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

Authors: Alvaro H. Salas; Simeon Casanova Trujillo
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 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 Section2, 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:(1)v″t+αvt+βvt3=0,β≠0.In the case whenβ>0, this oscillator may be interpreted as a forced oscillator having a spring whose restoring force F reads(2)F=−αv−βv3.This spring may be hardening or softening depending on the sign ofβ. If β>0, we have a hardening spring, while for β<0, we deal with a softening spring. This last interpretation is valid only for small v. 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(3)vt=c1cnα+βc12t+c2,β2α+βc12.The values of the constantsc1 and c2 are determined from the initial conditions. Let us consider the i.v.p(4)v″t+αvt+βvt3=0,subjected tov0=v0,v′0=v˙0.The numberΔ=α+βv022+2βv˙02 is called the discriminant of i.v.p (4). If α+βv022+2βv˙02>0 (or in the case when α,β are complex and α+βv022+2βv˙02≠0), the solution to initial value problem (4) is given by(5)vt=v0cnωtm+v˙0/ωsnωtmdnωtm1+μsnωtm2,where(6)ω=α+βv022+2βv˙02,m=12−α2α+βv022+2βv˙02,μ=α−α+βv022+2βv˙02+βv022α+βv022+2βv˙02.Here,cn, sn, and dn are the elliptic Jacobi functions. In the case when α+βv022+2βv˙02=0, the solution reads(7)vt=−−αβtanhα2t−tanh−1−βαu0,v′02=v˙02=α+βv022−2β.Finally, whenα+βv022+2βv˙02<0,(8)v=vt=c−2c1+dsct−1/ωsc−1a+v0/dv0−αmωm,where(9)ω=−α+c2β4d2,d=22c2βα2β−α−α+3c2βα+c2β,m=2+2d2α−3c2βα+c2β,c=sgny˙042αy02+βy04+2y˙02−β.


