Forced Nonlinear Schroedinger Equation with Arbitrary Nonlinearity (original) (raw)
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self interaction g^2/κ+1 (ψ^ψ)^κ+1 in the presence of the external forcing terms of the form r e^-i(kx + θ) -δψ. 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 → 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 q̇ (t) < 0, where p(t) is the normalized canonical momentum p(t) = 1/M(t)∂ L/∂q̇, and ...
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