Breathing pulses and beams innon-linear media

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Abstract

Oscillating solutions are known to exist for the (1+1)-dimensional non-linear Schrödinger equations (NLSE) with Kerr non-linearity and for (3+1)-dimensional and (2+1)-dimensional NLSE with saturable non-linearity. In them, there is an interplay between the amplitude and the widths of the solutions. Using the variational method, we show that non-linear Schrödinger equations with Kerr non-linearity support breather-like solutions in the form of oscillating spatiotemporal pulses and beams but unlike the cases mentioned above, the oscillating behaviour takes place between the widths, with the amplitude kept almost constant. For the space–time NLSE these solutions exist only in a medium with anomalous dispersion.

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