Stability of solitary waves in nonlinear Klein–Gordon equations

The stability of topological solitary waves and pulses in one-dimensional nonlinear Klein-Gordon systems is revisited. linearized equation describing small deviations around the static solution leads to a Sturm-Liouville problem, which solved systematic way for $-l\,(l+1)\,\sech^2(x)$-potential, showing orthogonality completeness relations fulfilled by set its solutions all values $l\in\mathbb{N}$. This approach allows determine linear kinks certain equations. Two families novel potentials are introduced. exact (kinks pulses) these exactly calculated, even when potential not explicitly known. models found be stable, whereas unstable. achieved introducing spatial inhomogeneities.