  We construct one soliton solutions for the nonlinear Schroedinger equation
with variable quadratic Hamiltonians in a unified form by taking advantage of a
complete (super) integrability of generalized harmonic oscillators. The soliton
wave evolution in external fields with variable quadratic potentials is totally
determined by the linear problem, like motion of a classical particle with
acceleration, and the (self-similar) soliton shape is due to a subtle balance
between the linear Hamiltonian (dispersion and potential) and nonlinearity in
the Schroedinger equation by the standards of soliton theory. Most linear
(hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second
Painleve transcendental) classical special functions of mathematical physics
are linked together through these solutions, thus providing a variety of
nonlinear integrable cases. Examples include bright and dark solitons, and
Jacobi elliptic and second Painleve transcendental solutions for several
variable Hamiltonians that are important for current research in nonlinear
optics and Bose-Einstein condensation. The Feshbach resonance matter wave
soliton management is briefly discussed from this new perspective.
