Forced Nonlinear Schroedinger Equation with Arbitrary Nonlinearity

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.