  A nonlinear modification of the Schr\"{o}dinger equation is proposed in which
the Lagrangian density for the Schr\"{o}dinger equation is extended by terms
polynomial in $\Delta^{m}\ln (\Psi^{*}/{\Psi})$ multiplied by $\Psi^{*}{\Psi}$.
This introduces a homogeneous nonlinearity in a Galilean invariant manner
through the phase $S$ rather than the amplitude $R$ of the wave function $\Psi
=R\exp (iS)$. From this general scheme we choose the simplest minimal model
defined in some reasonable way. The model in question offers the simplest way
to modify the Bohm formulation of quantum mechanics so as to allow a leading
phase contribution to the quantum potential and a leading quantum contribution
to the probability current removing asymmetries present in Bohm's original
formulation. It preserves most of physically relevant properties of the
Schr\"{o}dinger equation including stationary states of quantum-mechanical
systems. It can be thought of as the simplest model of nonlinear quantum
mechanics of extended objects among other such models that also emerge within
the general scheme proposed. The extensions of this model to $n$ particles and
the question of separability of compound systems are studied. It is noted that
there exists a weakly separable extension in addition to a strongly separable
one. The place of the general modification scheme in a broader spectrum of
nonlinear modifications of the Schr\"{o}dinger equation is discussed. It is
pointed out that the models it gives rise to have a unique definition of energy
in that the field-theoretical energy functional coincides with the
quantum-mechanical one. It is found that the Lagrangian for its simplest
variant represents the Lagrangian for a restricted version of the
Doebner-Goldin modification of this equation.
