# The Duffing Oscillator Equation and Its Applications in Physics

Authors: Alvaro Humberto Salas Salas; Jairo Ernesto Castillo Hernández; Lorenzo Julio Martínez Hernández
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.


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## Body

## 1. Introduction

The nonlinear equation describing an oscillator with a cubic nonlinearity is called the Duffing equation. Duffing [1], a German engineer, wrote a comprehensive book about this in 1918. Since then there has been a tremendous amount of work done on this equation, including the development of solution methods (both analytical and numerical) and the use of these methods to investigate the dynamic behavior of physical systems that are described by the various forms of the Duffing equation. Because of its apparent and enigmatic simplicity, and because so much is now known about the Duffing equation, it is used by many researchers as an approximate model of many physical systems or as a convenient mathematical model to investigate new solution methods [2–7]. This equation exhibits an enormous range of well-known behavior in nonlinear dynamical systems and is used by many educators and researchers to illustrate such behavior. Since the 1970s, it has become really popular with researchers into chaos, as it is possibly one of the simplest equations that describes chaotic behavior of a system. This equation is also useful in the study of soliton solutions to important physics models such as KdV equation, mKdV equation, sine-Gordon equation, Klein–Gordon equation, nonlinear Schrodinger equation, and shallow water wave equation [8–18].


## 2. Undamped and Unforced Duffing Equation

Letp, q, u0, and u˙0 be real numbers. The general solution to the undamped and unforced Duffing equation u″t+put+qu3t=0 may be expressed in terms of any of the twelve Jacobian elliptic functions, as shown in Table 1.Table 1
Equivalent solutions to a Duffing equation.
ut=c1cnqc12+pt+c2|qc12/2qc12+put=c1nc−qc12−pt+c2|qc12+2p/2qc12+put=c1snqc12+2p/2t+c2|−qc12/qc12+2put=c1ns−qc12/2t+c2|−2p/qc12−1ut=c1dnqc12/2t+c2|2qc12+p/qc12ut=c1nd−qc12−2p/2t+c2|2qc12+p/qc12+2put=c1scqc12−2p/2t+c2|2p−qc12/2p−qc12ut=c1cs−qc12/2t+c2|2qc12−p/qc12ut=c1sdqc12−q2c14+p2t+c2|qc12+p+q2c14+p2/2put=c1ds−qc12/2t+c2|−−qc12−2p/2qc12ut=c1dc−qc12/2t+c2|−qc12−2p/qc12ut=c1cdqc12+2p/2t+c2|−qc12/qc12+2pIn this section, we will solve the initial value problem(1)u″t+put+qu3t=0,u0=u0,u′0=u˙0,u02+u˙02q≠0.The number(2)Δ=p+qu022+2qu˙02is called the discriminant for problem (1).


