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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

Vol. 17 (2024), No. 6, 1887–1906
Abstract

We prove the uniqueness of several excited states to the ODE ÿ(t) + (2t)(t) + f(y(t)) = 0, y(0) = b, and (0) = 0, for the model nonlinearity f(y) = y3 y. The n-th excited state is a solution with exactly n zeros and which tends to 0 as t . These represent all smooth radial nonzero solutions to the PDE Δu + f(u) = 0 in H1. 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
Mathematical Subject Classification
Primary: 35B05, 35J60
Milestones
Received: 11 March 2021
Revised: 26 July 2022
Accepted: 26 January 2023
Published: 19 July 2024
Authors
Alex Cohen
Massachusetts Institute of Technology
Cambridge, MA
United States
Zhenhao Li
Massachusetts Institute of Technology
Cambridge, MA
United States
Wilhelm Schlag
Department of Mathematics
Yale University
New Haven, CT
United States

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