Application Of The Multiple-scales Method In Optoacoustic Coupled Mode Equations

The interaction of acoustic and optical signals are under intensive studies these years, One of the typical examples is the coupling of optical wave and acoustic wave in the the Bragg gratings. As we know, Bragg gratings have periodic variation of the refractive index, a very large group velocity dispersion and nonlinear effects. optical waves propagating in the Bragg gratings may form optical solitons. Moreover, light may drive sound through compressing the medium due to variations of the intensity of light, conversely, acoustic waves react into the optical wave through the dependence of the refractive index on the material density. So, considering the electrostriction, optical gap solitons may couple to acoustic waves, generate the optoacoustic solitons.Optoacoustic solitons were first found by Richard S. Tasgal in the Bragg gratings. And the optoacoustic coupled mode equations were given. But, Richard S. Tasgal only gave out the single-soliton solutions and tested solitons stability by numerical simulation. For the sake of completely investigating the properties of optoacoustic solitons, meanwhile because the coupled mode equations are non-integrable, and the multiple-scales method is effective to find the approximate solutions of non-integrable equations. So, In this paper, we use the multiple-scales method to discuss the optoacoustic coupled mode equations in the Bragg gratings.In this paper, using the multiple-scales method, we find that the optoacoustic coupled mode equations can be reduced to some simple models, such as the nonlinear Schrodinger equation. Then, from the solutions of the simple models, we obtain the approximate solutions of the optoacoustic coupled mode equations, such as the single-soliton solutions and two-soliton solutions. And that, we discuss the properties of the optoacoustic coupled mode equations in two aspects, such as neglecting phonon viscosity and considering phonon viscosity.