SPATIOTEMPORAL SOLITONS IN MULTIDIMENSIONAL OPTICAL MEDIA WITH A QUADRATIC NONLINEARITY

We consider solutions to the second-harmonic generation equations in t wo-and three-dimensional dispersive media in the form of solitons loca lized 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 sol ution is obtained in an approximate analytical form by means of a vari ational approach, which also allows the stability of the solutions to be predicted. Then, we directly simulate the two-dimensional case, tak ing the initial configuration as suggested by the variational approxim ation. We thus demonstrate that spatiotemporal solitons indeed exist a nd are stable. Furthermore, they are not, in the general case, equival ent to the previously known cylindrical spatial solitons. Direct simul ations generate solitons with some internal oscillations. However, the se oscillations neither grow nor do they exhibit any significant radia tive damping. Numerical solutions of the stationary version of the equ ations produce the same solitons in their unperturbed form, i.e., with out internal oscillations. Strictly stable solitons exist only if the system has anomalous dispersion at both the fundamental harmonic and s econd harmonic (SH), including the case of zero dispersion at SH. Quas istationary solitons, decaying extremely slowly into radiation, are fo und in the presence of weak normal dispersion at the second-harmonic f requency.