A Zakharov–Kuznetsov (ZK) equation, a modified ZK equation and a coupled ZK equation for small but finite amplitude dust acoustic waves in a magnetized two-ion-temperature dusty plasma with dust size distribution have been obtained in this paper. It seems that the soliton velocity for a dusty plasma with power law dust size distribution is larger than that of monosized dusty plasmas. The solitary waves with opposite polarity particles are investigated as well. It is found that rarefactive and compressive solitary waves exist in this system which depend on different conditions. On the other hand, the solitary wave is unstable at certain regions when the wave numbers of the transverse perturbations satisfies the conditions of 0<K<Kc and 0<L<Lc.

It is well known that the Korteweg–de Vires (KdV) equation can describe small but finite amplitude dust acoustic wavesin adustyplasmas. Inthispaper, weuse thereductive perturbationmethodandderive aKadomtsev–Petviashvili (KP) equation, a modified KP (MKP) equation and a coupled KP equation for unmagnetized, collisionless, cold, and two-ion-temperature dusty plasmas with N dierent species of dust grains. We find that if a solitary wave exist in this system, the smaller grains have larger velocities and propagate longer distances than that of larger particles. The com-parisons are given between the dusty plasma composed by di?erent dust particles and the mono-sized dusty plasma.

For two-ion-temperature dusty plasma with many dierent dust grains, we derived a Kadomtsev-Petviashvili (KP) equation for unmagnetized, collisionless, hot dusty plasma. We find that in a two-ion-temperature warm dusty plasmas with power law dust size distribution, the velocity of soliton is smaller than that for mono-sized dust plasma which has average dust size. Meanwhile as the temperature of dust plasmas increases, the amplitude of soliton decreases slower than that of mono-sized dust plasma with average dust size, while the width of soliton decreases quicker than that of mono-sized dust plasma with average dust size.