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Computation of excited states for the nonlinear Schr{ö}dinger equation: numerical and theoretical analysis

Christophe Besse, Romain Duboscq, Stefan Le Coz

Abstract

Our goal is to compute excited states for the nonlinear Schr{ö}dinger equation in the radial setting. We introduce a new technique based on the Nehari manifold approach and give a comparison with the classical shooting method. We observe that the Nehari method allows to accurately compute excited states on large domains but is relatively slow compared to the shooting method.

Computation of excited states for the nonlinear Schr{ö}dinger equation: numerical and theoretical analysis

Abstract

Our goal is to compute excited states for the nonlinear Schr{ö}dinger equation in the radial setting. We introduce a new technique based on the Nehari manifold approach and give a comparison with the classical shooting method. We observe that the Nehari method allows to accurately compute excited states on large domains but is relatively slow compared to the shooting method.
Paper Structure (7 sections, 7 theorems, 109 equations, 10 figures, 2 algorithms)

This paper contains 7 sections, 7 theorems, 109 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical approach for the minimization over the Nehari manifold
  3. The shooting method
  4. The Nehari method
  5. Some properties of the Nehari and shooting methods
  6. Numerical Experiments
  7. Quantitative Deformation Lemma

Key Result

Proposition 2.1

For every sequence $(u_n)\in \mathcal{N}$ such that there exist $u_\infty\in\mathcal{N}$ and $(y_n)\subset \mathbb{R}^d$ such that, up to a subsequence, Moreover, $u_\infty$ is a ground state solution of eq:snls.

Figures (10)

  • Figure 1: Bound states in dimension $d=3$
  • Figure 2: Computation of a ground state on large domain
  • Figure 3: Total number of iterations for the Nehari method depending on the number of nodes
  • Figure 4: Convergence for the Nehari method depending on the number of nodes
  • Figure 5: Convergence for the shooting method depending on the number of nodes
  • ...and 5 more figures

Theorems & Definitions (19)

  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • ...and 9 more