A variational approach to the stability of an embedded NLS soliton at the edge of the continuum

A Lagrangian-based method is used to construct embedded soliton solutions of a nonlinear Schrِdinger equation with higher order dispersive terms. It is shown that the embedded soliton consists of two widely separated peaks with linear, dispersive radiation of exponentially small amplitude between the peaks, this radiation acting to hold the pulses together. The Lagrangian-based technique determines the details of this embedded soliton, including the amplitude of the pulses, the separation of the pulses and the amplitude of the radiation. In addition, it is found that the embedded soliton exists for only discrete eigenvalues, related to the separation of the pulses. Moreover, since the embedded soliton solutions are steady, they are at the lower edge of the continuous spectrum. Finally, by allowing the parameters of the embedded soliton to be time dependent, it is found that the embedded soliton develops an oscillatory type of one-sided stability. This one-sided stability is different from the usual monotone one-sided stability found for solitons embedded inside the continuous spectrum.