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Effects of superthermal electron on the features of nonlinear acoustic waves in unmagnetized collisionless ion pair plasma with superthermal electrons have been examined. The system equations are reduced in the form of the nonlinear Schrodinger equation. The rogue wave characteristics dependences on the ionic density ratio (ν = n–0/n+0), ionic mass ratio (Q = m+/m−), and superthermality index (κ) are investigated. It is worth mentioning that the results present in this work could be applicable in the Earth's ionosphere plasmas.

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.