# A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

Authors: Alvaro H. Salas S; Jairo E. Castillo H; Darin J. Mosquera P
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the formz″=Fz in the sense that z=zt. In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.


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## Body

## 1. Introduction

Early descriptions of many of them date back to, at least, 1892, when the book by Greenhill [1] appeared, presenting a variety of such problems: a simple pendulum, catenaries, and a uniform chain that rotates steadily with a constant angular velocity about an axis to which the chain is fixed at two points. Later applications include nonlinear plasma oscillations [2], Duffing oscillators [3], rigid plates satisfying the Johansen yield criterion [4], nonlinear transverse vibrations of a plate carrying a concentrated mass [5], a beam supported at an axially oscillating mount [6], doubly periodic cracks subjected to concentrated forces [7], surface waves in a plasma column [8], coupled modes of nonlinear flexural vibrations of a circular ring [9], dual-spin spacecrafts [10], spacecraft motion about slowly rotating asteroids [11], nonlinear vibration of buckled beams [12], a nonlinear wave equation [13], deep-water waves with two-dimensional surface patterns [14], oscillations of a body with an orbital tethered system [15], and nonlinear mathematical models of DNA [16, 17]. Numerical studies of phase spaces, stability analysis, solution by means of finite differences, application of the Bernoulli wavelet method for estimating a solution of linear stochastic integral equations, existence of periodic solutions, and numerical simulations can be found in [18–22]. The search for new analytical methods that lead to the exact solution of the Helmholtz equations is the vital importance, since the developed methods can be applied to Schrodinger’s nonlinear differential equation, which is known to have different applications in nonlinear optics, plasma physics, fluid mechanics, and Bose–Einstein condensates [23, 24]. As a contribution to the literature, in this article, we present the exact solution to the Helmholtz Oscillator for the given arbitrary initial conditions by means of the Weierstrass elliptic function.In this paper, we will derive the exact solution to the Helmholtz oscillator:(1)q¨+α+βq+γq2=0,q0=q0,q′0=q˙0,where q≡qt denotes the displacement of the system, q′0≡∂tqt=0≡q˙0, q¨≡∂t2q, β is the natural frequency, γ is a nonlinear system parameter, and α is a system parameter independent of the time.Equation (1) is applied for mathematical modeling in physics and engineering like general relativity, betatron oscillations, vibrations of shells, vibrations of the acoustically driven human eardrum, and solid-state physics [24–28]. The Helmholtz oscillator can be interpreted as a particle moving in a quadratic potential field, and it has also been studied in a nonlinear circuit theory.One of the possible interesting interpretations of equation (1) is given by a simple electrical circuit.


## 2. The Physical Models and Helmholtz Equation

### 2.1. Alternation the LC Series Circuit

Let us consider alternating LC series circuit consisting of a linear inductor and a capacitor of two terminals as a dipole as shown in Figure1. Also, it is known that a functional relationship between the electric charge q, the capacitor voltage u≡uq, and the time charging t has the form as follows:(2)fq,u,t=0.Figure 1
The LC series circuit.The relationship between the chargeq of the nonlinear capacitor and the voltage drop across it may be approximated by the following quadratic equation [29]:(3)uc=sq+aq2,where uc gives the potential across the plates of the nonlinear capacitor and s and a are constants related to the capacitor.Now, Kirchhoff’s voltage law for the LC series circuit could be written as follows:(4)Li′+sq+aq2=VE,where i′≡∂ti, L is the inductor inductance measured by Henry unit, and VE gives the voltage of the battery which is constant. The relation between the current and charge reads i=dq/dt, and by rearrange equation (4), the Helmholtz equation is obtained as(5)q¨+α+βq+γq2=0,where q≡qt, q¨≡∂t2q, β=s/L=1/LC=ω02, where ω0=1/LC represents the resonant angular frequency, and γ=a/L=1/Cq0L, where C donates the capacitance of the capacitor in farad and q0 represent the initial value of charge on the capacitor plates which has minimum or zero value for the charging capacitor and for discharge, it has maximum values. The LC circuit is considered as an oscillating circuit which stores energy to oscillate at the resonant frequency f0=1/2πLC of the circuit where f0 could be calculated from the resonance condition: inductor impedance 2πf0L = capacitor impedance 1/2πf0C.


