Analytical Solutions of Some Strong Nonlinear Oscillators

Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing ’ s oscillator. Using the exact analytical solution to cubic Duffing and cubic-quinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form € x þ f x ð Þ ¼ 0 as well as € x þ 2 ε _ x þ f x ð Þ ¼ F t ð Þ , where x ¼ x t ð Þ and f ¼ f x ð Þ and F t ð Þ are continuous functions. In the present chapter, sometimes we will use f (cid:2) x ð Þ ¼ (cid:2) f x ð Þ and take the approximation f x ð Þ ≈ P Nj ¼ 1 p j x j , where j ¼ 1, 3, 5, ⋯ N only odd integer values and x ∈ (cid:2) A , A ½ (cid:3) . Moreover, we will take the approximation f x ð Þ ≈ P Nj ¼ 0 p j x j , where j ¼ 1, 2, 3, ⋯ N , and x ∈ (cid:2) A , A ½ (cid:3) . Arbitrary initial conditions are considered. The main idea is to approximate the function f ¼ f x ð Þ by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma physics, electronic circuits, soliton theory, and engineering are provided. Mixed parity oscillator, Damped Duffing equation, Damped Helmholtz equation, Forced Duffing


Abstract
Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing's oscillator. Using the exact analytical solution to cubic Duffing and cubicquinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form € x þf x ð Þ ¼ 0 as well as € ð Þ are continuous functions. In the present chapter, sometimes we will use f Àx ð Þ ¼ Àf x ð Þ and take the approximation f x ð Þ ≈ P N j¼1 p j x j , where j ¼ 1, 3, 5, ⋯N only odd integer values and x ∈ ÀA, A ½ . Moreover, we will take the approximation f x ð Þ ≈ P N j¼0 p j x j , where j ¼ 1, 2, 3, ⋯N, and x ∈ ÀA, A ½ . Arbitrary initial conditions are considered. The main idea is to approximate the function f ¼ f x ð Þ by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma

