Abstract

Too short optical solitons are destroyed most often due to influence of normal dispersion by virtue of Cherenkov radiation. However, sub-cycle solitons were never observed even for the most optimal dispersion profiles. In the absence of Cherenkov radiation another mechanism comes into play, namely, an unphysical cusp appears at the top of the pulse envelope and the soliton is destroyed. This happens at a critical duration of approximately one and a half optical cycles. The cusp generation can be explained by representing solitary solutions ϕ of corresponding short-pulse-equations as homoclinic trajectories of a reduced dynamical system $\frac{1}{2}{\left({\varphi }^{\prime }\right)}^2+U\left(\varphi \right)=\text{const}$. The effective potential U(ϕ) depends on the choice of the short-pulse-equation and on additional parameters such as pulse duration. As the latter decreases the potential evolves (Fig. 1a) and demonstrates singular behavior. Namely, a harmless singularity (Fig. 1a, green line) is finally replaced by a critical one (brown line) that destroys the soliton. This phenomenon was found both for several available short-pulse-equations [1] and for the generalized NLSE [2]. The shortest soliton (Fig. 1a, red effective potential) yields a cusp shape and exhibits a special power spectrum: standard exponential decay is replaced by an algebraic one [3]. We remark here, that such peaked solitons have originally been found outside optics, for shallow water waves [4], which indicates universal behavior of this mechanism.

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