A non-autonomous Gardner equation and its integrability: Solitons, positons and breathers

Abstract

The work studies some integrable properties and soliton type solutions of a non-autonomous Gardner equation with damping and forcing terms. A bilinear form, a bilinear Bäcklund transformation and a Lax pair are derived for the considered Gardner equation explicitly. K-Soliton solution with proper existence condition, smooth positons, breathers and their interaction solutions are presented via the bilinear form. Moreover, the amplitude as well as velocity of the soliton solutions are derived, and a first-order breather solution and a second-order smooth positon are generated from the two-soliton solution. The interaction between a single-breather solution and the single-soliton solution and the interaction of a second-order smooth positon and the single-soliton solution are studied analytically, based on the three-soliton solution. Profiles of various types of the obtained solutions and their interactions are illustrated graphically.