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A Korteweg-de Vries (KdV) equation including the effect of Landau damping isderived to study the propagation of weakly nonlinear and weakly dispersive ionacoustic waves in a collisionless unmagnetized plasma consisting of warmadiabatic ions and two different species of electrons at differenttemperatures. The hotter energetic electron species follows the nonthermalvelocity distribution of Cairns et al. [Geophys. Res. Lett. 22, 2709 (1995)]whereas the cooler electron species obeys the Boltzmann distribution. It isfound that the coefficient of the nonlinear term of this KdV like evolutionequation vanishes along different family of curves in different parameterplanes. In this context, a modified KdV (MKdV) equation including the effect ofLandau damping effectively describes the nonlinear behaviour of ion acousticwaves. It has also been observed that the coefficients of the nonlinear termsof the KdV and MKdV like evolution equations including the effect of Landaudamping, are simultaneously equal to zero along a family of curves in theparameter plane. In this situation, we have derived a further modified KdV(FMKdV) equation including the effect of Landau damping to describe thenonlinear behaviour of ion acoustic waves. In fact, different modified KdV likeevolution equations including the effect of Landau damping have been derived todescribe the nonlinear behaviour of ion acoustic waves in different region ofparameter space. The method of Ott & Sudan [Phys. Fluids 12, 2388 (1969)] hasbeen applied to obtain the solitary wave solution of the evolution equationhaving the nonlinear term $(\phi^{(1)})^{r}\frac{\partial \phi^{(1)}}{\partial\xi}$, where $\phi^{(1)}$ is the first order perturbed electrostatic potentialand $r =1,2,3$. We have found that the amplitude of the solitary wave solutiondecreases with time for all $r =1,2,3$.

Effect of Landau damping on ion acoustic solitary waves in a  multi-species collisionless unmagnetized plasma consisting of nonthermal and  isothermal electrons