Higher-order contributions in reductive perturbation theory are studied for small- but finite-amplitude ion-acoustic solitary waves in a warm plasma with negative-ion, positron and electron constituents traversed by a warm electron beam (with different temperatures and pressures). The basic set of fluid equations are reduced to a Korteweg–de Vries (KdV) equation for the first-order perturbed potential and a linear inhomogeneous KdV-type equation for the second-order perturbed potential. At the critical negative-ion density, the coefficient of the nonlinear term in the KdV equation vanishes. A new set of stretched coordinates is then used to derive a modified KdV equation and a linear inhomogeneous modified KdV-type equation at the critical density of negative ions for the second-order perturbed potential. Stationary solutions of the coupled equations, for both cases, are obtained using a renormalization method.
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