Introduction
Both the ordinary and partial differential equations have an important role in explaining many phenomena that occur in nature or in medical engineering, biotechnology, economic, ocean, plasma physics, etc. [1,2]. Duffing equation is considered one of the most important differential equations due to its ability for demonstrating the scenario and mechanism of various nonlinear phenomena that occur in nonlinear dynamic systems [3][4][5][6][7][8][9][10][11]. It is one of the most common models for analyzing and modeling many nonlinear phenomena in various fields of science such as the mechanical engineering [12], electrical engineering [13], plasma physics [14,15], etc. Mathematically, the Duffing oscillator is a second-order ordinary differential equation with a nonlinear restoring force of odd power where f Àx ð Þ ¼ Àf x ð Þ is a continuous function on some interval ÀA, A ½ with f 0 ð Þ ¼ 0, K i is a physical coefficient related to the physical problem under study, and i ¼ 1, 2, 3, ⋯∞. It is clear from Eq. (1) that there is no any friction/dissipation (this force arises either as a result of taking viscosity into account or the collisions between the oscillator and any other particle, etc.), and this only occurs in standardized systems such as superfluid (fluid with zero viscosity which it flows without losing any part from its kinetic energy sometimes like Bose-Einstein condensation) or the systems isolated from all the external force that resist the motion of the oscillator. The undamped Duffing equation [9] is considered one of the effective and good models for explaining many nonlinear phenomena that are created and propagated in optical fiber, Ocean, water tank, the laboratory and space collisionless and warm plasma (we will demonstrate this point below). As well known in fluid mechanics and in the fluid theory of plasma physics; the basic fluid equations of any plasma model can be reduced to a diverse series of evolution equations that can describe all phenomena that create and propagate in these physical models. For example, we can mention some of the most famous evolution equations that have been used to explain several phenomena in plasma physics and other fields of sciences; the family of one dimensional (1 À D) korteweg-de Vries equation (KdV) and it is higher-orders, including the KdV, KdV-Burgers (KdVB), modified KdV (mKdV), mKdV-Burgers (mKdVB), Gardner equation or called Extended KdV (EKdV), EKdV-Burgers (EKdVB), KdV-type equation with higherorder nonlinearity. All the above mentioned equations are partial differential equations and by using an appropriate transformation, we can convert them into ordinary differential equations of the second orders. If the frictional force is neglected, some of these equations can be converted into the undamped Duffing equation with  [17,18], and so on the other mentioned equations. However, these undamped models (without friction/dissipation) do not exist much in reality except under harsh conditions. In order to describe and simulate the natural phenomena that arise in many realistic physical models and dynamic systems, the friction/dissipation forces must be taken into account, as is the case in many plasma models and electronic systems. Accordingly, the following damped (non-conservative) Duffing equation will be devoted for this purpose If the frictional force does not neglect, so that all PDEs that have "Burgers ∂ 2 x Á ð Þ" term like KdVB-, mKdVB-, EKdVB-, KPB-, mKPB-, EKPB-, ZKB-, mZKB, EZKB-Eq. [1,2], etc. can be transformed to damped Duffing equation (2) without [19] and with [7,20,21] including damping term (2ε _ x) for f x ð Þ ≈ P x ð Þ ¼ K 1 x þ K 2 x 3 has been investigated and solved analytically and numerically by many authors using different approaches in order to understand its physical characters [22][23][24][25][26][27][28].
Many authors investigated the (un)damped Duffing equation, (un)damped Helmholtz Eq. [16,[29][30][31], and undamped H-D equation. On the contrary, there is a few numbers of published papers about damped Duffing-Helmholtz equation [32,33]. For example, Zúñiga [32] derived a semi-analytical solution to the damped Duffing-Helmholtz equation in the form of Jacobian elliptic functions, but he putted some restrictions on the coefficient of the linear term, and then obtained a solution that gives good results compared to numerical solutions. Also, it is noticed that Zúñiga solution [32] is very sensitive to the initial conditions. Gusso and Pimentel [33] obtained obtain improved approximate analytical solution to the forced and damped Duffing-Helmholtz in the form of a truncated Fourier series utilizing the harmonic balance method.
In this chapter, we display some novel semi-analytical (approximate analytical) solutions to the strong higher-order nonlinear damped oscillators of the following initial value problem (i.v.p) and its family ( Our new semi-analytical solution to Eq. (3) is derived in terms of Weierstrass and Jacobian elliptic functions. Also, we will solve Eq. (3) numerically using Runge-Kutta 4th (RK4) and make a comparison between both the semi-analytical and numerical solutions. Moreover, as some realistic physical application to the problem (3) and its family will be investigated.

Duffing equation
Let us consider the standard (undamping) Duffing equation in the absence both friction (2ε _ x) and excitation (F t ð Þ) forces [34,35] which is subjected to the following initial conditions The general solution of Eq. (4) maybe written in terms of any of the twelve Jacobian elliptic functions.
For example, let us assume By inserting solution (6) in Eq. (4), we get where cn ¼ cn ffiffiffi ffi ω p t þ c 2 , m ð Þ : Equating to zero the coefficients of cn j gives an algebraic system whose solution gives Thus, the general solution of Eq. (4) reads The values of the constants c 1 and c 2 could be determined from the initial conditions given in Eq. (5). (5). Below three cases will be discussed depending on the sign of the discriminant Δ.

First case: Δ > 0
For Δ > 0, the solution of the i.v.p. (4)-(5) is given by Making use of the additional formula the solution (10) could be expressed as where Solution (12) is a periodic solution with period Example 1.
Let us consider the i.v.p.
Using formula (10) and its periodicity is given by In Figure 1, the comparison between the exact analytical solution (17) and the approximate numerical RK4 solution is presented. Full compatibility between the two analytical and numerical solutions is observed.