### 2.1. First Case:Δ>0

In the case, whenu˙0=0, we get Δ=p+qu022 so that p+qu02≠0 and the problem reduces to(3)u″t+put+qu3t=0,subject tou0=u0,u′0=0.Its solution is given by(4)ut=u0cnp+qu02t,qu022p+qu02.Letu˙0≠0. First of all, observe that if u=ut is a solution to the ode u″t+put+qu3t=0, then Ut=ut+C is also a solution for any constant C. Secondly, let yt=c1cnω,m (c1 = a nonzero constant) be the Jacobi elliptic function cn with modulus m and parameter k defined by k2=m. We have(5)y″t+1−2mωyt+2mωc12y3t=0,for anyt,anyc1≠0.Therefore, comparing (1) and (5) gives(6)m=qc122p+qc12,ω=p+qc12,and we conclude that the analytic function(7)u=ut=c1cnp+qc12t+c2,qc122p+qc12,c1≠0,is the general solution to the Duffing equation u″t+put+qu3t=0 for arbitrary constants c1 and c2. The values of these constants are determined from the initial conditions u0=u0 and u′0=u˙0.We have(8)cncn−1u0,m,m=u0,so thatc2=cn−1u0,m,c2=cn−1u0,qc122p+qc12,cnc2,qc122p+qc12=u0,and the value of c1 results from solving the equation u′0=u˙0, i.e.,(9)−c1snc2,mdnc2,m=u˙0.Squaring this last equation and taking into account relations (6) and the identities(10)dn2c2,m=1−msn2c2,m=1−m1−cn2c2,m=1−m1−u02,sn2c2,m=1−cn2c2,m=1−u02,we arrive at the equation(11)qc14+2pc12−2pu02+qu04+2u˙02=0.Solving equation (11) for c1 gives(12)c1=±−p±p+qu022+2qu˙02q=±−p±Δq.To avoid the ambiguity with plus-minus signs, we define(13)c1=p+qu022+2qu˙02−pq,ifu˙0<0,c1=−p+qu022+2qu˙02−pq,ifu˙0≥0,c2=cn−1u0c1∣qc122qc12+p,ifu0≠0,c2=Kqc122qc12+p,ifu0=0.Making use of the addition formula(14)cnu+v=cnucnv−snudnusnvdnv1−msn2usn2v,the solution may also be written in the form(15)ut=y0cnωt∣m+y˙0/ωsnωt∣mdnωt∣m1+bsnωt∣m2,where(16)ω=Δ,m=121−pΔ,b=12p+qu02Δ−1.The solution is periodic and its main period equals(17)T=4Kmω,for0≤m<1.In the case, whenm>1, we make use if the identities(18)cnt,m=dnmt,1m,snt,m=1msnmt,1m,dnt,m=cnmt,1m.The main period will then be(19)T=2K1/mmω,m>0.Ifm<0, we transform the solution by means of the following identities:(20)cnt∣m=cd1−mt∣mm−1,snt∣m=11−msd1−mt∣mm−1,dnt∣m=nd1−mt∣mm−1.Remember that cd = cn/dn, sd = sn/dn, and nd = 1/dn. For reference, Tables2–4 give useful conversion formulas.Table 2
Conversion formulas form > 1.
cnt∣m=dnmt∣1/msnt∣m=snmt∣1/m/mdnt∣m=cnmt∣1/mnct∣m=1/dnmt∣1/mnst∣m=m/snmt∣1/mndt∣m=1/cnmt∣1/msdt∣m=snmt∣1/m/mcnmt∣1/mdst∣m=mcnmt∣1/m/snmt∣1/msct∣m=snmt∣1/m/mdnmt∣1/mcst∣m=mdnmt∣1/m/snmt∣1/mcdt∣m=dnmt∣1/m/cnmt∣1/mdct∣m=cnmt∣1/m/dnmt∣1/mTable 3
Conversion formulas form < 0.
cnt∣m=cd1−mt∣m/m−1snt∣m=sd1−mt∣m/m−1/1−mdnt∣m=nd1−mt∣m/m−1nst∣m=1−m/sd1−mt∣m/m−1ndt∣m=1/nd1−mt∣m/m−1sdt∣m=sd1−mt∣m/m−1/1−mnd1−mt∣m/m−1nct∣m=1/cd1−mt∣m/m−1dst∣m=1−mnd1−mt∣m/m−1/sd1−mt∣m/m−1sct∣m=sd1−mt∣m/m−1/1−mcd1−mt∣m/m−1cst∣m=1−mcd1−mt∣m/m−1/sd1−mt∣m/m−1cdt∣m=cd1−mt∣m/m−1/nd1−mt∣m/m−1dct∣m=nd1−mt∣m/m−1/cd1−mt∣m/m−1Table 4
Jacobi imaginary transformation.
cnit,m=nct,1−msnit,m=isnt,1−m/cnt,1−mdnit,m=dnt,1−m/cnt,1−mdnit,m=dnt,1−m/cnt,1−mncit,m=cnt,1−mndit,m=cnt,1−m/dnt,1−msdit,m=isnt,1−m/dnt,1−mdsit,m=−idnt,1−m/snt,1−mcdit,m=1/dnt,1−mdcit,m=dnt,1−mscit,m=isnt,1−mcsit,m=−i/snt,1−mExample 1.
Letp=1, q=1, y0=0, and y˙0=2. The solution to the i.v.p.(21)u″t+ut+u3t=0,u0=0,u′0=2,is given by(22)ut=−23dn3t∣1/3sn3t∣1/3sn3t∣1/32−3=233sd3t∣13.
This solution is periodic with main periodT=4K1/3/3. See Figure 1 for a comparison with Runge–Kutta numerical solution (dashed curve).Figure 1
Comparison between the exact and the numerical solution.Example 2.
Letp=−5, q=1, y0=1, and y˙0=2. The solution to the i.v.p.,(23)u″t−5ut+u3t=0,u0=1,u′0=2,reads(24)ut=−5+26cn23/443t+cn−1−15+26∣12+546∣12+546=−3.14626cn−2.21336t−5.50011−3.13354−1∣1.01031.
An equivalent expression without the imaginary unit is(25)ut=dnwt+26−2cnwt∣snwt1−26−2snwt2,beingw=1+32,m=206−48≈0.98979.
See Figure2 for a comparison with Runge–Kutta numerical solution (dashed curve).Figure 2
Comparison with Runge–Kutta numerical solution (dashed curve).