### 2.2. Electronegative Plasmas and the KdV–Helmholtz Equation

Let us consider the propagation of electrostatic nonlinear ion-acoustic structures in a collisionless electronegative plasma consisting of thermal particles (including Maxwellian electrons and light negative ions) and fluid cold positive ions. The dynamics of nonlinear electrostatic structures are governed by the following dimensionless fluid equations [30–33]:(6)∂tn+∂xnu=0,(7)∂tu+u∂xu+∂xϕ=0,(8)∂x2ϕ=μeexpϕ+μnexpσnϕ−n,where n and u are the normalized positive ion number density and fluid velocity, respectively, and ϕ gives the normalized electrostatic potential [30]. Here, σn=Te/Tn gives the temperature ratio of electron-to-negative ion, μe=ne0/n0 gives the electron concentration, and μn=nn0/n0 refers to the negative concentration, where n0, ne0, and nn0 are the unperturbed equilibrium number densities of the positive ion, electron, and negative ion, respectively. Accordingly, the neutrality condition at equilibrium reads μe+μn=1, with μn=αn/1+αn and μe=1/1+αn, where αn=nn0/ne0.By applying the RPT, the independent variables are given byζ=ε1/2x−Vpht and τ=ε3/2t and the dependent quantities are expanded as n=1+εn1+ε2n2+⋯, u=εu1+ε2u2+⋯, and ϕ=εϕ1+ε2ϕ2+⋯. Here, Vph is the normalized wave phase velocity and ε is a real and small parameter ε≪1. Inserting stretching and expansion values of the independent and dependent quantities into equations (6)–(8), we get a system of reduced equations with different orders of ε. The first-order dependent quantities n1,u1,ϕ1 could be obtained from solving the system of the lowest orders in ε as n1=ϕ1/Vph2, u1=ϕ1/Vph, and Vph=1/S1. By solving the system of reduced equations to the next order in ε, the following KdV equation is obtained:(9)∂τφ+K1φ∂ζφ+K2∂ζ3φ=0,where the coefficient of the nonlinear term K1=Vph33/Vph4−2S2/2 and the coefficient of the dispersion term K2=Vph3/2, where S1=μe+μnσ and S2=μe+μnσ2/2. Note that, for K1>0K1<0, the positive (negative) pulses can exist and propagate in the present plasma model. The sign and the values of K1 are related to the relevant plasma parameters.Using the travelling wave transformξ=ζ+λτ and φζ,τ=qξ into equation (9) and integrating the obtained result over ξ, we get(10)q″+α+βq+γq2=0,where q≡qξ, q˙≡∂ξq, q¨≡∂ξ2q, and λ represents the normalized velocity of the moving frame. Here, α=ck/K2, where ckis the integrating constant, β=λ/K2, and γ=K1/2K2.(11)q″+αq+βq2=0.


## 2.1. Alternation the LC Series Circuit

Let us consider alternating LC series circuit consisting of a linear inductor and a capacitor of two terminals as a dipole as shown in Figure1. Also, it is known that a functional relationship between the electric charge q, the capacitor voltage u≡uq, and the time charging t has the form as follows:(2)fq,u,t=0.Figure 1
The LC series circuit.The relationship between the chargeq of the nonlinear capacitor and the voltage drop across it may be approximated by the following quadratic equation [29]:(3)uc=sq+aq2,where uc gives the potential across the plates of the nonlinear capacitor and s and a are constants related to the capacitor.Now, Kirchhoff’s voltage law for the LC series circuit could be written as follows:(4)Li′+sq+aq2=VE,where i′≡∂ti, L is the inductor inductance measured by Henry unit, and VE gives the voltage of the battery which is constant. The relation between the current and charge reads i=dq/dt, and by rearrange equation (4), the Helmholtz equation is obtained as(5)q¨+α+βq+γq2=0,where q≡qt, q¨≡∂t2q, β=s/L=1/LC=ω02, where ω0=1/LC represents the resonant angular frequency, and γ=a/L=1/Cq0L, where C donates the capacitance of the capacitor in farad and q0 represent the initial value of charge on the capacitor plates which has minimum or zero value for the charging capacitor and for discharge, it has maximum values. The LC circuit is considered as an oscillating circuit which stores energy to oscillate at the resonant frequency f0=1/2πLC of the circuit where f0 could be calculated from the resonance condition: inductor impedance 2πf0L = capacitor impedance 1/2πf0C.


