Self-trapped states in a saturable Klein-Gordon equation

We present numerical and theoretical results for self-trapped states in the lossless, saturably nonlinear Klein-Gordon equation $u_{\mathrm{tt}}$-$u_{\mathrm{xx}}$=-u/(1+$u^2$). A simple approximate analytic theory is developed which agrees well with self-trapped states found in simulations to emerge from certain types of localized, stationary, one-sided ‘‘displacements,’’ u(x,0)≥0, $u_t$(x,0)=0. The stability of theses states to strong perturbations is studied by pulse-collision simulations, using for the perturbation one of the two traveling-wave pulses generated in the fast dissociation of a highly unstable initial displacement. The self-trapped states are highly stable, exhibiting a shape change and centroid shift after collision, but little energy loss or change of period.