**9.2 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 [37–41]. 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*ð Þ <sup>0</sup> <sup>2</sup> <sup>þ</sup> *<sup>n</sup>*ð Þ <sup>0</sup> *<sup>e</sup>* <sup>¼</sup> *<sup>n</sup>*ð Þ <sup>0</sup> <sup>1</sup> where *<sup>n</sup>*ð Þ <sup>0</sup> *<sup>s</sup>*,*<sup>e</sup>* 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 *x*�directional which means that the fluid equations of the plasma particles become perturbed only in *x*�directional. If the effect of the ionic kinematic viscosities *η<sup>s</sup>* for both positive *η*<sup>1</sup> ð Þ and negative *η*<sup>2</sup> 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: *<sup>∂</sup>xns* <sup>þ</sup> *<sup>∂</sup>t*ð Þ¼ *nsus* 0, *<sup>∂</sup>tus* <sup>þ</sup> *us∂xus* <sup>þ</sup> *<sup>δ</sup>=Qs* ð Þ*∂x<sup>ϕ</sup>* � *<sup>η</sup>s∂*<sup>2</sup> *xus* <sup>¼</sup> 0, and *<sup>∂</sup>*<sup>2</sup> *<sup>x</sup><sup>ϕ</sup>* � *<sup>n</sup>*ð Þ <sup>0</sup> *<sup>d</sup>* � *n*<sup>2</sup> � *ne* þ *n*<sup>1</sup> ¼ 0, where *ne* ¼ *<sup>μ</sup>* <sup>1</sup> � *βϕ* <sup>þ</sup> *βϕ*<sup>2</sup> � � exp *<sup>ϕ</sup>*. Here, *ns* donates the normalized number density of positive and negative ions, and *us* represents the normalized fluid velocity of positive and negative ions, and *ϕ* is the normalized electrostatic wave potential. The mass ratio is defined as: *Qs* ¼ *m*1*=ms* (note that *Q*<sup>1</sup> ¼ *m*1*=m*<sup>1</sup> ¼ 1 and *Q*<sup>2</sup> � *Q* ¼ *m*1*=m*2), where *ms* is the ionic mass, *δ* ¼ 1ð Þ �1 for positive (negative) ion, and *β* illustrates nonthermality parameter. The quasi-neutrality condition in the normalized form reads: *<sup>μ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>α</sup>*, where *<sup>α</sup>* <sup>¼</sup> *<sup>n</sup>*ð Þ <sup>0</sup> <sup>2</sup> *<sup>=</sup>n*ð Þ <sup>0</sup> <sup>1</sup> gives the the negative ion concentration and *<sup>μ</sup>* <sup>¼</sup> *<sup>n</sup>*ð Þ <sup>0</sup> *<sup>e</sup> =n*ð Þ <sup>0</sup> <sup>1</sup> is the electron concentration.

Now, the RPM is introduced to reduce the fluid plasma equations to the evolution equation. According to the RPM, the independent quantities ð Þ *x*, *t* are stretched as: *<sup>ξ</sup>* <sup>¼</sup> *<sup>ε</sup> <sup>x</sup>* � *Vpht* � �, *<sup>τ</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>3</sup>*t*, and *<sup>η</sup><sup>s</sup>* <sup>¼</sup> *εη*ð Þ <sup>0</sup> *<sup>s</sup>* where *Vph* 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 <sup>Π</sup>ð Þ� *<sup>x</sup>*, *<sup>t</sup>* ð Þ *<sup>n</sup>*1, *<sup>n</sup>*2, *<sup>u</sup>*1, *<sup>u</sup>*2, *<sup>ϕ</sup> <sup>T</sup>* are expanded as: <sup>Π</sup> <sup>¼</sup> <sup>Π</sup>ð Þ <sup>0</sup> <sup>þ</sup> <sup>P</sup><sup>∞</sup> *<sup>j</sup>*¼<sup>1</sup>*<sup>ε</sup> <sup>j</sup>* <sup>Π</sup>ð Þ *<sup>ξ</sup>*, *<sup>τ</sup>* ð Þ*<sup>j</sup>* , where <sup>Π</sup>ð Þ <sup>0</sup> <sup>¼</sup> ð Þ 1, *<sup>α</sup>*, 0, 0, 0 *<sup>T</sup>* and *<sup>T</sup>* 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