## 2.2. Electronegative Plasmas and the KdV–Helmholtz Equation

Let us consider the propagation of electrostatic nonlinear ion-acoustic structures in a collisionless electronegative plasma consisting of thermal particles (including Maxwellian electrons and light negative ions) and fluid cold positive ions. The dynamics of nonlinear electrostatic structures are governed by the following dimensionless fluid equations [30–33]:(6)∂tn+∂xnu=0,(7)∂tu+u∂xu+∂xϕ=0,(8)∂x2ϕ=μeexpϕ+μnexpσnϕ−n,where n and u are the normalized positive ion number density and fluid velocity, respectively, and ϕ gives the normalized electrostatic potential [30]. Here, σn=Te/Tn gives the temperature ratio of electron-to-negative ion, μe=ne0/n0 gives the electron concentration, and μn=nn0/n0 refers to the negative concentration, where n0, ne0, and nn0 are the unperturbed equilibrium number densities of the positive ion, electron, and negative ion, respectively. Accordingly, the neutrality condition at equilibrium reads μe+μn=1, with μn=αn/1+αn and μe=1/1+αn, where αn=nn0/ne0.By applying the RPT, the independent variables are given byζ=ε1/2x−Vpht and τ=ε3/2t and the dependent quantities are expanded as n=1+εn1+ε2n2+⋯, u=εu1+ε2u2+⋯, and ϕ=εϕ1+ε2ϕ2+⋯. Here, Vph is the normalized wave phase velocity and ε is a real and small parameter ε≪1. Inserting stretching and expansion values of the independent and dependent quantities into equations (6)–(8), we get a system of reduced equations with different orders of ε. The first-order dependent quantities n1,u1,ϕ1 could be obtained from solving the system of the lowest orders in ε as n1=ϕ1/Vph2, u1=ϕ1/Vph, and Vph=1/S1. By solving the system of reduced equations to the next order in ε, the following KdV equation is obtained:(9)∂τφ+K1φ∂ζφ+K2∂ζ3φ=0,where the coefficient of the nonlinear term K1=Vph33/Vph4−2S2/2 and the coefficient of the dispersion term K2=Vph3/2, where S1=μe+μnσ and S2=μe+μnσ2/2. Note that, for K1>0K1<0, the positive (negative) pulses can exist and propagate in the present plasma model. The sign and the values of K1 are related to the relevant plasma parameters.Using the travelling wave transformξ=ζ+λτ and φζ,τ=qξ into equation (9) and integrating the obtained result over ξ, we get(10)q″+α+βq+γq2=0,where q≡qξ, q˙≡∂ξq, q¨≡∂ξ2q, and λ represents the normalized velocity of the moving frame. Here, α=ck/K2, where ckis the integrating constant, β=λ/K2, and γ=K1/2K2.(11)q″+αq+βq2=0.


