A Mathematical Modelling with Zakharov System for Langmuir Wave in Unmagnetized Plasma

In this article we present an discussion and overview of mathematical result of the self- focusing of a Langmuir wave which governs Zakharov system and has studied the self- focusing of a Langmuir wave following by gaussian distribution. Langmuir wave propagates through uncharged plasma which governed by Zakharov systems. The phenomenon plays a vital role in the Dynamics. We present the article mathematical model with effect of Landou damping. Relativistic mass oscillation and ponderomotive force on electrons of the ionized plasma encouraged the Langmuir wave which resists the self-focusing effect when damping is ignored. The Beam radius gets narrow. when it further propagates considering the paraxial rays approximation, the self-focusing length Rn. It shows that characteristics of varying bandwidth distance of propagation in relativistic plasma.