  Dispersive PDEs are important both in applications (wave phenomena e.g. in
hy- drodynamics, nonlinear optics, plasma physics, Bose-Einstein
condensates,...) and a mathematically very challenging class of partial
differential equations, especially in the time dependent case. An important
point with respect to applications is the stability of exact solutions like
solitons. Whereas the linear or spectral stability can be addressed
analytically in some situations, the proof of full nonlinear (in-)stability
remains mostly an open question. In this paper, we numerically investi- gate
the transverse (in-)stability of the solitonic solution to the one-dimensional
cubic NLS equation, the well known isolated soliton, under the time evolution
of several higher dimensional models, being admissible as a tranverse
perturbation of the 1d cubic NLS. One of the recent work in this context [42]
allowed to prove the instability of the soliton, under the flow of the
classical (elliptic) 2d cubic NLS equation, for both localized or periodic
perturbations. The characteristics of this instability stay however unknown. Is
there a blow-up, dispersion..? We first illustrate how this instability occurs
for the elliptic 2d cubic NLS equation and then show that the elliptic-elliptic
Davey Stewartson system (a (2+1)-dimensional generalization of the cubic NLS
equation) behaves as the former in this context. Then we investigate hyperbolic
variants of the above models, for which no theory in this context is available.
Namely we consider the hyperbolic 2d cubic NLS equation and the
Davey-Stewartson II equations. For localized perturbations, the isolated
soliton appears to be unstable for the former case, but seems to be orbitally
stable for the latter. For periodic perturbations the soliton is found to be
unstable for all transversally perturbed models considered.
