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The cubic nonlinear Schrödinger equation with rough potential

Norbert J. Mauser, Yifei Wu, Xiaofei Zhao

Abstract

We consider the cubic nonlinear Schrödinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the well-posedness analysis on the PDE level, and subsequently a new efficient numerical method, its convergence analysis and simulations that illustrate our analytical results. In the analysis part, our results focus on understanding how the regularity of the solution is influenced by the regularity of the potential, where we provide quantitative and explicit characterizations. Ill-posedness results are also established to demonstrate the sharpness of the obtained regularity characterizations and to indicate the minimum regularity required from the potential for the NLS to be solvable. Building upon the obtained regularity results, we design an appropriate numerical discretization for the model and establish its convergence with an optimal error bound. The numerical experiments in the end not only verify the theoretical regularity results, but also confirm the established convergence rate of the proposed scheme. Additionally, a comparison with other existing schemes is conducted to demonstrate the better accuracy of our new scheme in the case of a rough potential.

The cubic nonlinear Schrödinger equation with rough potential

Abstract

We consider the cubic nonlinear Schrödinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the well-posedness analysis on the PDE level, and subsequently a new efficient numerical method, its convergence analysis and simulations that illustrate our analytical results. In the analysis part, our results focus on understanding how the regularity of the solution is influenced by the regularity of the potential, where we provide quantitative and explicit characterizations. Ill-posedness results are also established to demonstrate the sharpness of the obtained regularity characterizations and to indicate the minimum regularity required from the potential for the NLS to be solvable. Building upon the obtained regularity results, we design an appropriate numerical discretization for the model and establish its convergence with an optimal error bound. The numerical experiments in the end not only verify the theoretical regularity results, but also confirm the established convergence rate of the proposed scheme. Additionally, a comparison with other existing schemes is conducted to demonstrate the better accuracy of our new scheme in the case of a rough potential.
Paper Structure (27 sections, 18 theorems, 330 equations, 8 figures)

This paper contains 27 sections, 18 theorems, 330 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Well-posedness theory
  3. Ill-posedness theory
  4. Numerical counterpart
  5. Preliminaries
  6. Basic notations
  7. Fourier transform
  8. Some operators
  9. Some tool lemmas
  10. Proof for well-posedness theory
  11. Proof of \ref{['main:thm1']}
  12. Estimates on $I_{21,k}$.
  13. Estimates on $I_{22,k}$.
  14. Estimates on $I_{23,k}$.
  15. Proof of \ref{['main:thm3-smoothdata']}
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\xi\in \hat{b}^{s,p}$ for $s\ge 0,2<p\le \infty$, so $\gamma_p\in(\frac{3}{2},2)$. Then, model is locally well-posed in $H^{s+\gamma_p-}(\mathbb{T})$ (notation $\gamma_p-$ explained in sec2:subsec1 (iii), (iv).

Figures (8)

  • Figure 1: Test of \ref{['main:thm1']} for $s=0,p=\infty$: modulus of Fourier coefficient of the solution (left) and the generated potential $\xi\in\hat{l}^\infty$ (right).
  • Figure 2: Test of \ref{['main:thm1']} for $s=0,p=4$: modulus of Fourier coefficient of the solution (left) and the generated potential $\xi\in\hat{l}^\infty$ (right).
  • Figure 3: Test of \ref{['main:thm3-smoothdata']} for $s=0,p=2$: modulus of Fourier coefficient of the solution (left) and the generated potential $\xi\in L^2$ (right).
  • Figure 4: Results of \ref{['ex:err1']}: profiles of the potential $\xi(x)$ and the solution $|u_0(x)|$, $|u(t=1,x)|$ (1st row); error \ref{['err def']} of LRI \ref{['NuSo-NLS']} (2nd row).
  • Figure 5: Results of \ref{['ex:err2']} under $\xi\in\hat{b}^{0,2}$: profiles of $\xi(x)$, $|u_0(x)|$ and $|u(t=1,x)|$ (1st row); error \ref{['err def']} of LRI \ref{['NuSo-NLS']} (2nd row).
  • ...and 3 more figures

Theorems & Definitions (41)

  • Definition 1.1: Well-posedness
  • Remark 1.1
  • Theorem 1.1: Well-posedness for $\hat{b}^{s,p}$-potential
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2: Well-posedness for $H^{s,p'}$-potential
  • Definition 1.2: Ill-posedness
  • Theorem 1.3: Ill-posedness for $\hat{b}^{s,p}$-potential
  • Remark 1.4
  • Theorem 1.4: Ill-posedness for $H^s$-potential
  • ...and 31 more