### 2.2. Second Case:Δ<0

Define(26)δ=2pu02+qu04+2u˙02.SinceΔ<0, necessarily q<0. From the equality(27)δ=p2−Δ−q,it is evident that δ>0. We seek a solution to the i.v.p. (1) in the ansatz form(28)ut=A−2A1+vt,where v=vt is the solution to some Duffing equations(29)v″t+αvt+βv3t=0,v0=v0≔A+u0A−u0,v˙0≔2Au˙0A−u02.We have(30)v″t=−αvt−βv3t,(31)v′t2=D−αvt2−12βvt4,D=2A4α+A4β+4A3u0β−4A2u02α+6A2u02β+8A2u˙02+4Au03β+2u04α+u04β2A−u04.Inserting the ansatz (30) into the ode u″t+put+qu3t=0 and taking into account (30) and (31), we get(32)u″t+put+qu3t=A1+vt3A2q+p−2βvt3+p+2α−3A2qvt2−p+2α−3A2qvt−A2q+4D+p.Equating to zero the coefficients ofv0t,v1t,v1t, and v3t in (32), we obtain an algebraic system. Solving it gives(33)α=123A2q−p,β=12A2q+p,A=δ−q4.Observe that(34)α+v02β2+2v˙02β=2p2−qqδ+−qδ>0.Thus, the Duffing equation (29) has a positive discriminant. The solution to the i.v.p. (3) is then given by(35)ut=δ−q4−2δ/−q41+v0cnω¯x∣m¯+v˙0/ωsnω¯x∣m¯dnω¯x∣m¯/1+b¯snω¯x∣m¯2,where(36)ω¯=Δ¯,m¯=12−α2Δ¯,b¯=α+βv022Δ¯−12,v0=A+u0A−u0,v˙0=2Au˙0A−u02,Δ¯=α+v02β2+2v˙02β.The values ofa,β, and A are found from (33).


### 2.3. Third Case:Δ=0

When the discriminant vanishes, thenq<0 and the only solution to problem (1) with u′02=u˙02 is(37)ut=−−pqtanhp2t−tanh−1−qpu0,which may be verified by direct computation.