$$
\partial\_{\tau} \rho + \left(P\_1 \rho + P\_2 \rho^2\right) \partial\_{\xi} \rho + P\_3 \partial\_{\xi}^3 \rho + P\_4 \partial\_{\xi}^2 \rho = \mathbf{0},\tag{118}
$$

with the coefficients of the quadratic nonlinear, cubic nonlinear, dispersion, and dissipation terms *P*1, *P*2, *P*3, and *P*4, respectively,

$$P\_1 = \frac{3}{2} P\_3 \left[ \frac{1}{V\_{ph}^4} - \frac{3a}{V\_{ph}^4 Q^2} - \frac{2h\_2}{3} \right], \\ P\_2 = \frac{3}{4} P\_3 \left[ \frac{5}{V\_{ph}^6} - \frac{5a}{V\_{ph}^6 Q^3} - 2h\_3 \right],$$

$$P\_3 = \frac{V\_{ph}^3 Q}{2(Q+a)}, \\ P\_4 = -P\_3 \left[ \frac{\eta\_1^{(0)}}{V\_{ph}^4} + \frac{a \eta\_2^{(0)}}{Q V\_{ph}^3} \right] \text{ kg } V\_{ph} = \sqrt{\frac{(Q+a)}{Qh\_1}},$$

where *<sup>φ</sup>* � *<sup>ϕ</sup>*ð Þ<sup>1</sup> , *<sup>h</sup>*<sup>1</sup> <sup>¼</sup> *<sup>μ</sup>*ð Þ <sup>1</sup> � *<sup>β</sup>* , *<sup>h</sup>*<sup>2</sup> <sup>¼</sup> *<sup>μ</sup>=*2, and *<sup>h</sup>*<sup>3</sup> <sup>¼</sup> *<sup>μ</sup>*ð Þ <sup>1</sup> <sup>þ</sup> <sup>3</sup>*<sup>β</sup> <sup>=</sup>*6*:*

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<sup>c</sup>* or *α<sup>c</sup>* (critical value of negative ion concentration), the coefficient *P*<sup>1</sup> 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

$$
\partial\_{\tau} \varrho + P\_2 \varrho^2 \partial\_{\xi} \varrho + P\_3 \partial\_{\xi}^3 \varrho + P\_4 \partial\_{\xi}^2 \varrho = 0,\tag{119}
$$

To convert EKdVB Eq. (118) to the damped H-D Eq. (4), the traveling wave transformation *φ ξ*ð Þ! , *τ φ*ð Þ *X* with *X* ¼ ð Þ *ξ* þ *λτ* should be inserted into Eq. (118) and integrate once over *η*, and by applying the boundary conditions: *φ*, *φ*<sup>0</sup> , *φ*<sup>00</sup> ð Þ! 0 as ∣*X*∣ ! ∞, the constant forced damped following constant forced damped Duffing-Helmholtz equation is obtained

$$
\rho \rho'' + 2\epsilon \rho' + p\rho + q\rho^2 + r\rho^3 + D = 0,\tag{120}
$$

where *λ* represents the reference frame speed, *φ*<sup>0</sup> and *φ*<sup>00</sup> denote the first and second ordinary derivative of regarding *X*, *ε* ¼ *P*4*=*ð Þ 2*P*<sup>3</sup> , *p* ¼ *λ=P*3, *q* ¼ *P*1*=*ð Þ 2*P*<sup>3</sup> , *r* ¼ *P*2*=*ð Þ 3*P*<sup>3</sup> , 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*<sup>4</sup> 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.

## **10. 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 ð Þ Δ > 0, Δ <0, andΔ ¼ 0 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.

*Analytical Solutions of Some Strong Nonlinear Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97677*
