Abstract
The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.
1. Introduction
It is known that most phenomena in nature have a nonlinear character, i.e., their laws of evolution are governed by either nonlinear ordinary or nonlinear partial differential equations. In many situations, it is desiderable to make an analytical study of the behavior of the equation solutions by means of the stability analysis of some associated linear systems (for example, for hyperbolic equilibria, Hartman–Grobman Theorem). This “linearization” is not possible in all cases. This is the reason why analytical techniques are required to analyze the behavior of these solutions. There are analytical methods that give necessary and sufficient conditions for the existence and uniqueness of solution to nonlinear equations (say Lie Groups, Sobolev spaces, etc.). However, we are investigating analytical methods that allow us to obtain exact solutions to this type of equations. In that sense, we meet in literature different techniques for integrating nonlinear equations, such as parameter perturbation techniques and homotopic perturbation methods, among others. As a contribution to the literature, in this article, we present the exact solution to the cubic-quintic Duffing oscillator equation by means of the famous Weierstrass elliptic function. The approach we present here is different from other approaches known in the literature [1–3]. A Mathematica code is included in Appendix at the end.
This paper is organized as follows. In Section 2, we give the solution to the cubic Duffing equation in terms of Jacobi elliptic functions. In Section 3, we give the solution to the cubic Duffing equation by means of the Weierstrass elliptic function. Section 4 is related to the solution of the cubic-quintic Duffing equation for given initial conditions. Section 5 is related to applications of the obtained theoretical results for solving the nonlinear cubic-quintic nonlinear Schrodinger equation and the nonlinear cubic-quintic reaction-diffusion equation. A PHP script for solving the damped cubic-quintic equation may be found at http://fizmako.com/duffing35.php.
2. The Cubic Duffing Oscillator Equation
The equation is as follows:
In the case when , this oscillator may be interpreted as a forced oscillator having a spring whose restoring force F reads
This spring may be hardening or softening depending on the sign of . If , we have a hardening spring, while for , we deal with a softening spring. This last interpretation is valid only for small . In this last case, the Duffing oscillator describes the dynamics of a point mass in a double-well potential. Chaotic motions can be observed in this case [4, 5]. Duffing equation is closely related to the pendulum equation [6, 7], and it has many important applications in soliton theory [8]. Other physical interpretations may be found in [9]. [10, 11] describe stability analysis for the Duffing equation.
The general solution to this ode reads
The values of the constants and are determined from the initial conditions. Let us consider the i.v.p
The number is called the discriminant of i.v.p (4). If (or in the case when are complex and ), the solution to initial value problem (4) is given bywhere
Here, cn, sn, and dn are the elliptic Jacobi functions. In the case when , the solution reads
Finally, when ,where
3. Solution to the Duffing Equation in terms of the Weierstrass Elliptic Function
Our next aim is to solve initial value problem (1). Letwhere , , , , and are some constants to be determined. Here, stands for the elliptic Weierstrass function. This function satisfies the ode
From (10), it is clear that
Inserting ansatz (10) into the equation giveswhere .
Equating the coefficients of to zero gives a nonlinear system of algebraic equations. Solving it gives
In expression (14), the quantity is arbitrary withy .
Thus, the solution to the initial value problemis given for by
Observe that the functionis also a solution to the equation for any constant (real or complex). Our aim is to solve initial value problem (4).To this end, we will make use of the addition formulaand then
We already know the values of the constants , , , and (from (14) and (15)). We must find the values of the constants , , and . We will determine them from the conditions
Solving the last system gives
In the case of periodic solution, the period of oscillations is that of the Weierstrass function , and it may be calculated by means of the formulawhere is the first root to the cubic
We have proved the following.
Theorem 1. The solution to initial value problem (4) is given by
The respective constants are evaluated by formulas (??), (15), and (22). The flow of the nonlinear dynamical system,reads
There is a more general equation called the generalized Duffing equation or Helmholtz–Duffing equation:with real or complex constant coefficients. The solution to this equation may be found in [8].
4. The Analytic Solution to the Complex Cubic-Quintic Equation for Given Initial Conditions
In this section, we make use of results in Section 2 in order to solve the cubic-quintic oscillator equation. We show that the cubic-quintic Duffing equation is reduced to the cubic Duffing equation. That is, knowing the flow of the dynamical system associated with the cubic oscillator is enough to find that of the cubic-quintic. Indeed, let , , , , and be arbitrary complex numbers with . We will solve the initial value problem
Letwhere the function is the solution to some Duffing equations given by (4). For small , we may consider that equation (29) represents a small perturbation of equation (1). In that sense, equation (29) has a physical meaning similar to that of (1).
Multiplying equation (29) by and integrating it with respect to give
In a similar way, from equation , we obtain
Let
Inserting ansatz (30) into (33) and taking into account (32) give
Equating to zero, the coefficients of give the following nonlinear algebraic system:
We now eliminate the variables and taking into account that and , and we obtain
From the first two equations of system (36), it follows that
The number is obtained by solving the cubic equation
The values of and are found from the equations and , and they read
We have proved the following.
Theorem 2. The solution to the initial value problemis given bywhere
The respective constants are evaluated by formulas (16), (23), and (37)–(39) - (??).
5. Applications
5.1. Nonlinear Cubic-Quintic Nonlinear Schrodinger (CQNLSE) Equation
This equation reads
In the case when , , and , the number represents a dimensionless positive parameter characterizing the medium that describes wave propagation in fluids, plasmas, and nonlinear optics, while is the wave number of propagating waves. Let
This transformation giveswhich is a cubic-quintic Duffing equation.
5.2. Nonlinear Reaction-Diffusion (NLRD) Equation
The dimensionless form of the variable coefficients in the nonlinear reaction-diffusion (NLRD) equation iswhere is the concentration or density variable depending on the phenomena under study; is the diffusion coefficient; is the convection term coefficient; and , , and are the reaction term coefficients. Making the traveling wave transformation and letting give
By selecting = constant and , equation (47) turns out to bewhere
In the case , we have a cubic-quintic Duffing equation. If andthenis a solution to equation (48) for any constants , , and . Finally, when and , we have the damped Duffing equation
Using Lie group theory, it is possible to prove that equation (52) admits a solution of the formwhere the function is a solution to the quintic Duffing equation .
Equation (52) may be solved numerically for given initial conditions and at the authors’ website http://fizmako.com/duffing35.php.
We think that some formulas given here are new in the literature. The Mathematica code for solving either symbolically or numerically both cubic and cubic-quintic oscillator complex oscillator equations is given in Appendix.
Appendix
Mathematica code:
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.