Flatons: Flat-top solitons in extended Gardner-like equations

We present and study an extended Gardner-like family of equations, G(m,n;k):u_t+(c₊u^m−c₋uⁿ)_x+(u^k)_xxx=0,c_±>0, n > m > 1, k ≥ 1, endowed with a non-convex convection which may be due to two opposing mechanisms that bound the range of velocities of both solitons and compactons beyond which they dissolve, and kink and/or antikink form. Close to solitons and compactons barrier, there is a narrow strip of velocities where the wave shape undergoes a structural change and rather than grow with velocity, their top flattens and they widen rapidly with minute changes in velocity. These waves, referred to as flatons, may be viewed as an approximate amalgam of a kink and anti kink placed at any distance from each other. Typical of solitons, once flatons form they are very robust with their domain of attraction being sensitive to the amplitude at which convection reverses its direction. A multi-dimensional extension of these equations unfolds a plethora of flatons which, unless m is even and n is odd, for every admissible velocity may span an entire sequence of multi-nodal radially symmetric flatons.