Abstract
We consider a class of nonlinear Schrödinger equations (conservative and dispersive systems) with localized and dispersive solutions. We obtain a class of initial conditions, for which the asymptotic behavior (t→±∞) of solutions is given by a linear combination of nonlinear bound state (time periodic and spatially localized solution) of the equation and a purely dispersive part (decaying to zero with time at the free dispersion rate). We also obtain a result ofasymptotic stability type: given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general, the time-period (and energy) of the localized part is different fort→+∞ from that fort→−∞. Moreover the solution acquires an extra constant asymptotic phasee iy ±.
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Communicated by J. Fröhlich
This research was supported in part by grants from the National Science Foundation
The results of this paper were announced in a lecture (June, 1988) on which the Proceedings article [Sof-We] is based
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Soffer, A., Weinstein, M.I. Multichannel nonlinear scattering for nonintegrable equations. Commun.Math. Phys. 133, 119–146 (1990). https://doi.org/10.1007/BF02096557
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DOI: https://doi.org/10.1007/BF02096557