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