Stability analysis solutions for nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a magnetized electron–positron plasma

•The hydrodynamic model is applied to three-dimensional magnetized electron–positron plasma waves.•New exact solutions for the modified Korteweg–de Vries–Zakharov–Kuznetsov equation, wave solutions, modified direct algebraic method.•We will present three traveling-wave solutions to modified Korteweg–de Vries–Zakharov–Kuznetsov equation.•We discussed the stability analysis for these solutions.

The nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov ​(mKdV–ZK) equation governs the behavior of weakly nonlinear ion-acoustic waves in magnetized electron–positron plasma which consists of equal hot and cool components of each species. By using the reductive perturbation procedure leads to a mKdV–ZK equation governing the oblique propagation of nonlinear electrostatic modes. The stability of solitary traveling wave solutions of the mKdV–ZK equation to three-dimensional long-wavelength perturbations is investigated. We found the electrostatic field potential and electric field in form traveling wave solutions for three-dimensional mKdV–ZK equation. The solutions for the mKdV–ZK equation are obtained precisely and efficiency of the method can be demonstrated.