Journal Article

Relaxation of excited states in nonlinear Schrödinger equations

Tai-Peng Tsai and Horng-Tzer Yau

in International Mathematics Research Notices

Volume 2002, issue 31, pages 1629-1673
Published in print January 2002 | ISSN: 1073-7928
Published online January 2002 | e-ISSN: 1687-0247 | DOI: http://dx.doi.org/10.1155/S1073792802201063
Relaxation of excited states in nonlinear Schrödinger equations

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We consider a nonlinear Schrödinger equation with a bounded local potential in ℝ3. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear excited state. We give a sufficient condition on the initial data so that the solution to the Schrödinger equation approaches to certain nonlinear ground state as the time tends to infinity. Our proof is based on a notion of outgoing estimate which measures the time-direction related information of the wave functions for the nonlinear Schrödinger equations.

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Subjects: Mathematics

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