Second case: Δ < 0
For Δ < 0, in this case q < 0 and then, δ ¼ where y ¼ y t ð Þ is a solution of some Duffing equation with initial conditions Inserting ansatz (18) into Eq. (4) and taking the below relation into account we get Equating the coefficients of y j t ð Þ to zero, gives an algebraic system. A solution to this system gives has a positive discriminant and it is given by Then the problem reduces to the first case. Accordingly, the solution of the i.v.p. (4)-(5) maybe written in the form, where The solution (23) is unbounded and its periodicity is given by

Example 2.
Let us assume the following i.v.p.
The solution of the i.v.p. (26) and the periodicity of this solution is given by Solution (27) is displayed in Figure 2.

Third case: Δ ¼ 0
If the discriminant vanishes Δ ¼ 0 ð Þ, then q < 0 and the only solution of problem (4) with which may be verified by direct computation.
is given by is given by Remark 3. According to the following identity with the solution of the i.v.p. (4)-(5) could be written in terms of the Weierstrass elliptic function ℘ ℘ t; with and The solution (36) is periodic with period where ρ is the greatest real root of the cubic 4x 3 À g 2 x À g 3 ¼ 0. Remark 4. An approximate analytic solution of the i.v.p. (31) is given by where and λ is a root of the cubic

Example 3.
Let us consider the i.v.p.
The approximate solution in trigonometric form is given by The exact solution reads with period The error on the interval 0 ≤ t ≤ T equals 0:025: The comparison between the approximate analytic solution (44) and the exact analytic solution (45) is illustrated in Figure 3.

Remark 5.
An approximate analytical solution of the i.v.p. (33) is given by where and λ is a solution of the quintic The approximate trigonometric solution of The exact solution is with period The error on the interval 0 ≤ t ≤ T equals 0:00018291:

An analytical solution of the undamped Duffing-Helmholtz Equation
The undamped Duffing-Helmholtz equation reads We will give a solution to the i.v.p. (53) in terms of Weierstrass elliptic functions. For solving this problem the following ansatz is considered where BC 6 ¼ 0: Substituting the ansatz (54) into the ordinary differential equation (ode) with Equating the coefficients K j to zero will give us an algebraic system. Solving this system, we finally get The values of t 0 and A could be determined from the initial conditions x 0 We have The number A is a solution to the quartic according to the relation (54) is given by

The solution of the forced undamped Duffing-Helmholtz equation
Suppose that the physical system to be studied is under the influence of some constant external/excitation force, so the standard Duffing-Helmholtz equation can be reformulated to the following constant forced Duffing-Helmholtz i.v.p.
For solving the i.v.p. (62), the following assumption is introduced where ζ is a solution to the cubic algebraic equation Note that the constant forced Duffing-Helmholtz Eq. (62) has been reduced to the standard Duffing-Helmholtz Eq. (65) with the following new initial conditions

Example 6.
Suppose that we have the following i.v.p. and we want to solve it The comparison between the solution (68) and the RK4 solution is introduced in Figure 6. The periodicity of solution (68) is given by

An approximate analytic solution of the forced damped Duffing-Helmholtz equation
Let us define the following i.v.p.
then the first equation in system (69) can be written as For solving the i.v.p. (69), the following ansatz is assumed where the function y y t ð Þ represents the exact solution to the following i.v.p.
Let us define the following residual and by applying the condition R 0 0 ð Þ ¼ 0, we obtain By solving this equation we can get the value of ρ.
The distance error as compared to the RK4 numerical solution is given by Also, the comparison between solution (78) and RK4 solution is presented in Figure 7.
Remark 5. For the damped and constant forced Helmholtz equation The value of d can be determined from: pd þ qd 2 ¼ F: However, if this equation has no real solutions we can choose d ¼ 0. Remark 6. Letting q ¼ 0, we obtain the damped and constant forced Duffing equation In this case, the number d must be a root to the cubic pd þ rd 3 ¼ F:

Approximate analytic solution of the damped and trigonometric forced Duffing-Helmholtz equation
Let us define the following new i.v.p. We suppose that q 2 À 4pr < 0, and the following residual is defined Let us define the solution of i.v.p. (82) as follows where The function y y t ð Þ is a solution to the i.v.p. wherep The value of ρ can be determined from the following equation The approximate analytic solution of the i.v.p. (89) is given by The distance error according to the RK4 numerical solution is calculated as max 0 ≤ t ≤ 60 Moreover, solution (90) is compared with RK4 solution as shown in Figure 8.

