  We consider the nonlinear Schr{\"o}dinger equation (NLSE) in 1+1 dimension
with scalar-scalar self interaction $\frac{g^2}{\kappa+1} (\psi^\star
\psi)^{\kappa+1}$ in the presence of the external forcing terms of the form $r
e^{-i(kx + \theta)} -\delta \psi$. We find new exact solutions for this problem
and show that the solitary wave momentum is conserved in a moving frame where
$v_k=2 k$. These new exact solutions reduce to the constant phase solutions of
the unforced problem when $r \rightarrow 0.$
  In particular we study the behavior of solitary wave solutions in the
presence of these external forces in a variational approximation which allows
the position, momentum, width and phase of these waves to vary in time. We show
that the stationary solutions of the variational equations include a solution
close to the exact one and we study small oscillations around all the
stationary solutions.
  We postulate that the dynamical condition for instability is that $ dp(t)/d
\dot{q} (t) < 0$, where $p(t)$ is the normalized canonical momentum $p(t) =
\frac{1}{M(t)} \frac {\partial L}{\partial {\dot q}}$, and $\dot{q}(t)$ is the
solitary wave velocity. Here $M(t) = \int dx \psi^\star(x,t) \psi(x,t)$.
Stability is also studied using a "phase portrait" of the soliton, where its
dynamics is represented by two-dimensional projections of its trajectory in the
four-dimensional space of collective coordinates.
  The criterion for stability of a soliton is that its trajectory is a closed
single curve with a positive sense of rotation around a fixed point.
  We investigate the accuracy of our variational approximation and these
criteria using numerical simulations of the NLSE.