## 2.1. First Case:Δ>0

In the case, whenu˙0=0, we get Δ=p+qu022 so that p+qu02≠0 and the problem reduces to(3)u″t+put+qu3t=0,subject tou0=u0,u′0=0.Its solution is given by(4)ut=u0cnp+qu02t,qu022p+qu02.Letu˙0≠0. First of all, observe that if u=ut is a solution to the ode u″t+put+qu3t=0, then Ut=ut+C is also a solution for any constant C. Secondly, let yt=c1cnω,m (c1 = a nonzero constant) be the Jacobi elliptic function cn with modulus m and parameter k defined by k2=m. We have(5)y″t+1−2mωyt+2mωc12y3t=0,for anyt,anyc1≠0.Therefore, comparing (1) and (5) gives(6)m=qc122p+qc12,ω=p+qc12,and we conclude that the analytic function(7)u=ut=c1cnp+qc12t+c2,qc122p+qc12,c1≠0,is the general solution to the Duffing equation u″t+put+qu3t=0 for arbitrary constants c1 and c2. The values of these constants are determined from the initial conditions u0=u0 and u′0=u˙0.We have(8)cncn−1u0,m,m=u0,so thatc2=cn−1u0,m,c2=cn−1u0,qc122p+qc12,cnc2,qc122p+qc12=u0,and the value of c1 results from solving the equation u′0=u˙0, i.e.,(9)−c1snc2,mdnc2,m=u˙0.Squaring this last equation and taking into account relations (6) and the identities(10)dn2c2,m=1−msn2c2,m=1−m1−cn2c2,m=1−m1−u02,sn2c2,m=1−cn2c2,m=1−u02,we arrive at the equation(11)qc14+2pc12−2pu02+qu04+2u˙02=0.Solving equation (11) for c1 gives(12)c1=±−p±p+qu022+2qu˙02q=±−p±Δq.To avoid the ambiguity with plus-minus signs, we define(13)c1=p+qu022+2qu˙02−pq,ifu˙0<0,c1=−p+qu022+2qu˙02−pq,ifu˙0≥0,c2=cn−1u0c1∣qc122qc12+p,ifu0≠0,c2=Kqc122qc12+p,ifu0=0.Making use of the addition formula(14)cnu+v=cnucnv−snudnusnvdnv1−msn2usn2v,the solution may also be written in the form(15)ut=y0cnωt∣m+y˙0/ωsnωt∣mdnωt∣m1+bsnωt∣m2,where(16)ω=Δ,m=121−pΔ,b=12p+qu02Δ−1.The solution is periodic and its main period equals(17)T=4Kmω,for0≤m<1.In the case, whenm>1, we make use if the identities(18)cnt,m=dnmt,1m,snt,m=1msnmt,1m,dnt,m=cnmt,1m.The main period will then be(19)T=2K1/mmω,m>0.Ifm<0, we transform the solution by means of the following identities:(20)cnt∣m=cd1−mt∣mm−1,snt∣m=11−msd1−mt∣mm−1,dnt∣m=nd1−mt∣mm−1.Remember that cd = cn/dn, sd = sn/dn, and nd = 1/dn. For reference, Tables2–4 give useful conversion formulas.Table 2
Conversion formulas form > 1.
cnt∣m=dnmt∣1/msnt∣m=snmt∣1/m/mdnt∣m=cnmt∣1/mnct∣m=1/dnmt∣1/mnst∣m=m/snmt∣1/mndt∣m=1/cnmt∣1/msdt∣m=snmt∣1/m/mcnmt∣1/mdst∣m=mcnmt∣1/m/snmt∣1/msct∣m=snmt∣1/m/mdnmt∣1/mcst∣m=mdnmt∣1/m/snmt∣1/mcdt∣m=dnmt∣1/m/cnmt∣1/mdct∣m=cnmt∣1/m/dnmt∣1/mTable 3
Conversion formulas form < 0.
cnt∣m=cd1−mt∣m/m−1snt∣m=sd1−mt∣m/m−1/1−mdnt∣m=nd1−mt∣m/m−1nst∣m=1−m/sd1−mt∣m/m−1ndt∣m=1/nd1−mt∣m/m−1sdt∣m=sd1−mt∣m/m−1/1−mnd1−mt∣m/m−1nct∣m=1/cd1−mt∣m/m−1dst∣m=1−mnd1−mt∣m/m−1/sd1−mt∣m/m−1sct∣m=sd1−mt∣m/m−1/1−mcd1−mt∣m/m−1cst∣m=1−mcd1−mt∣m/m−1/sd1−mt∣m/m−1cdt∣m=cd1−mt∣m/m−1/nd1−mt∣m/m−1dct∣m=nd1−mt∣m/m−1/cd1−mt∣m/m−1Table 4
Jacobi imaginary transformation.
cnit,m=nct,1−msnit,m=isnt,1−m/cnt,1−mdnit,m=dnt,1−m/cnt,1−mdnit,m=dnt,1−m/cnt,1−mncit,m=cnt,1−mndit,m=cnt,1−m/dnt,1−msdit,m=isnt,1−m/dnt,1−mdsit,m=−idnt,1−m/snt,1−mcdit,m=1/dnt,1−mdcit,m=dnt,1−mscit,m=isnt,1−mcsit,m=−i/snt,1−mExample 1.
Letp=1, q=1, y0=0, and y˙0=2. The solution to the i.v.p.(21)u″t+ut+u3t=0,u0=0,u′0=2,is given by(22)ut=−23dn3t∣1/3sn3t∣1/3sn3t∣1/32−3=233sd3t∣13.
This solution is periodic with main periodT=4K1/3/3. See Figure 1 for a comparison with Runge–Kutta numerical solution (dashed curve).Figure 1
Comparison between the exact and the numerical solution.Example 2.
Letp=−5, q=1, y0=1, and y˙0=2. The solution to the i.v.p.,(23)u″t−5ut+u3t=0,u0=1,u′0=2,reads(24)ut=−5+26cn23/443t+cn−1−15+26∣12+546∣12+546=−3.14626cn−2.21336t−5.50011−3.13354−1∣1.01031.
An equivalent expression without the imaginary unit is(25)ut=dnwt+26−2cnwt∣snwt1−26−2snwt2,beingw=1+32,m=206−48≈0.98979.
See Figure2 for a comparison with Runge–Kutta numerical solution (dashed curve).Figure 2
Comparison with Runge–Kutta numerical solution (dashed curve).


