Soliton–mean flow interaction problem for a non-convex system with quadratic-cubic nonlinearity in the stratified fluid

The interaction problem between solitons and mean flows including the dispersive shock wave, rarefaction wave, kink and contact dispersive shock wave, is discussed in the framework of the Gardner equation, a generic non-convex mathematical model containing quadratic-cubic nonlinear terms, describing the dynamics of internal waves for an arbitrary vertical stratification of the density field. Based on the augmented solitonic modulation system constructed in a Riemann invariant form, two main interaction regimes are identified: soliton tunneling and soliton trapping. Furthermore, kinks and flat-top solitons caused by the non-convexity, play dual roles as both mean flows and localised waves, imparting polarity reversal to solitons and mean flows. Specially, flat-top solitons formed by the quadratic-cubic nonlinear competition, can be considered as a combination of kinks and antikinks, and induce polarity reversal to solitons and mean flows twice, which is an evidence of elastic collisions between flat-top solitons and bell solitons. In addition, the shape of flat-top solitons is changed after the interaction with mean flows. A full classification of the interaction problem is given for both signs of the cubic nonlinear term in the Gardner equation, and numerical simulations agree with modulation theory predictions.