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A new set of nonlinear coupled equations is derived in the context of smallamplitude limit of the general wave equations in a fluid type warmelectrons/cold ions plasma irradiated by a continuous laser beam. This limit isproved to be integrable by means of the spectral transform theory with singulardispersion relation. An exact asymptotic solution is obtained. This modelaccounts for a nonlinear mode coupling of the electrostatic waves with the ionsound wave, and is shown to be unstable and does not propagate any stable smallamplitude solution. This instability is understood as a continuous seculartransfer of energy from the electrostatic wave to the ion sound wave throughthe ponderomotive force. The exact mechanism of this transfer is exposed. Thedynamics of this energy transfer results in a singular asymptotic behavior ofthe ion sound wave which explains the low penetration of the incident laserbeam.

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