Integrable generalizations of the sine-Gordon, short pulse, and reduced Maxwell–Bloch equations

Abstract

The multi-parameter generalizations of the sine-Gordon (SG), short pulse (SP), and reduced Maxwell–Bloch (RMB) equations are considered. These generalizations are integrable by the inverse scattering transformation method and connected with the modified SG equation, its limiting case, and the modified RMB equations by the changes of the dependent and independent variables expressed in terms of the conserved densities and fluxes of the latter. The particular cases of the generalizations considered include the Rabelo–Fokas (RF) equation, the modified SP equation, and some other equations. The properties of the soliton solutions of some of the generalizations are studied. It is revealed that the interaction of the well-defined solitons of the RF equation and the corresponding version of the RMB equations leads to an appearance of the intervals in some regions of the soliton parameters, where the solution becomes multi-valued, and to the blow-up of the solution. In addition, the compacton-like soliton solutions can exist in some cases.