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

• If the forces governing the dynamics of a nonlinear dispersive system set an upper bound on the speeds at which traveling waves are admissible, as happens in the Gardner equation or the presented Gardner-like systems where the convection change its direction at a critical amplitude, close to the speed limit solitons top turns flat-like and solitary or compact flatons emerge. • Unless convection retains its direction within flatons range, their domain of attraction is quite narrow. However, once they form they are very robust. • Spherically symmetric flatons are shown to be far more prevalent than their 1D siblings. Typically, every speed may support an entire sequence of multi-modal flatons. We present and study an extended Gardner-like family of equations, G (m , n ; k) : u t + (c + u m − c − u n) 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. [ABSTRACT FROM AUTHOR]