Novel solitary patterns in a class of Klein–Gordon equations

We study the emergence, stability and evolution of solitons and compactons in a class of Klein–Gordon equations u_tt−u_xx+u=u¹⁺ⁿ−κ_1+2nu¹⁺²ⁿ,−1/2<n,endowed with both trivial and non-trivial stable equilibria, and demonstrate that similarly to the classical κ_1+2n=0 cases, solitons are linearly unstable, but the instability weakens as κ_1+2n↑, and vanishes at [Formula presented], where solitons disappear and kink forms. As the growing unstable soliton approaches the non-trivial equilibrium, it morphs into a ’mesaton’, a robust box shaped sharp pulse with a flat-top plateau, which expands at a sonic speed. In the κ_1+2n^crit vicinity, where instability is suppressed, whereas the internal modes have hardly changed, solitons persist for a very long time but then, rather than turn into mesaton, convert into a breather-like formation. Linear damping tempers the conversion and slows mesaton's propagation. When −1/2<n<0, compactons emerge and being unstable morph either into a mesaton or into a breather-like formation.