Radial solutions with bounded H^1 norm decompose asymptotically into a free wave and a weakly localized component, for a broad class of nonlinear and time-dependent interactions.
Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces
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abstract
The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through $\overline{\partial}$-derivatives. To extend the asymptotics to solutions with initial data in lower regularity spaces, we apply a global approximation via PDE techniques.
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The large time asymptotics of nonlinear multichannel Schroedinger equations
Radial solutions with bounded H^1 norm decompose asymptotically into a free wave and a weakly localized component, for a broad class of nonlinear and time-dependent interactions.