  The phase space dynamics generated by different orthogonal polynomial
self-interactions exhibited in higher order nonlinear Schr\"{o}dinger equation
(NLSE) are often less intuitive than those ofcubic and quintic nonlinearities.
Even for nonlinearities as simple as a cubic in NLSE, the dynamics for generic
initial states shows surprising features. In this Letter, for the first time,
we identify the higher-order nonlinearities in terms of orthogonal polynomials
in the generalized NLSE/GPE. More pertinently, we explicate different exotic
phase space structures for three specific examples: (i) Hermite, (ii)
Chebyshev, and (iii) Laguerre polynomial self-interactions. For the first two
self-interactions, we exhibit that the alternating signs of the various
higher-order nonlinearities are naturally embedded in these orthogonal
polynomials that confirm to the experimental conditions. To simulate the
phase-space dynamics that bring about by the Laguerre self-interactions, a
source term should {\it necessarily} be included in the modified NLSE/GPE.
Recent experiments suggest that this modified GPE captures the dynamics of
self-bound quantum droplets, in the presence of external source.
