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

+e nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubicquintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. +e 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.


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 eorem). is "linearization" is not possible in all cases.
is is the reason why analytical techniques are required to analyze the behavior of these solutions. ere 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. e approach we present here is different from other approaches known in the literature [1][2][3]. A Mathematica code is included in Appendix at the end. is 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 reactiondiffusion equation. A PHP script for solving the damped cubic-quintic equation may be found at http://fizmako.com/ duffing35.php.

The Cubic Duffing Oscillator Equation
(1) In the case when β > 0, this oscillator may be interpreted as a forced oscillator having a spring whose restoring force F reads is spring may be hardening or softening depending on the sign of β. If β > 0, we have a hardening spring, while for β < 0, we deal with a softening spring. is last interpretation is valid only for small v. In this last case, the Duffing oscillator describes the dynamics of a point mass in a doublewell 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.

Solution to the Duffing Equation in terms of the Weierstrass Elliptic Function
Our next aim is to solve initial value problem (1). Let where A, B, C, g 2 , and g 3 are some constants to be determined. Here, ℘(t, g 2 , g 3 ) stands for the elliptic Weierstrass function. is function satisfies the ode From (10), it is clear that Inserting ansatz (10) into the equation u ″ (t) + αu(t) + βu 3 (t) � 0 gives Equating the coefficients of ℘ j to zero (j � 0, 1, 2, 3) gives a nonlinear system of algebraic equations. Solving it gives In expression (14), the quantity A is arbitrary withy us, the solution to the initial value problem is given for α + 3A 2 β ≠ 0 by Observe that the function is also a solution to the equation u ″ (t) + αu(t) + βu 3 (t) � 0 for any constant t 0 (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 Mathematical Problems in Engineering 3 We already know the values of the constants B, C, g 2 , and g 3 (from (14) and (15)). We must find the values of the constants A, λ 1 , and λ 2 . We will determine them from the conditions (21) Solving the last system gives In the case of periodic solution, the period of oscillations is that of the Weierstrass function ℘(t; g 2 , g 3 ), and it may be calculated by means of the formula where e 1 is the first root to the cubic We have proved the following.

Theorem 1. e solution to initial value problem (4) is given by
e respective constants are evaluated by formulas (??), (15), and (22). e flow ϕ t (x, v) of the nonlinear dynamical system, ere is a more general equation called the generalized Duffing equation or Helmholtz-Duffing equation: v ″ (t) + n + αv(t) + βv(t) 2 + cv(t) 3 with real or complex constant coefficients. e solution to this equation may be found in [8].

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. at is, knowing the flow of the dynamical system associated with the cubic oscillator is enough to find that of the cubic-quintic. Indeed, let p, q, r, y 0 , and _ y 0 be arbitrary complex numbers with r ≠ 0. We will solve the initial value problem 4 Mathematical Problems in Engineering y ″ (t) + py(t) + qy(t) 3 + ry(t) 5 � 0, subjected to y(0) � y 0 , y ′ (0) � _ y 0 .

(29)
Let where the function v � v(t) is the solution to some Duffing equations given by (4). For small r, 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 y ′ (t) and integrating it with respect to t give In a similar way, from equation Inserting ansatz (30) into (33) and taking into account (32) give Equating to zero, the coefficients of y(t) j (j � 0, 2, 4, 6) give the following nonlinear algebraic system: We now eliminate the variables v 0 and _ v 0 taking into account that v(0) � v 0 and v ′ (0) � _ v 0 , and we obtain 2α + λ 6py 2 0 + 3qy 4 0 + 2ry 6 0 + 6 _ y 2 0 � 2p, β + λ 2 6py 2 0 + 3qy 4 0 + 2ry 6 0 + 6 _ y 2 0 � 4λp + q, From the first two equations of system (36), it follows that e number λ is obtained by solving the cubic equation e values of v 0 and � v 0 are found from the equations y(0) � y 0 and y ′ (0) � _ y 0 , and they read We have proved the following.

Nonlinear Cubic-Quintic Nonlinear Schrodinger (CQNLSE) Equation.
is equation reads In the case when A � − (k/2), B � − (k/2)χ, and C � 0, the number χ represents a dimensionless positive parameter characterizing the medium that describes wave propagation in fluids, plasmas, and nonlinear optics, while k is the wave number of propagating waves. Let is transformation gives which is a cubic-quintic Duffing equation.
We think that some formulas given here are new in the literature. e Mathematica code for solving either symbolically or numerically both cubic and cubic-quintic oscillator complex oscillator equations is given in Appendix. 6 Mathematical Problems in Engineering