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Uniqueness of excited
states to $-\Delta u + u -u^3=0$ in three dimensions
Alex Cohen, Zhenhao Li and Wilhelm Schlag |
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Vol. 17 (2024), No. 6, 1887–1906
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Abstract |
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We prove the uniqueness of several excited states to the ODE , , and , for the model nonlinearity . The -th excited state is a solution with exactly zeros and which tends to as . These represent all smooth radial nonzero solutions to the PDE in . We interpret the ODE as a damped oscillator governed by a double-well potential, and the result is proved via rigorous numerical analysis of the energy and variation of the solutions. More specifically, the problem of uniqueness can be formulated entirely in terms of inequalities on the solutions and their variation, and these inequalities can be verified numerically. |
Keywords
Klein–Gordon, interval arithmetic, soliton, excited state
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Mathematical Subject Classification
Primary: 35B05, 35J60
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Milestones
Received: 11 March 2021
Revised: 26 July 2022
Accepted: 26 January 2023
Published: 19 July 2024
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Authors |
| Alex Cohen | |
| Massachusetts Institute of
Technology Cambridge, MA United States |
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| Zhenhao Li | |
| Massachusetts Institute of
Technology Cambridge, MA United States |
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| Wilhelm Schlag | |
| Department of Mathematics Yale University New Haven, CT United States |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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