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Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: II. The Derivative NLS

David Damanik, Yong Li, Fei Xu

TL;DR

The paper addresses the local well-posedness and asymptotic behavior of the derivative NLS with spatially quasi-periodic initial data, focusing on retaining the same frequency vector under the evolution. It develops an explicit combinatorial framework based on an infinite system of ODEs for Fourier coefficients, Picard iteration, Feynman diagrams, and a labeled complex-conjugate (pcc) scheme to manage derivative-loss issues. The main results prove existence, uniqueness, and uniform exponential decay of the quasi-periodic solution, and establish that in a weakly nonlinear regime the nonlinear solution converges to the linear solution in both $L_x^\infty$ and analytic Sobolev norms on the relevant time scale. The methodology extends to a generalized gdNLS model, showing the robustness of the approach for quasi-periodic data and high-nonlinearity powers, with the Appendix supplying essential technical bounds.

Abstract

This is the second part of a two-paper series studying the nonlinear Schrödinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schrödinger equation. Under the assumption that the Fourier coefficients of the initial data obey an exponential upper bound, we establish local existence of a solution that retains quasi-periodicity in space with a slightly weaker Fourier decay. Moreover, the solution is shown to be unique within this class of quasi-periodic functions. Also, we prove that, for the derivative nonlinear Schrödinger equation in a weakly nonlinear setting, within the time scale, as the small parameter of nonlinearity tends to zero, the nonlinear solution converges asymptotically to the linear solution in the sense of both sup-norm and analytic Sobolev-norm. The proof proceeds via a consideration of an associated infinite system of coupled ordinary differential equations for the Fourier coefficients and an explicit combinatorial analysis for the Picard iteration with the help of Feynman diagrams and the power of $\ast^{[\cdot]}$ labelling the complex conjugate.

Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: II. The Derivative NLS

TL;DR

The paper addresses the local well-posedness and asymptotic behavior of the derivative NLS with spatially quasi-periodic initial data, focusing on retaining the same frequency vector under the evolution. It develops an explicit combinatorial framework based on an infinite system of ODEs for Fourier coefficients, Picard iteration, Feynman diagrams, and a labeled complex-conjugate (pcc) scheme to manage derivative-loss issues. The main results prove existence, uniqueness, and uniform exponential decay of the quasi-periodic solution, and establish that in a weakly nonlinear regime the nonlinear solution converges to the linear solution in both and analytic Sobolev norms on the relevant time scale. The methodology extends to a generalized gdNLS model, showing the robustness of the approach for quasi-periodic data and high-nonlinearity powers, with the Appendix supplying essential technical bounds.

Abstract

This is the second part of a two-paper series studying the nonlinear Schrödinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schrödinger equation. Under the assumption that the Fourier coefficients of the initial data obey an exponential upper bound, we establish local existence of a solution that retains quasi-periodicity in space with a slightly weaker Fourier decay. Moreover, the solution is shown to be unique within this class of quasi-periodic functions. Also, we prove that, for the derivative nonlinear Schrödinger equation in a weakly nonlinear setting, within the time scale, as the small parameter of nonlinearity tends to zero, the nonlinear solution converges asymptotically to the linear solution in the sense of both sup-norm and analytic Sobolev-norm. The proof proceeds via a consideration of an associated infinite system of coupled ordinary differential equations for the Fourier coefficients and an explicit combinatorial analysis for the Picard iteration with the help of Feynman diagrams and the power of labelling the complex conjugate.
Paper Structure (14 sections, 20 theorems, 112 equations)

This paper contains 14 sections, 20 theorems, 112 equations.

Table of Contents

  1. Introduction and Main Results
  2. Proof of Theorem \ref{['dnlsth']}
  3. Preliminaries
  4. Power of $\ast^{[\cdot]}$ for the Complex Conjugate
  5. Combinatorial Structure and Some Basic Concepts
  6. Main Steps of Proof
  7. Infinite-Dimensional ODEs, Picard Iteration and Combinatorial Tree
  8. The Picard Sequence is Exponentially Decaying
  9. The Picard Sequence is a Cauchy Sequence
  10. Existence and Convergence
  11. Uniqueness
  12. Asymptotic Dynamics
  13. gdNLS
  14. Appendix

Key Result

Theorem 1.1

If the initial Fourier data ${\mathfrak{c}}$ is $\mathtt \kappa$-exponentially decaying in the sense of ed, then the following statements hold:

Theorems & Definitions (35)

  • Theorem 1.1: dNLS
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 25 more