## 2.2. Second Case:Δ<0

Define(26)δ=2pu02+qu04+2u˙02.SinceΔ<0, necessarily q<0. From the equality(27)δ=p2−Δ−q,it is evident that δ>0. We seek a solution to the i.v.p. (1) in the ansatz form(28)ut=A−2A1+vt,where v=vt is the solution to some Duffing equations(29)v″t+αvt+βv3t=0,v0=v0≔A+u0A−u0,v˙0≔2Au˙0A−u02.We have(30)v″t=−αvt−βv3t,(31)v′t2=D−αvt2−12βvt4,D=2A4α+A4β+4A3u0β−4A2u02α+6A2u02β+8A2u˙02+4Au03β+2u04α+u04β2A−u04.Inserting the ansatz (30) into the ode u″t+put+qu3t=0 and taking into account (30) and (31), we get(32)u″t+put+qu3t=A1+vt3A2q+p−2βvt3+p+2α−3A2qvt2−p+2α−3A2qvt−A2q+4D+p.Equating to zero the coefficients ofv0t,v1t,v1t, and v3t in (32), we obtain an algebraic system. Solving it gives(33)α=123A2q−p,β=12A2q+p,A=δ−q4.Observe that(34)α+v02β2+2v˙02β=2p2−qqδ+−qδ>0.Thus, the Duffing equation (29) has a positive discriminant. The solution to the i.v.p. (3) is then given by(35)ut=δ−q4−2δ/−q41+v0cnω¯x∣m¯+v˙0/ωsnω¯x∣m¯dnω¯x∣m¯/1+b¯snω¯x∣m¯2,where(36)ω¯=Δ¯,m¯=12−α2Δ¯,b¯=α+βv022Δ¯−12,v0=A+u0A−u0,v˙0=2Au˙0A−u02,Δ¯=α+v02β2+2v˙02β.The values ofa,β, and A are found from (33).


## 2.3. Third Case:Δ=0

When the discriminant vanishes, thenq<0 and the only solution to problem (1) with u′02=u˙02 is(37)ut=−−pqtanhp2t−tanh−1−qpu0,which may be verified by direct computation.


