Investigation of cubic non‐linearity‐driven electrostatic structures in the presence of double spectral index distribution function

In this article, modified Korteweg‐de Vries equation (which involves cubic non‐linearity) was derived to study non‐linear ion acoustic waves in a plasma in which electrons follow the double spectral index distribution function. The double spectral index distribution successfully apes the distribution functions that have been frequently observed in space plasmas. The spectral index r moulds the distribution function at low energy and by increasing its value, flatness of the distribution enhances. The spectral index r can also have negative values due to which distribution becomes spiky at low energies. The index q, on the other hand, controls the shape of the tail of distribution function. It has been shown that propagation of the solitary structures gets significantly altered by the choice of the double spectral indices, namely, r and q. A comparison was also made, using the parameters of the auroral zone, between the quadratic and cubic non‐linearities‐driven non‐linear structures and it was shown that the solitary structures form on a much shorter scale for cubic non‐linearity compared to their quadratic counterpart. In this article, we studied non‐linear ion acoustic waves in the presence of cubic non‐linearity and the electrons that follow double spectral (r, q) distribution. The present investigation may be helpful to develop an in‐depth understanding of the quadratic and cubic non‐linearity‐driven non‐linear structures with reference to the non‐thermal behaviour of electrons in regions of high and low phase space densities.