## 3. Solution to the Duffing Equation in terms of the Weierstrass Elliptic Function

Our next aim is to solve initial value problem (1). Let(10)ut=limτ−>tA+B1+C℘τ;g2,g3,where A, B, C, g2, and g3 are some constants to be determined. Here, ℘t,g2,g3 stands for the elliptic Weierstrass function. This function satisfies the ode(11)℘′t,g2,g32=4℘t,g2,g33−g2℘t,g2,g3−g3.From (10), it is clear that(12)u0=A,u′0=0.Inserting ansatz (10) into the equation u″t+αut+βu3t=0 gives(13)2C22B+ACα+A3Cβ℘3+2C−6B+3ACα+BCα+3A3Cβ+3A2BCβ℘2+C6Aα+4Bα+6A3β+12A2Bβ+6AB2β−3BCg2℘+2Aα+2Bα+2A3β+6A2Bβ+6AB2β+2B3β+BCg2−4BC2g3=0,where ℘=℘t=limτ−>t℘τ;g2,g3.Equating the coefficients of℘j to zero j=0,1,2,3 gives a nonlinear system of algebraic equations. Solving it gives(14)B=−6AA2β+α3A2β+α,C=123A2β+α.In expression (14), the quantity A is arbitrary withy 3A2β+α≠0.(15)g2=112−3A4β2−6A2αβ+α2,g3=1216α9A4β2+18A2αβ+α2.Thus, the solution to the initial value problem(16)u″t+αut+βut3=0,subjected tou0=A,u′0=0,is given for α+3A2β≠0 by(17)ut=limτ⟶tA−6Aα+A2β/α+3A2β1+12/α+3A2β℘τ;1/12α2−6A2αβ−3A4β2,1/216αα2+18A2αβ+9A4β2.Observe that the function(18)vt=ut+t0is also a solution to the equation u″t+αut+βu3t=0 for any constant t0 (real or complex). Our aim is to solve initial value problem (4).To this end, we will make use of the addition formula(19)℘τ+t0;g2,g3=14℘′τ;g2,g3−℘′t0;g2,g3℘τ;g2,g3−℘t0;g2,g32−℘τ;g2,g3−℘t0;g2,g3,and then(20)vt=limτ−>tA+B1+C1/4℘′τ;g2,g3−λ2/℘τ;g2,g3−λ12−℘τ;g2,g3−λ1.We already know the values of the constantsB, C, g2, and g3 (from (14) and (15)). We must find the values of the constants A, λ1, and λ2. We will determine them from the conditions(21)v0=v0,v′0=v0,v″0+αv0+βv03=0.Solving the last system gives(22)λ1=3A3β+3A2βv0+5Aα+αv012A−12v0,λ2=Av0A2β+α2A−v02,A=α+βv022+2βv02−αβ.In the case of periodic solution, the period of oscillations is that of the Weierstrass function℘t;g2,g3, and it may be calculated by means of the formula(23)T=2∫e1∞14t3−g2t−g3,where e1 is the first root to the cubic(24)z3−g2z−g3=0.We have proved the following.Theorem 1.
The solution to initial value problem (4) is given by(25)vt=vv0,v0t=limτ−>tA+B1+C1/4℘′τ;g2,g3−λ2/℘τ;g2,g3−λ12−℘τ;g2,g3−λ1.The respective constants are evaluated by formulas (??), (15), and (22). The flow ϕtx,v of the nonlinear dynamical system,(26)x˙=−αv−βv3,v=x,reads(27)ϕtx,v=ddtvx,vtvx,vt.There is a more general equation called the generalized Duffing equation or Helmholtz–Duffing equation:(28)v″t+n+αvt+βvt2+γvt3=0,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 Section2 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 p, q, r, y0, and y˙0 be arbitrary complex numbers with r≠0. We will solve the initial value problem(29)y″t+pyt+qyt3+ryt5=0,subjected toy0=y0,y′0=y˙0.Let(30)yt=vt1+λvt2,where the function v=vt 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(31)12y′t2+p2yt2+q4yt4+r6yt6=12y˙02+p2y02+q4y04+r6y06.In a similar way, from equationv″t+αvt+βvt3=0, we obtain(32)v′t2=αv02+βv042+v˙02−αvt2−12βvt4.Let(33)Rt=12y′t2+p2yt2+q4yt4+r6yt6−12y˙02+p2y02+q4y04+r6y06:residual.Inserting ansatz (30) into (33) and taking into account (32) give(34)Rt=6λ2p−6λ3py02+3λq−3λ3qy04−2λ3ry06+2r−6λ3y˙02yt6−3β−4λp+6λ2py02+3λ2qy04−q+2λ2ry06+6λ2y˙02yt4−32α+6λpy02−2p+3λqy04+2λry06+6λy˙02yt2−6py02−3qy04−2ry06+6αv02+3βv04+6v˙02−6y˙02.Equating to zero, the coefficients ofytj j=0,2,4,6 give the following nonlinear algebraic system:(35)6λ2p−6λ3py02+3λq−3λ3qy04−2λ3ry06+2r−6λ3y˙02=0,β−4λp+6λ2py02+3λ2qy04−q+2λ2ry06+6λ2y˙02=0,2α+6λpy02−2p+3λqy04+2λry06+6λy˙02=0,−6py02−3qy04−2ry06+6αv02+3βv04+6v˙02−6y˙02=0.We now eliminate the variablesv0 and v˙0 taking into account that v0=v0 and v′0=v˙0, and we obtain(36)2α+λ6py02+3qy04+2ry06+6y˙02=2p,β+λ26py02+3qy04+2ry06+6y˙02=4λp+q,6py02+3qy04+2ry06+6y˙02λ3=3λ2λp+q+2r.From the first two equations of system (36), it follows that(37)α=p−12λ6py02+3qy04+2ry06+6y˙02,β=4λp+q−6py02+3qy04+2ry06+6y˙02λ2.The numberλ is obtained by solving the cubic equation(38)6py02+3qy04+2ry06+6y˙02λ3−6pλ2−3qλ−2r=0.The values ofv0 and v0∘ are found from the equations y0=y0 and y′0=y˙0, and they read(39)v0=±y01−λy02,v˙0=±y˙0−λ3y06+3λ2y04−3λy02+1.We have proved the following.Theorem 2.
The solution to the initial value problem(40)y″t+pyt+qyt3+ryt5=0,subjected toy0=y0,y′0=y˙0,is given by(41)yt=vt1+λvt2,where(42)vt=limτ−>tA+B1+C1/4℘′τ;g2,g3−λ2/℘τ;g2,g3−λ12−℘τ;g2,g3−λ1.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(43)−1∂φ∂Z+A∂2φ∂T2+Bφ2φ+Cφ4φ=0.In the case whenA=−k/2, B=−k/2χ, 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(44)φZ,T=exp−1cT+bZyaT−2aAcZ.This transformation gives(45)y′′ξ−b+Ac2a2Ayξ+Ba2Ayξ3+Ca2Ayξ5=0,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(46)ut+vtux=Duxx+αu−βu3+γu5,where u=ux,t is the concentration or density variable depending on the phenomena under study; D is the diffusion coefficient; vt is the convection term coefficient; and α, β, and γ are the reaction term coefficients. Making the traveling wave transformation ξ=kx+ηt and letting uξ=atyξ give(47)aty+aη′t+vaky′ξ=Dak2y″ξ+αayξ−βa3y3ξ+γa5y5ξ.By selectinga = constant and ηt=2δ/a−k∫vtdt, equation (47) turns out to be(48)y″ξ+2δy′ξ+pyξ+qy3ξ+ry5ξ=0,where(49)p=αDk2,q=βa2Dk2,r=γa4Dk2.In the caseδ=0, we have a cubic-quintic Duffing equation. If δ≠0 and(50)p=−14kk−2n,r=−3k24A4,then(51)yξ=A1+Bexpkξis a solution to equation (48) for any constants A, B, and k. Finally, when q=0 and p=3/4δ2, we have the damped Duffing equation(52)y″ξ+2δy′ξ+3/4δ2yξ+ry5ξ=0.Using Lie group theory, it is possible to prove that equation (52) admits a solution of the form(53)yξ=4γrc1e−nξ/4υc2−2c1ne−nξ/2,where the function υ=υt is a solution to the quintic Duffing equation υ″ξ+γυξ5=0.Equation (52) may be solved numerically for given initial conditions y0=y0 and y′0=y0′ 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.


