Internal vibrations of a vector soliton in the coupled nonlinear Schrödinger equations

Boris A. Malomed, Richard S. Tasgal

Research output: Contribution to journalArticlepeer-review


Static and dynamic properties of a two-component soliton are studied via the variational approximation (VA), consideration of the radiation spectrum, and direct numerical simulations. The VA, based on a Gaussian Ansatz, proves to be (as compared to the direct simulations) fairly accurate in some respects and inaccurate in others—in particular, the predictions for the widths of the stationary states are about a sixth part greater than the actual widths. We formulate an empirically modified version of the variational approximation: at the end of the analysis, the Gaussian is replaced by sech with properly rescaled widths. This hybrid VA yields extremely accurate predictions for the stationary states. The error in the width prediction is ≲1%, and simulations demonstrate minuscule radiation losses. The VA model predicts three eigenmodes of the soliton’s internal vibrations, all of which are observed numerically. Oscillation of the separation between the two components is found to be the most persistent mode, and in-phase oscillation of the two widths is the next most persistent one; in contrast, the out-of-phase width oscillations are unstable, quickly rearranging themselves into the stable in-phase mode. These features are easily explained by comparing the corresponding vibrational eigenfrequencies to the spectral gaps which isolate oscillations localized at the soliton from delocalized radiation modes. For vector solitons with energy nearly equally divided between the components, the analysis reveals a remarkable feature: saturation of the separation oscillations, with the radiative decay virtually ceasing at a finite level of the mode’s amplitude. The relatively stable in-phase width-oscillation mode decays indefinitely, but according to a very slow power law rather than exponentially. Lastly, for large-amplitude vibrations, the VA models predict dynamical chaos, but, due to the quick decay of the large oscillations, direct simulations show no chaos.

Original languageEnglish
Pages (from-to)2564-2575
Number of pages12
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Issue number2
StatePublished - 1998


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