Cylindrical solitary waves and their interaction in Benard-Marangoni layers

We consider a motionless Newtonian and incompressible shallow liquid layer bounded below by a bottom plate where temperature is held fixed and above with a free deformable surface whose tension is linearly temperature dependent and on which the heat flux is fixed. When the dimensionless surface tension gradient, measured by a Marangoni number, slightly exceeds its critical value, radially symmetric, long-wavelength excitations obey a dissipative cylindrical Korteweg-de Vries (DCKdV) equation. A dissipative cylindrical Kadomtsev-Petviashvili (DCKP) equation is also derived for nearly radially symmetric disturbances. Exact solitary wave solutions of the DCKP equation are found and the solitary wave solutions of the DCKdV equation are also discussed. Finally the head-on collision between two concentric cylindrical solitary waves is considered and its solitonic character is displayed.