## 5.1. Nonlinear Cubic-Quintic Nonlinear Schrodinger (CQNLSE) Equation

This equation reads(43)−1∂φ∂Z+A∂2φ∂T2+Bφ2φ+Cφ4φ=0.In the case whenA=−k/2, B=−k/2χ, 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(44)φZ,T=exp−1cT+bZyaT−2aAcZ.This transformation gives(45)y′′ξ−b+Ac2a2Ayξ+Ba2Ayξ3+Ca2Ayξ5=0,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(46)ut+vtux=Duxx+αu−βu3+γu5,where u=ux,t is the concentration or density variable depending on the phenomena under study; D is the diffusion coefficient; vt is the convection term coefficient; and α, β, and γ are the reaction term coefficients. Making the traveling wave transformation ξ=kx+ηt and letting uξ=atyξ give(47)aty+aη′t+vaky′ξ=Dak2y″ξ+αayξ−βa3y3ξ+γa5y5ξ.By selectinga = constant and ηt=2δ/a−k∫vtdt, equation (47) turns out to be(48)y″ξ+2δy′ξ+pyξ+qy3ξ+ry5ξ=0,where(49)p=αDk2,q=βa2Dk2,r=γa4Dk2.In the caseδ=0, we have a cubic-quintic Duffing equation. If δ≠0 and(50)p=−14kk−2n,r=−3k24A4,then(51)yξ=A1+Bexpkξis a solution to equation (48) for any constants A, B, and k. Finally, when q=0 and p=3/4δ2, we have the damped Duffing equation(52)y″ξ+2δy′ξ+3/4δ2yξ+ry5ξ=0.Using Lie group theory, it is possible to prove that equation (52) admits a solution of the form(53)yξ=4γrc1e−nξ/4υc2−2c1ne−nξ/2,where the function υ=υt is a solution to the quintic Duffing equation υ″ξ+γυξ5=0.Equation (52) may be solved numerically for given initial conditions y0=y0 and y′0=y0′ 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.



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