The nonlinear equation is obtained describing the dynamics of nonlinear wave structures in the dusty plasma above the illuminated surface of the Moon in the case of low frequencies and pancake-like shape of wave packet in the direction along the external magnetic field. This equation is the modified Zakharov-Kuznetsov equation. The analytical formula for the one-dimensional soliton solution is derived. The analysis of the stability of one-dimensional soliton solution was performed.

Modified Kadomtsev-Petviashvili equation describing nonlinear dynamics of nearly one-dimensional wave structures in dusty plasma above illuminated part of the Moon in the situation in which localization along magnetic-field vector is much stronger than along other directions is derived. The equation differs from the ordinary Kadomtsev-Petviashvili equation by the nonlinear term being non-analytical. Modified Kadomtsev-Petviashvili differs from generalizations of the Kadomtsev-Petviashvili equation in which nonlinearity retains the same form as in the ordinary Kadomtsev-Petviashvili equation but higher-order corrections for dispersion are taken into account. An analytical expression governing one-dimensional soliton solution to the modified Kadomtsev-Petviashvili equation is obtained. The solution differs from the well-known one-dimensional soliton solutions to the Korteweg-De Vries and ordinary Kadomtsev-Petviashvili equations. Stability analysis of the one-dimensional soliton solution showed that it is stable. Possible applications of the discussed solitons from the point of view of description of the so-called transient lunar phenomena representing short-lived light, changes in color or appearance on the surface of the Moon are discussed.