We consider solutions to the second-harmonic generation equations in two-and three-dimensional dispersive media in the form of solitons localized in space and time. As is known, collapse does not take place in these models, which is why the solitons may be stable. The general solution is obtained in an approximate analytical form by means of a variational approach, which also allows the stability of the solutions to be predicted. Then, we directly simulate the two-dimensional case, taking the initial configuration as suggested by the variational approximation. We thus demonstrate that spatiotemporal solitons indeed exist and are stable. Furthermore, they are not, in the general case, equivalent to the previously known cylindrical spatial solitons. Direct simulations generate solitons with some internal oscillations. However, these oscillations neither grow nor do they exhibit any significant radiative damping. Numerical solutions of the stationary version of the equations produce the same solitons in their unperturbed form, i.e., without internal oscillations. Strictly stable solitons exist only if the system has anomalous dispersion at both the fundamental harmonic and second harmonic ͑SH͒, including the case of zero dispersion at SH. Quasistationary solitons, decaying extremely slowly into radiation, are found in the presence of weak normal dispersion at the second-harmonic frequency. ͓S1063-651X͑97͒14009-0͔PACS number͑s͒: 42.65.Tg, 42.65.Ky

We determine the power dependence of dispersion-managed solitons on map strength and average dispersion, using a combination of numerical simulations and the variational approach. In particular, we investigate the behavior near zero dispersion and identify the region of existence of dispersion-managed solitons in the average normal-dispersion regime.

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A tutorial review is presented of the use of direct variational methods based on Rayleigh-Ritz optimization for finding approximate solutions to various nonlinear evolution equations. The practical application of the approach is demonstrated by some illustrative examples in connection with the nonlinear Schrödinger equation.