Parametric Dissipative Nonlinear Schrödinger Equation

Nonlinear Schrödinger equation with complex-conjugate-type (parametric) forcing and linear damping is studied both analytically and numerically. Bifurcations of stationary and time-dependent solutions are investigated. A periodic motion arising via a Hopf bifurcation of the cnoidal solution explains well the soliton-oscillation phenomena of water wave in a vertically forced long container discovered by Wu et al . The analogy is pointed out in an explicit way between the mode competition of surface waves in a container and the soliton-oscillation. The existence of spatiotemporal chaotic states is illustrated by numerical simulations.