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Multichannel nonlinear scattering theory for nonintegrable equations. (English) Zbl 0712.35074

Integrable systems and applications, Proc. Workshop, Oléron/Fr. 1988, Lect. Notes Phys. 342, 312-327 (1989).
Summary: [For the entire collection see Zbl 0698.00037.]
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 \(t\to \pm \infty)\) 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 \(t\to +\infty\) from that for \(t\to -\infty\). Moreover, the solution acquires an extra constant phase \(e^{i\gamma \pm}\).

MSC:

35P25 Scattering theory for PDEs
35F20 Nonlinear first-order PDEs

Citations:

Zbl 0698.00037