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The homotopy analysis method is used to obtain analytical solutions of the Rayleigh equation for the radial oscillations of a multielectron bubble in liquid helium. The small order approximations for amplitude and frequency fit well with those computed numerically. The results confirm that the homotopy analysis method is a powerful and manageable tool for finding analytical solutions of strongly nonlinear dynamical systems.

Nonlinear equations are widely used for modeling complex phenomena in various fields of sciences and engineering. Nonlinear problems are in general difficult to solve analytically. In recent years, the tool for finding analytical solutions of nonlinear equations known as the homotopy analysis method (HAM) has been developed by Liao [

HAM method is applied in this paper for finding a periodic solution of the Rayleigh equation, which describes the radial free oscillations of a multielectron bubble (MEB) in liquid Helium [

Another aim of this work is to obtain analytical solutions of the Rayleigh equation (^{6} m/s^{2}) resulting in pulses of electromagnetic radiation or sonoluminescence.

The text sequence used in [

Applying the transformation

Differentiating the zero-order deformation equation with respect to

A nondimensional radius

In terms of the dimensionless variables (

From (

At this point, it is worth comparing (

According to the property (

From (

The Rayleigh equation (

An additional simplification is found when (

From now on, the system represented by (

The oscillation of the conservative system (

From (

Two sets of results are presented. The first one is for free oscillations (Figures

The

Comparison between 2nd-order HAM approximations (solid lines) and Runge-Kutta numerical solutions (open symbols) for:

The relative error between HAM approximations and numerical solutions for the radius

Comparison between HAM approximations (hollow symbols) and Runge-Kutta numerical solutions (solid lines) for

Comparison of the 1st, 2nd, and 5th-order HAM approximations of

The relative error for the frequency

(a) Comparison between 5th-order HAM solution (hollow symbols) and Runge-Kutta numerical solution (solid lines) of

Comparison between 4th-order HAM solutions (hollow symbols) and Runge-Kutta numerical solutions (solid lines) for: ^{4} electrons with an equilibrium radius

Relative error between 4th-order HAM solution and the corresponding numerical integration for displacement

The curve of

Relative error between 4th-order HAM approximation and numerical solutions for the frequency

The following characteristic scales were chosen to study the effect of varying both the initial radius of the MEB (by changing

The 2nd-order approximation for a proper value of

Solutions in Figure ^{4} electrons, as that numerically simulated by Salomaa and Williams [

The maximum relative error, indicated with a filled triangle in Figure

The effect of incrementing the initial radius of the MEB is depicted in Figure ^{4} electrons is considered but now the initial dimensionless radius is fixed to

Figure

Figure ^{8} electrons attains at the first main collapse a minimum dimensionless radius ^{8} electrons.

In Figure ^{6} m/s^{2}) and a minimum radius more and more close to zero. This strong collapse conditions might involve sonoluminescence phenomenon as suggested by Tempere et al. [

In order to study the effect of varying the driving pressure

Figure

In this study the HAM has been used to obtain analytical solutions of the Rayleigh equation for the radial oscillations of a MEB in liquid helium. The small-order HAM approximations for freely oscillating bubbles agree very well with numerical solutions even for bubbles with initial radial amplitudes as high as 1.6 times the equilibrium Coulomb radius. The analytical solutions for radius, velocity and acceleration of the freely oscillating bubble wall are accurate enough to accomplish surface stability studies (both parametric and Rayleigh-Taylor instabilities could be computed) with the possibility of both saving calculations and giving a bigger understanding of bubble shape instabilities when compared to solutions from a numerical scheme.

In the case of forced oscillations, the fourth-order HAM solutions for displacement and velocity of the bubble wall agree well with those computed numerically. Nevertheless, when the magnitude of pressure step is large enough (

The authors give thanks to Instituto de Ingeniería-UNAM for providing computing resources. This work was supported in part by DGAPA-PAPIIT-UNAM under Grant IN107509 and by II-FI UNAM under Grant 1135. The authors also wish to thank Julian Espinosa García for his useful comments and suggestions to improve the quality of the paper.