The dynamical behaviour of weakly nonlinear, low-frequency sound waves are investigated in a plasma composed of only positive and negative ions incorporating the effects of a weak external uniform magnetic field. In the plasma model the mass (temperature) of the positive ions is smaller (larger) than that of the negative ions. The dynamics of the nonlinear wave is shown to be governed by a novel nonlinear equation. The stationary plane wave (analytical and numerical) nonlinear analysis on the basis of experimental parameters reveals that the nonlinear wave does have quasi-periodic and chaotic solutions. The Poincarè return map analysis confirms these observed complex structures. Copyright © The Author(s), 2020. Published by Cambridge University Press.