## 3. New Trigonometric Jacobian Functions

Define the generalized cosine and sine functions as follows:(38)cosλt=1+λcos1+λt1+λcos21+λt,sinλt=sin1+λt1+λcos21+λt.Our aim is to find someλ so that(39)cnt,m≈cosλt.Define(40)Rt=12mxt4+121−2mxt2+12x′t2+12m−1.Observe thatRt=0 when xt=cnt,m. Let xt=cosλt. We have(41)Rt=12λcos2λ+1t+1313214λ2+2m+12λ+12m+132−17λ2−32λ−λm−16mcos2λ+1t+1322λ2+8λ−2λm+4mcos4λ+1t+132λ2+λmcos6λ+1t.We will chooseλ so that(42)14λ2+2m+12λ+12m=0.Define(43)λ=114m2−144m+144−m+12.The obtained approximations are good. This is seen from Tables5 and 6.Table 5
Errors for approximations (38) and (39).
mmax−T/2≤t≤T/2cnt,m−cosmtmax−T/2≤t≤T/2snt,m−sinmt−1.000.006720.00638−0.950.006240.00598−0.900.005740.00557−0.850.005260.00516−0.800.004780.00476−0.750.004320.00435−0.700.003850.00395−0.650.003410.00355−0.600.003010.00316−0.550.002610.00277−0.500.002220.00240−0.450.001850.00203−0.400.001510.00168−0.350.001190.00135−0.300.000910.00104−0.250.000650.00076−0.200.000430.00051−0.150.000250.00030−0.100.000110.00014−0.050.000030.00003Table 6
Errors for approximations (38) and (39).
mmax−T/2≤t≤T/2cnt,m−cosmtmax−T/2≤t≤T/2snt,m−sinmt0.000.000000.000000.050.000030.000040.100.000130.000180.150.000320.000440.200.000610.000850.250.001000.001440.300.001520.002250.350.002190.003330.400.003040.004760.450.004100.006610.500.005410.009030.550.007040.012160.600.009050.016250.650.011560.021640.700.014720.028820.750.018800.038560.800.024130.052130.850.031390.071610.900.041900.100310.950.059530.13719We now will introduce new Jacobian “trigonometric functions” as follows:(44)cnmt=cosλt,for−1≤m≤0.85,cnmt=18m−1sinh2t−2ttanhtsecht+secht,for0.85<m≤1,(45)λ=114m2−144m+144−m+12,snmt=sinλt,for−1≤m≤0.85,snmt=14m−1tsech2t−tanht+tanht,for0.85<m≤1,λ=114m2−144m+144−m+12.Define(46)dnmt=1−msnm2t,ncmt=1cnmt,nsmt=1snmt,ndmt=1dnmt,(47)scmt=snmtcnmt,csmt=cnmtsnmt,sdmt=snmtdnmt,dsmt=dnmtsnmt,cdmt=cnmtdnmt,dcmt=dnmtcnmt.We extend the new functions (44)–(47) cnmt and snmt for m>1 and m<0 and imaginary argument it using Tables 1–3 replacing the cnt,m with cnmt and cnmt with cnmt and so on.


## 4. Applications in Physics

Many partial differential equations arising in soliton theory may be reduced to odes or systems of odes by means of a traveling wave transformation. These odes are generally nonlinear and some of them are Duffing type equations. Let us consider some important models of soliton theory.


### 4.1. The Klein–Gordon–Zakharov (KGZ) Equation in Plasmas

The KGZ equation reads(48)qtt−k2qxx+aq+brq+cq2q=0,rtt−k2rxx=dq2xx.We transform the KGZ by means of the traveling wave substitution(49)qx,t=ux−λtexp−1−κx+ωt+θ,rx,t=vx−λt,to obtain the system(50)u″ξ−uξa+bvξ+κ2k2−ω2+cuξ3+2iu′ξκk2−λωk−λk+λ=0,ξ=x−λt,(51)v″ξ+2duξu″ξ+u′ξ2k−λk+λ=0,ξ=x−λt.We chooseλ so that κ=λω/k2 and integrating the equation (51) twice taking null integration constants, we obtain(52)vξ=dλ2−k2uξ2,u″ξ+1k2−λ2a+ω2λ2k2−1uξ+ck2−λ2−bdk2−λ22uξ3=0,and the problem reduces to solve a Duffing equation.


