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Nonlinear Schrödinger equation on a unit ball in one and two dimensions

Christian Klein, Svetlana Roudenko, Nikola Stoilov

Abstract

We consider the nonlinear Schrödinger equation on a unit ball in one and two dimensions with Dirichlet boundary conditions, which have stabilizing effect on solutions behavior. In particular, we confirm that the ground state solutions are stable in subcritical and critical cases, and in the supercritical case the ground state solutions split into a stable and an unstable branch. Perturbations of a ground state on the stable branch keep solutions near a corresponding ground state with very small oscillation around it, while perturbations of the unstable branch make solutions either blow up in finite time, if perturbations have an amplitude large than the height of the ground state, or oscillate between two states, if perturbations have an amplitude smaller than the original ground state. We also observe that this equation does not have any scattering or radiation, and thus, the soliton resolution holds for all data, splitting solutions into coherent structures such as ground state solutions even for very small initial data.

Nonlinear Schrödinger equation on a unit ball in one and two dimensions

Abstract

We consider the nonlinear Schrödinger equation on a unit ball in one and two dimensions with Dirichlet boundary conditions, which have stabilizing effect on solutions behavior. In particular, we confirm that the ground state solutions are stable in subcritical and critical cases, and in the supercritical case the ground state solutions split into a stable and an unstable branch. Perturbations of a ground state on the stable branch keep solutions near a corresponding ground state with very small oscillation around it, while perturbations of the unstable branch make solutions either blow up in finite time, if perturbations have an amplitude large than the height of the ground state, or oscillate between two states, if perturbations have an amplitude smaller than the original ground state. We also observe that this equation does not have any scattering or radiation, and thus, the soliton resolution holds for all data, splitting solutions into coherent structures such as ground state solutions even for very small initial data.

Paper Structure

This paper contains 20 sections, 2 theorems, 37 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Solitary waves in a ball
  4. Properties of ground states
  5. Pokhozhaev identities
  6. Changes of mass and energy with respect to $b$
  7. Two regimes of convergence of ground states
  8. Numerical approach
  9. Numerical construction of ground states in 1D
  10. Numerical construction of ground states in 2D
  11. Time integration
  12. Numerical results
  13. Convergence as $b \to \infty$
  14. Convergence as $b \to -\lambda$
  15. 1D case
  16. ...and 5 more sections

Key Result

Theorem 1.1

For $d \geq 1$ and $Q_b$ being the least-energy (ground state) solution, the following holds: I. Orbital stability. (a.1) Let $0 < \alpha \leq \frac{4}{d}$. The standing wave $e^{ib t } Q_b$ is orbitally stable in $H^1_0 (B_1)$ for any $b \in (-\lambda_1, -\lambda_1 + \epsilon) \cup (b_1, \infty)$,

Figures (16)

  • Figure 1: Stability of ground states $Q_b$ in critical and supercritical cases. The axes labels on the left are from the example of 1D quintic NLS, Figure \ref{['F:ME-1D-p3-5']}(D) and on the right are from the example of 2D quintic NLS, Figure \ref{['F:ME-2D']}(I).
  • Figure 2: The profiles of $\mathcal{R}_{\alpha}$ for various values of $\alpha$ (left) and mass $M(\mathcal{R}_{\alpha})$ (right).
  • Figure 3: 1D cubic NLS. Convergence of the rescaled ground states $Q_b$ (solid blue) to the 1D ground state $\mathcal{R} = \sqrt 2 \, \textmd{sech} \, x$ (dotted red), rescaled to the interval $[-1,1]$ as in \ref{['E:R-rescaled']}, as $b \to \infty$: $b=10$ (left), $b=100$ (middle), difference for $b=100$ (right).
  • Figure 4: 2D cubic NLS. Convergence of the rescaled ground states $Q_b$ (solid blue) to the 2D ground state $\mathcal{R}$ (computed numerically and rescaled to $|r| \leq 1$ as in \ref{['E:R-rescaled']}, dotted red) as $b \to \infty$: $b=10$ (left), $b=100$ (middle), difference for $b=100$ (right).
  • Figure 5: 1D cubic NLS (subcritical): convergence of normalized ground states to $\chi_1$ as $b \to -\lambda_1$, which is indicated by the value $-\pi^2/4$ (light blue); zoom-in of the peak on the right.
  • ...and 11 more figures

Theorems & Definitions (5)

  • Theorem 1.1: FM2001, FHK2012
  • Conjecture 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1