Hermite-Gaussian stationary solutions in strongly nonlocal nonlinear optical media

Approximate analytical stationary solutions (SSs) of a cluster of Hermite-Gaussian (HG) shape is obtained in strongly nonlocal nonlinear media by the variational approach. The evolution of the HG SSs shows that when the order n⩽3, they propagate stably and form solitons; otherwise, when n⩾4, they always propagate unstably and evolve into self-trapped speckle-like beams. However, all these SSs maintain nearly invariant statistic beam-width during their propagation. Furthermore, when the input power deviates from the so-called critical power, the unstable HG beam will adjust its beam-width to form a new self-trapped beam, unlike the soliton which will turn to be a breather. But the average beam-widths are independent of the stability of the propagation of the HG SSs.