### 4.2. The Sine-Gordon Equation

This is the equation(53)vtt=αvxx+βsinv.This important model appears in differential geometry and relativistic field theory. It is denominated following its similar form to the Klein–Gordon equation. The equation, as well as several solution techniques, was known in the 19th century, but the equation grew greatly in importance when it was realized that it led to solutions (“kink” and “antikink”) with the collision properties of solitons.The sine-Gordon equation is widely applied in physical and engineering applications, including the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendular attached to a stretched wire, and dislocations in crystals. It also arises in nonlinear optics. We apply the traveling wave transformation(54)vx,t=2arcsinux−λt,and the sine-Gordon equation converts into(55)2α−λ2uξ2−1u″ξ+2uξλ2−αu′ξ2−βuξ2−12=0.It may be easily verified that equation (55) holds for any solution u=uξ of the Duffing equation(56)u″ξ+βα−λ2uξ+2βλ2−βu3ξ=0.


### 4.3. The Pendulum Equation

This equation reads(57)θ″t+ω2sinθt=0,θ0=θ0,θ′0=θ˙0.Let(58)θt=2arcsinut.Inserting ansatz (58) into (57) gives(59)utu′t2+ω2ut2−12−ut2−1u″t=0.Equation (59) is satisfied for any solution u=ut to Duffing equation obeying(60)u″t+pu02+12qu04+u˙02+ω2ut−2ω2ut3=0,u0=u0,u′0=u˙0,whereu0=u0=sinθ02,u˙0=u′0=θ˙02cosθ02.


### 4.4. The KdV Equation

This equation originated from soliton theory. It reads(61)∂tu+∂xxxu+au∂xu=0,a=non zero constant.Letu=vx+λt=uξ. The traveling wave substitution gives λv′ξ+avξv′ξ+v‴ξ=0. Integrating once, we obtain(62)λvξ+a2vξ2+v″ξ=C,where C is the constant of integration. We seek a solution to the nonlinear ode (62) in the ansatz form(63)vξ=R+Sy2ξ,wherey″ξ+pyξ+qy3ξ=0,where the constants p, q, R, and S are to be determined. Since ∫y′ξy″ξ+pyξ+qy3ξdξ=∫0dξ=D, it is clear that(64)y′ξ2=D−pyξ2−q2yξ4.Inserting the ansatz (63) into (62) and taking into account (63), we obtain(65)aS22−3qSyξ4+aRS−4pS+λSyξ2+aR22+C+2DS+λR=0.Equating the coefficients ofyξ4, yξ2, and C+2DS+λR to zero gives an algebraic system. Solving it, we arrive at the expressions(66)p=14aR+λ,q=aS6,S≠0.


### 4.5. The Nonlinear Schrodinger Equation

The nonlinear Schrödinger equation is among the most prominent equations in nonlinear physics, especially in nonlinear optics. The nonlinear Schrödinger equation is of particular importance in the description of nonlinear effects in optical fibers. The nonlinear Schrödinger equation is a central model of nonlinear science, applying to hydrodynamics, plasma physics, molecular biology, and optics. It has been studied for more than 40 years, and it is employed in numerous fields well beyond plasma physics and nonlinear optics, where it originally appeared. The nonlinear Schrödinger equation (NLSE) is in the following form:(67)−1∂u∂t+∂2u∂x2+γu2u=0,where γ is a nonzero real constants and u=ux,t is a complex valued function of two real variables x,t. The Schrödinger equations occur in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. The NLSE (67) exhibits soliton and periodic cnoidal wave solutions. Let(68)ut,x=eiαx+βtvx−2αt.Under this transformation, the NLSE (67) takes the form(69)v″ξ−α2+βvξ+2γv3ξ=0,which is a Duffing equation.


## 4.1. The Klein–Gordon–Zakharov (KGZ) Equation in Plasmas

The KGZ equation reads(48)qtt−k2qxx+aq+brq+cq2q=0,rtt−k2rxx=dq2xx.We transform the KGZ by means of the traveling wave substitution(49)qx,t=ux−λtexp−1−κx+ωt+θ,rx,t=vx−λt,to obtain the system(50)u″ξ−uξa+bvξ+κ2k2−ω2+cuξ3+2iu′ξκk2−λωk−λk+λ=0,ξ=x−λt,(51)v″ξ+2duξu″ξ+u′ξ2k−λk+λ=0,ξ=x−λt.We chooseλ so that κ=λω/k2 and integrating the equation (51) twice taking null integration constants, we obtain(52)vξ=dλ2−k2uξ2,u″ξ+1k2−λ2a+ω2λ2k2−1uξ+ck2−λ2−bdk2−λ22uξ3=0,and the problem reduces to solve a Duffing equation.


