In this paper, we present numerical methods
for the determination of solitons, that consist in spatially localized
stationary states of nonlinear scalar equations or coupled systems
arising in nonlinear optics.
We first use the well-known shooting method in order to find
excited states (characterized by the number k of nodes) for the
classical nonlinear Schrödinger equation. Asymptotics can then
be derived in the limits of either large k are large nonlinear
exponents σ.
In a second part, we compute solitons for a nonlinear
system governing the propagation of two coupled
waves in a quadratic media in any
spatial dimension, starting from one-dimensional states obtained
with a shooting method and considering the dimension as a
continuation parameter. Finally, we investigate the case of three wave
mixing, for which the shooting method is not relevant.