Multichannel nonlinear scattering theory for nonintegrable equations

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Abstract

We consider a class of nonlinear equations with localized and dispersive solutions; we show that for a ball in some Banach space of initial conditions, the asymptotic behavior (as t → ±∞) of such states is given by a linear combination of a periodic (in time), localized (in space) solution (nonlinear bound state) of the equation and a purely dispersive part (with free dispersion). We also show that 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 for t → ±∞from that for t → −∞.Moreover, the solution acquires an extra constant phase eiγ±.