An analytic solution of cubic-quintic Duffing equation
Let us consider the following ordinary differential equation [36] € which is subjected to the following initial conditions Theorem 1. a. Suppose that x 0 6 ¼ 0, then the solution of the i.v.p. (92)-(93) is given by where the function v v t ð Þ is the solution to the following Duffing equation The values of the coefficients p and q are given by and the value of the quantity λ is a solution of the cubic The solution to the the i.v.p. (95) is obtained from the formulas in the first section. b. Suppose that x 0 ¼ 0, in this case, the solution of the i.v.p. (92)-(93) is given by where the function v v t ð Þ is the solution of the following Duffing equation The values of the coefficients p and q are expressed as and the value of λ is a solution of the cubic Note that the solution of the i.v.p. (100) could be obtained from the formulas in the first section.
Equating the coefficients H j to zero gives an algebraic system: H 1 ¼ 0, H 3 ¼ 0, and H 5 ¼ 0. Solving H 1 ¼ 0 and H 3 ¼ 0 will give the values of p and q that are given in Eqs. (101)-(102). Finally, by inserting the values of p and q into H 1 ¼ 0, we obtain the cubic Eq. (103). Likewise, the case (b) can be proved.

Damped Cubic-Quintic Oscillator
Let us define the following i.v.p.
We seek approximate analytic solution in the ansatz form where y y t ð Þ is the exact solution to the i.v.p.

Realistic physical applications
The above solutions could be applied to various fields of physics and engineering such as they could be used for describing the behavior of oscillations in RLC electronic circuits, plasma physics etc. In the below section, the above solution will be devoted for studying oscillations in various plasma models.

Nonlinear oscillations in RLC series circuits with external source
In the RLC series circuits consisting of a linear resistor with resistance R in Ohm unit, a linear inductor with inductance L in Henry unit, and nonlinear capacitor with capacitance C in Farady unit as well as external applied voltage E in voltage unit, the Kirchhoff's voltage law (KVL) could be written as where the relation between the current the charge is given by i ¼ ∂ t q _ q, i 0 ∂ t i, the coefficients a, s ð Þ are related to the nonlinear capacitor, and E represents the voltage of the battery which is constant. By reorganizing Eq. (116), the following constant forced and damped Helmholtz equation could be obtained as t q, and _ q ∂ t q. The solution of Eq. (117) can be devoted for interpreting and analyzing the oscillations that can generated in the RLC circuit.

