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Uniqueness of excited states to $-Δu+u-u^3=0$ in three dimensions

Alex Cohen, Zhenhao Li, Wilhelm Schlag

Abstract

We prove the uniqueness of several excited states to the ODE $\ddot y(t) + \frac{2}{t} \dot y(t) + f(y(t)) = 0$, $y(0) = b$, and $\dot y(0) = 0$ for the model nonlinearity $f(y) = y^3 - y$. The $n$-th excited state is a solution with exactly $n$ zeros and which tends to $0$ as $t \to \infty$. These represent all smooth radial nonzero solutions to the PDE $Δu + f(u)= 0$ in $H^1$. 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.

Uniqueness of excited states to $-Δu+u-u^3=0$ in three dimensions

Abstract

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.

Paper Structure

This paper contains 17 sections, 8 theorems, 56 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview of approach
  3. Toy example: finding zeros of a function
  4. Finding and isolating excited states
  5. Approximating solutions via interval arithmetic
  6. Analytical description of the damped oscillator dynamics
  7. Basic properties of the ODE
  8. The equation of variation
  9. Picard approximation
  10. The equation at infinity
  11. Limit sets and convergence theorems
  12. Passing over the saddle
  13. Proof of Theorem \ref{['thm:first_excited_state_unique']}
  14. Outline of proof
  15. Planning section
  16. ...and 2 more sections

Key Result

Theorem 1

The first twenty excited states of ODE (ODE:eq-ODE:init_val) are unique for $f(y)=-y+y^3$.

Figures (5)

  • Figure 1: Potential function $V(y) = \frac{1}{4}y^4 - \frac{1}{2}y^2$
  • Figure 2: Limiting position $y_b(T)$ as $T\to \infty$, plotted as a function of the initial condition $b$ up to $b = 1200$. The solid red dots represent bound states, and this graph holds due to Theorem \ref{['thm:first_excited_state_unique']} and the rigorous numerical work done in this paper.
  • Figure 3: VNODE-LP numerical integration with solution intervals scaled up $\times 100$. The "pinch points" of near zero $y$-uncertainty occur when $\delta = y_b(t) \sim 0$, and although it is not shown here, the $\dot y$-uncertainty is larger at these points.
  • Figure 4: Graph showing first three excited states, and how the $b$-axis is partitioned by different proof methods.
  • Figure :

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 5 more