A four-component quantum plasma consisting of electrons, positrons, negative heavy ions and positive light ions is proposed in this work. The dynamics of nonlinear and supernonlinear ion-acoustic waves are studied in the framework of the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations which are derived employing the reductive perturbation technique. Using Galilean transformation these two evolution equations are transformed into planar dynamical systems. All possible phase portraits and corresponding small-amplitude Sagdeev’s pseudopotential of these dynamical systems are presented graphically. The unique topology of phase portrait and a maxima in between two minimas in pseudopotential curves clearly establish quantum ion-acoustic superperiodic waves. Solitary, periodic and superperiodic wave solutions corresponding respectively to homoclinic, periodic and superperiodic orbits in phase portraits are obtained numerically and the influence of different parameters on these waves is observed. Further, various kinds of analytical wave solutions for the two evolution equations are discussed.