Duffing-Helmholtz equation for modeling the oscillations in a plasma
For studying the plasma oscillations using fluid theory, the basic equations of plasma particles using the reductive perturbation method (RPM) will be reduced to some evolution equations such as KdV equation and its family [37][38][39][40][41]. Let us consider a collisionaless and unmagnetized electronegative complex plasma, consisting of inertialess cold positive and negative ion species, inertia non-Maxwellian electrons in addition to stationary negative dust impurities [42]. Thus, the quasi-neutrality condition reads: n s,e donates the unperturbed number density of the plasma particles (here, the index "s" ¼ "1" and "2" point out the positive ion and negative ion, and "e" refers to the electron, respectively). It is assumed that the plasma oscillations take place only in xÀdirectional which means that the fluid equations of the plasma particles become perturbed only in xÀdirectional. If the effect of the ionic kinematic viscosities η s for both positive η 1 ð Þ and negative η 2 ð Þ ions are included in the present investigation, as a source of damping/dissipation, in this case we will get a new evolution equation governs the dynamics of damping pulses. The dynamics of plasma oscillations are governed by the following fluid equations: Here, n s donates the normalized number density of positive and negative ions, and u s represents the normalized fluid velocity of positive and negative ions, and ϕ is the normalized electrostatic wave potential. The mass ratio is defined as: ð Þ for positive (negative) ion, and β illustrates nonthermality parameter. The quasi-neutrality condition in the normalized form reads: where V ph is the wave phase velocity of the ion-acoustic waves and ε is a real and small parameter (0 < ε < < 1). The dependent perturbed quantities Π x, t ð Þ n 1 , n 2 , u 1 , u 2 , ϕ ð Þ T are expanded as: Π ¼ , 0, 0 ð Þ T and T represents the transpose of the matrix. Inserting both the stretching and expansions of the independent and dependent quantities into the basic fluid equations and after boring but straightforward calculations, the Gardner-Burgers/EKdVB equation is obtained with the coefficients of the quadratic nonlinear, cubic nonlinear, dispersion, and dissipation terms P 1 , P 2 , P 3 , and P 4 , respectively, It is shown that the coefficients P 1 , P 2 , P 3 , and P 4 , are functions in the physical plasma parameters namely, negative ion concentration α, the mass ratio Q, and the electron nonthermal parameter β. It is known that at the critical plasma compositions say β c or α c (critical value of negative ion concentration), the coefficient P 1 vanishes and in this case Eq. (118) will be reduced to the following mKdVB equation which is used to describe the damped wave dynamics at critical plasma compositions To convert EKdVB Eq. (118) to the damped H-D Eq. (4), the traveling wave transformation φ ξ, τ ð Þ ! φ X ð Þ with X ¼ ξ þ λτ ð Þshould be inserted into Eq. (118) and integrate once over η, and by applying the boundary conditions: φ, φ 0 , φ 00 ð Þ!0 as |X| ! ∞, the constant forced damped following constant forced damped Duffing-Helmholtz equation is obtained where λ represents the reference frame speed, φ 0 and φ 00 denote the first and second ordinary derivative of regarding X, ε ¼ P 4 = 2P 3 ð Þ, p ¼ λ=P 3 , q ¼ P 1 = 2P 3 ð Þ, r ¼ P 2 = 3P 3 ð Þ, and D ¼ C=P 3 . Note that the coefficient q may be positive or negative according to the values of plasma parameters and for studying oscillations using (120), solution (72) can be devoted for this purpose. In the absence of the ionic kinematic viscosity (P 4 ¼ 0 or ε ¼ 0), then Eq. (120) reduces to the constant forced undamping Duffing-Helmholtz equation and in this case the solution (63) can be applied for investigating the undamped oscillations in the present plasma model. Also, for q ¼ 0, Eq. (120) reduces to the constant forced damped Helmholtz equation. Moreover, the constant forced damped Duffing equation can be obtained for p ¼ 0.

Conclusion
The analytical and semi-analytical solutions for nonlinear oscillator integrable and non-integrable equations have been investiagted. First, the standard integrable Duffing equation has been analyized and its solutions have been obtained depending on the sign of its discriminant Δ. Accordingly, three cases Δ > 0, Δ < 0, and Δ ¼ 0 ð Þ have been discussed in details and the solutions of each case has be obtained. Second, the analytical and semi-analytical solutions of the integrable Duffing-Helmholtz equation and its non-integrable family including the damped Duffing-Helmholtz equation, forced undamped Duffing-Helmholtz equation, forced damped Duffing-Helmholtz equation, and the damped and trigonometric forced Duffing-Helmholtz equation have been obtained and discussed in details. Third, the solutions to the intgrable cubic-quintic Duffing equation and the non-intgrable damped cubic-quintic Duffing equation have been investigated. Moreover, some realistic applications reaslted to the RLC circuits and physics of plasmas have been introduced and discussed depending on the solutions of the mentioned evolution equations.