Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 3985975, 8 pages
https://doi.org/10.1155/2020/3985975

Research Article

A New Approach for Solving the Complex Cubic-Quintic Duffing Oscillator Equation for Given Arbitrary Initial Conditions

Alvaro H. Salas¹ and Simeon Casanova Trujillo²

¹Department of Mathematics and Statistics, Universidad Nacional de Colombia-Sede Manizales-Nubia Campus, Fizmako Research Group, Manizales, Colombia
²Department of Mathematics and Statistics, Universidad Nacional de Colombia-Sede Manizales-Nubia Campus, Manizales, Colombia

Correspondence should be addressed to Alvaro H. Salas; ahsalass@unal.edu.co

Received 8 December 2019; Revised 29 January 2020; Accepted 18 March 2020; Published 28 May 2020

Academic Editor: Francesco Pellicano

Copyright © 2020 Alvaro H. Salas and Simeon Casanova Trujillo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 by

where

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). Let

where , , , , 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 gives

where .

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 problem

is given for by

Observe that the function

is 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 formula

and 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 formula

where 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

Let

where 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 problem

is given by

where

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 gives

which 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 is

where 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 be

where

In the case , we have a cubic-quintic Duffing equation. If and

then

is 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 form

where 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.

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