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The fourth-order Schrödinger equation on lattices

Jiawei Cheng

Abstract

In this paper, we study the fourth-order Schrödinger equation \begin{equation*} i \partial_t u + Δ^2 u - γΔu = \pm |u|^{s-1}u \end{equation*} on the lattice $\mathbb{Z}^d$ with dimensions $d=1,2$ and parameter $γ\in \mathbb{R}$. In order to establish sharp dispersive estimates, we consider the fundamental solution as an oscillatory integral and analyze the Newton polyhedron of its phase function. Furthermore, we prove Strichartz estimates which yield the existence of global solutions to nonlinear equations with small data.

The fourth-order Schrödinger equation on lattices

Abstract

In this paper, we study the fourth-order Schrödinger equation \begin{equation*} i \partial_t u + Δ^2 u - γΔu = \pm |u|^{s-1}u \end{equation*} on the lattice with dimensions and parameter . In order to establish sharp dispersive estimates, we consider the fundamental solution as an oscillatory integral and analyze the Newton polyhedron of its phase function. Furthermore, we prove Strichartz estimates which yield the existence of global solutions to nonlinear equations with small data.
Paper Structure (13 sections, 19 theorems, 114 equations)

This paper contains 13 sections, 19 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics on the discrete setting and the DFS
  4. Uniform estimates for oscillatory integrals
  5. Integrals in one variable
  6. Newton polyhedra
  7. Integrals in two variables
  8. Proof of Theorem \ref{['theorem-main conclusion']}
  9. $\gamma \in (-\infty,-16) \cup (0,\infty)$
  10. $\gamma \in (-16,-8) \cup (-8,0)$
  11. $\gamma \in \{0,-16\}$
  12. $\gamma = -8$
  13. Strichartz estimates and nonlinear equations

Key Result

Theorem 1.1

There exists $C=C(d,\gamma)>0$ independent of $x$, such that

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 26 more