## 3. Exact Solution

In this section, we are going to solve, in general, the equation:(12)q¨+α+βq+γq2=0,q0=q0,q′0=q˙0.To do that, the following ansatz is suggested:(13)qt=A+B1+C℘t−t0;g2,g3,where BC≠0, ℘t stands for the Weierstrass elliptic function. This function satisfies the following ordinary differential equation (ODE):(14)℘′ℱ;g2,g32=4℘3ℱ;g2,g3−g1℘ℱ;g2,g3−g3,(15)℘″ℱ;g2,g3=−g22+6℘2ℱ;g2,g3.Inserting ansatz (13) into (12) and making use of (15) gives Rt=0, where(16)Rt=C2A2Cγ+ACβ+2B+Cα℘3t−t0;g2,g3+C3A2Cγ+2ABCγ+3ACβ+BCβ−6B+3Cα℘2t−t0;g2,g3+12C6A2γ+8ABγ+6Aβ+2B2γ−3BCg2+4Bβ+6α℘t−t0;g2,g3+A2γ+2ABγ+Aβ+B2γ−2BC2g3+12BCg2+Bβ+α.Equating the coefficients of℘jt−t0;g2,g3 to zero gives an algebraic system, and by solving it, we obtain(17)B=−62Aγ+βA2γ+Aβ+α,C=122Aγ+β,g2=112β2−4αγ,g3=12162Aγ+β−2A2γ2−2Aβγ−6αγ+β2,,and thus, we get(18)qt=A−6AAγ+β+α12℘t−t0;1/12β2−4αγ,1/216β+2Aγβ2−2Aγβ−2γγA2+3α+2Aγ+β.Now, for determining the values oft0 and A, the initial conditions q0=q0andq′0=q˙0 must be applied in addition to the following formula:(19)℘z+y;g2,g3=14℘′z;g2,g3−℘′y;g2,g3℘z;g2,g3−℘y;g2,g32−℘z;g2,g3−℘y;g2,g3,where z=t and y=−t0 for our case. According to this relation, we get(20)℘t−t0;g2,g3=14℘′t;g2,g3−c1℘t;g2,g3−c02−℘t;g2,g3−c0,where c0=℘−t0;g2,g3 and c1=℘′−t0;g2,g3.The initial conditionsq0=q0andq′0=q˙0 give(21)c0=−−q02A+B+AA+B+Bq˙0+q02CA−q02,c1=−Bq˙0CA−q02,where the values of B and C are defined in equation (17).Finally, the value ofA can be estimated from the condition(22)q¨0+α+βq0+γq20=0,which gives(23)22γA3+3βA2+6αA−6αq0−3βq02−2γq03−3q023A−q0=0.It is clear thatA must obey the cubic equation:(24)γA3+3βA2+6αA−6αq0+3βq02+2γq03+3q02=0.We will choose the first root of equation (24), i.e.,(25)A=12γΔ27+6γ22αq0+βq02+q02+6αβγ−β3+4γ3q033+β2−4αγΔ/27+6γ22αq0+βq02+q02+6αβγ−β3+4γ3q033−β,with(26)Δ=729β+2γq0−6αγ+β2−2γq0β+γq0−6γ2q022+36αγ−9β23.Forq˙0=0, i.e., q0=q0 and q′0=0, the following solution is obtained:(27)qt=q0−6α+βq0+γq02β+2γq0+12℘q;1/12β2−4αγ,1/216β+2γq0β2−2γq0β−2γ2q02−6αγ.In the case, whenq˙0≠0, the solution to initial value problem (1) is given by(28)qt=A+B1+C1/4c1−℘′t;g2,g3/c0−℘t;g2,g32−c0−c1−℘t;g2,g3,or, in a more compact form,(29)qt=A+B1+C℘t−t0;g2,g3,where t0=−℘−1c0;g2,g3.Expression℘−1c0;g2,g3 stands for the inverse of the Weierstrass elliptic function which is defined as(30)℘−1x;g2,g3=∫∞x14ζ3−g2ζ−g3dζ,for real x and ℜe4ζ3−g2ζ−g3>0.The obtained solution (29) is periodic with period(31)T=±2∫e1∞14z3−g2z−g3dz,where e1 is the greatest real root of the cubic(32)4z3−g2z−g3=0.Taking (30) into account, the period T in (29) may also be expressed as(33)T=±2℘−1e1;g2,g3.Remark 1.
The Weierstrass elliptic function℘z;g2,g3 is related to the Jacobian elliptic functions sn,cd,and dn as follows: (34)dnωzm2=1−3mω3℘z;4/3m2−m+1ω2,4/27m−2m+12m−1ω3,(35)℘z;g2,g3=−mωdnωzm2−1−13m+1ω,=ωsnωzm2−13m+1ω,=ω1−cnωzm2−13m+1ω,where(36)16ω6−24ω4g2+9ω2g22−g23+27g32=0,(37)4g23−27g32ω6−12g23−27g32ω5−3g23+216g32ω4+213g23+378g32ω3−3g23+216g32ω2−12g23−27g32ω+4g23−108g32=0.
In view of (35), the fundamental period of the Weierstrass function ℘z;g2,g3 may also be expressed in the following form:(38)T=2Kmω.
Further properties of the Weierstrass function with applications in quantum theory of Lame potential may be found in [26].


