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 $u_{tt}-u_{xx}+u=u^{1+n}-{\kappa }_{1+2n}{u}^{1+2n},-1/2<n,$endowed with both trivial and non-trivial stable equilibria, and demonstrate that similarly to the classical ${\kappa }_{1+2n}=0$ cases, solitons are linearly unstable, but the instability weakens as ${\kappa }_{1+2n}\uparrow$, and vanishes at ${\kappa }_{1+2n}^{crit}=\frac{1+n}{{(2+n)}^2}$, 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 ${\kappa }_{1+2n}^{crit}$ 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.