Higher order stability of dust ion acoustic solitary wave solution described by the KP equation in a collisionless unmagnetized nonthermal plasma in presence of isothermal positrons

Sardar et al. [Phys. Plasmas 23, 073703 (2016)] have studied the stability of small amplitude dust ion acoustic solitary waves in a collisionless unmagnetized electron - positron - ion - dust plasma. They have derived a Kadomtsev Petviashvili (KP) equation to investigate the lowest - order stability of the solitary wave solution of the Korteweg-de Vries (KdV) equation for long-wavelength plane-wave transverse perturbation when the weak dependence of the spatial coordinates perpendicular to the direction of propagation of the wave is taken into account. In the present paper, we have extended the lowest - order stability analysis of KdV solitons given in the paper of Sardar et al. [Phys. Plasmas 23, 073703 (2016)] to higher order with the help of multiple-scale perturbation expansion method of Allen and Rowlands [J. Plasma Phys. 50, 413 (1993); 53, 63 (1995)]. It is found that solitary wave solution of the KdV equation is stable at the order k^2, where k is the wave number for long-wavelength plane-wave perturbation.