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The stability and dynamical properties of the so-called resonant nonlinearSchr\"odinger (RNLS) equation, are considered. The RNLS is a variant of thenonlinear Schr\"odinger (NLS) equation with the addition of a perturbation usedto describe wave propagation in cold collisionless plasmas. We first examinethe modulational stability of plane waves in the RNLS model, identifying themodifications of the associated conditions from the NLS case. We then move tothe study of solitary waves with vanishing and nonzero boundary conditions.Interestingly the RNLS, much like the usual NLS, exhibits both dark and brightsoliton solutions depending on the relative signs of dispersion andnonlinearity. The corresponding existence, stability and dynamics of thesesolutions are studied systematically in this work.

Solitary waves in the resonant nonlinear Schr\"odinger equation: stability and dynamical properties