the particular solution u(t) = t, while the general solution is given by
where F(x)=e-x2∫x0ey2dy is the Dawson integral, and k1 is an integration constant. Again, because of the rational term, this solution is singular at -et(α+βt)β-1/2F(α+2βt/2√β)=k1.
4.3.Exponential source term
For the source term of exponential form, I(t)=κeλt, and for c1 = 0, the Riccati equation is
The solution is given by
where k4 is an integration constant, ̃t=2(√κβ/λ√λ-a2)eλt/2, and ¯F(̃t) is the following combination of hypergeometric functions
¯F(˜t)=˜t20F1(;2+αλ;-˜t2)+αλ0F1(;1+αλ;-˜t2)+0F1(;αλ;-˜t2) .
For k4 = 0, we have the simpler solution
The case corresponding to α=-1 simplifies further to
4.4.Back to the constant source case
We return to the constant source term case since we wish to point out the interesting feature that it is more general than the exponential case. Indeed, let us take the source term as I(t)=ϵ, an arbitrary constant, and a2=1. This leads to the Riccati equation
which is similar to the Riccati equation for the exponential case unless for ϵ. The general solution of (35) is a rational expression in terms of Bessel functions given by
where m=√α2+4βϵ, ̃t=2√βc1et/2, and k3, an integration constant. It displays singularities at the zeros of its denominators.
When k3 = 0, this solution takes the simpler form
Notice that in the particular case of c 1 = 0, the exponential scaling of time is annihilated, and the Riccati equation is of constant coefficients having the well-known regular kink solution
which is also a Liénard kink. If in the expression for the parameter m, we substitute ϵ by (25) for a 2 = 1, we obtain m = ip, and (38) becomes the solution (26).
5.Conclusion
The nonlinear factorization method developed in 4-6,9 has been used to obtain closed-form solutions of certain types of inhomogeneous Liénard equations. The conditions imposed upon the nonlinear coefficients of the equations by the factorization method and the insertion of the forcing term in the factorization scheme act as designing tools of specific forms of the forcing terms to generate integrable cases by these means. The illustrative examples have been chosen from the class of polynomial (up to cubic) and exponential forcing terms similar to a recent study of inhomogeneous Airy equations 12. However, the obtained Liénard solutions have rational parts, which make them prone to the presence of singularities. The only regular solutions we have obtained by employing this simple factorization method are the usual tanh kinks. Finally, the scheme presented here is bounded to constant factorization functions f2, since only in this case equation (5) can be turned into the linear (7) in the independent variable t.