## 4. Results and Discussions

Let us consider a nonlinearLC series circuit with L=1H, C=0.1F, qo=1C, and let the circuit without external battery, i.e., VE=0α=0. Thus, the exact solution to the initial value problem(39)q¨+10q+q2=0,q0=q0=1,q′0=q˙0=0,according to formula (27), is given by(40)qt=1−1121+℘t;25/3,13/3.This solution is periodic, and to find its period according to equation (32), the following cubic equation is solved:(41)4z3−253z−133=0,which is given the following three real roots:(42)z1=−1,z2=3+436,z3=3−436.The greatest root ise1=z2.Using formula (31), we obtain(43)T=∫z2∞14z3−25/3z−13/3dz,=22119+43ℜeK11143+243.Also, we can use equations (34)–(38) to obtain(44)qt=−42+3+9+43dn32+23t11143−2432,and then, the period of the solution is given by(45)T=2K1/1143−2433/2+2/3.From equation (43) and (45), the period T is obtained as(46)T=22119+43ℜeK11143+243,=2K1/1143−2433/2+2/3,≈1.99590288250676993.In Figure2, the analytical solution (40) is compared with the Runge–Kutta (RK) numerical solution, and the agreement between the two solutions is found to be very good. If the initial current value of the LC circuit is not equal to zero, i.e, i=q′0=q˙0≠0, but has a specified value q˙0=0.3A, in this case, we can consider solution (28) or (29) instead of the first one (27). It is observed that the results obtained by the exact solution are in an excellent agreement with the RK numerical solution, as shown in Figure 3. For the plasma application, our analyses are based on the experimental observations data where electron and negative ion temperatures are, respectively, given by Te≈0.69eV and Tn≈0.06±0.02eV and ne=3.8×109cm−3 gives the electron number density. According to these data, we get σn=8.7, 11.5, and 17.25 and the negative ions concentration 0≤μn≤1 [26, 27, 30]. Using these data, we get β,γ=4.433,0.44, and by plotting these data according to solution (27), we note, from Figure 4, the excellent matching between our solution and the RK numerical solution.Figure 2
(Color online). A comparison between the analytical solution (40) (dashed curve) and the Runge–Kutta (RK) numerical solution (dotted curve) for the LC series circuit with L=1H, C=0.1F, q0=1C, and q˙0=0.Figure 3
(Color online). A comparison between the analytical solution (28) (dashed curve) and the Runge–Kutta (RK) numerical solution (dotted curve) for the LC series circuit with L=1H, C=0.1F, q0=1, and q˙0=0.3A.Figure 4
(Color online). A comparison between the analytical solution (28) (dashed curve) and the Runge–Kutta (RK) numerical solution (dotted curve) for the electronegative plasma data αn=0.1, σn=8.7, q0=0.51, and q˙0=0.1.


## 5. Conclusions

The quadratic nonlinear Helmholtz differential equation is solved analytically for any arbitrary initial conditions through the Weierstrass elliptic function. Accordingly, some new analytical solutions are obtained for the first time using ansatz (12). The proposed methodology is of great importance in solving several physics problems in plasma physics and electronic circuits. For instance, our obtained solutions are devoted for describing the dynamics of nonlinear oscillations that propagate in electronegative complex plasmas. Moreover, our solutions are applied for studying the characteristic behavior of signal oscillations in the LC series circuits.



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