New example of integrable nonlinear coupled equations with exact asymptotic singular solution in the context of laser-plasma interaction

A new set of nonlinear coupled equations is derived in the context of small amplitude limit of the general wave equations in a fluid type warm electrons/cold ions plasma irradiated by a continuous laser beam. This limit is proved to be integrable by means of the spectral transform theory with singular dispersion relation. An exact asymptotic solution is obtained. This model accounts for a nonlinear mode coupling of the electrostatic waves with the ion sound wave, and is shown to be unstable and does not propagate any stable small amplitude solution. This instability is understood as a continuous secular transfer of energy from the electrostatic wave to the ion sound wave through the ponderomotive force. The exact mechanism of this transfer is exposed. The dynamics of this energy transfer results in a singular asymptotic behavior of the ion sound wave which explains the low penetration of the incident laser beam.