## 4.2. The Sine-Gordon Equation

This is the equation(53)vtt=αvxx+βsinv.This important model appears in differential geometry and relativistic field theory. It is denominated following its similar form to the Klein–Gordon equation. The equation, as well as several solution techniques, was known in the 19th century, but the equation grew greatly in importance when it was realized that it led to solutions (“kink” and “antikink”) with the collision properties of solitons.The sine-Gordon equation is widely applied in physical and engineering applications, including the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendular attached to a stretched wire, and dislocations in crystals. It also arises in nonlinear optics. We apply the traveling wave transformation(54)vx,t=2arcsinux−λt,and the sine-Gordon equation converts into(55)2α−λ2uξ2−1u″ξ+2uξλ2−αu′ξ2−βuξ2−12=0.It may be easily verified that equation (55) holds for any solution u=uξ of the Duffing equation(56)u″ξ+βα−λ2uξ+2βλ2−βu3ξ=0.


## 4.3. The Pendulum Equation

This equation reads(57)θ″t+ω2sinθt=0,θ0=θ0,θ′0=θ˙0.Let(58)θt=2arcsinut.Inserting ansatz (58) into (57) gives(59)utu′t2+ω2ut2−12−ut2−1u″t=0.Equation (59) is satisfied for any solution u=ut to Duffing equation obeying(60)u″t+pu02+12qu04+u˙02+ω2ut−2ω2ut3=0,u0=u0,u′0=u˙0,whereu0=u0=sinθ02,u˙0=u′0=θ˙02cosθ02.


## 4.4. The KdV Equation

This equation originated from soliton theory. It reads(61)∂tu+∂xxxu+au∂xu=0,a=non zero constant.Letu=vx+λt=uξ. The traveling wave substitution gives λv′ξ+avξv′ξ+v‴ξ=0. Integrating once, we obtain(62)λvξ+a2vξ2+v″ξ=C,where C is the constant of integration. We seek a solution to the nonlinear ode (62) in the ansatz form(63)vξ=R+Sy2ξ,wherey″ξ+pyξ+qy3ξ=0,where the constants p, q, R, and S are to be determined. Since ∫y′ξy″ξ+pyξ+qy3ξdξ=∫0dξ=D, it is clear that(64)y′ξ2=D−pyξ2−q2yξ4.Inserting the ansatz (63) into (62) and taking into account (63), we obtain(65)aS22−3qSyξ4+aRS−4pS+λSyξ2+aR22+C+2DS+λR=0.Equating the coefficients ofyξ4, yξ2, and C+2DS+λR to zero gives an algebraic system. Solving it, we arrive at the expressions(66)p=14aR+λ,q=aS6,S≠0.


## 4.5. The Nonlinear Schrodinger Equation

The nonlinear Schrödinger equation is among the most prominent equations in nonlinear physics, especially in nonlinear optics. The nonlinear Schrödinger equation is of particular importance in the description of nonlinear effects in optical fibers. The nonlinear Schrödinger equation is a central model of nonlinear science, applying to hydrodynamics, plasma physics, molecular biology, and optics. It has been studied for more than 40 years, and it is employed in numerous fields well beyond plasma physics and nonlinear optics, where it originally appeared. The nonlinear Schrödinger equation (NLSE) is in the following form:(67)−1∂u∂t+∂2u∂x2+γu2u=0,where γ is a nonzero real constants and u=ux,t is a complex valued function of two real variables x,t. The Schrödinger equations occur in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. The NLSE (67) exhibits soliton and periodic cnoidal wave solutions. Let(68)ut,x=eiαx+βtvx−2αt.Under this transformation, the NLSE (67) takes the form(69)v″ξ−α2+βvξ+2γv3ξ=0,which is a Duffing equation.



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*Source: 9994967-2021-11-30.xml*