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On the well-posedness of the periodic fractional Schrödinger equation

Beckett Sanchez, Oscar Riaño, Svetlana Roudenko

Abstract

We consider the periodic fractional nonlinear Schrödinger equation $$ iu_t -(-Δ)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two ways: the first one $\mathcal{N}\in C^J(\mathbb R^+)$, whose derivatives have a certain polynomial decay, e.g., $\mathcal{N}(|u|)=\log(|u|)$; the second one is given by a sum of powers, possibly infinite, $$ \mathcal{N}(|u|) = \sum a_k |u|^{γ_k}, \quad γ_k \in \mathbb{R}, ~~ a_k \in \mathbb{C}, $$ which includes examples such as $\mathcal{N}(|u|) \, u =\frac{u}{|u|^γ},$ $γ>0$. By using standard properties of periodic Sobolev spaces $H^J(\mathbb{T}^N)$, $J>0$, we study the local well-posedness for the Cauchy problems of the above equations when initial data satisfy a non-vanishing condition $\inf\limits_{x\in \mathbb{T}^N}|u_0(x)|>0$.

On the well-posedness of the periodic fractional Schrödinger equation

Abstract

We consider the periodic fractional nonlinear Schrödinger equation where the nonlinearity term is expressed in two ways: the first one , whose derivatives have a certain polynomial decay, e.g., ; the second one is given by a sum of powers, possibly infinite, which includes examples such as . By using standard properties of periodic Sobolev spaces , , we study the local well-posedness for the Cauchy problems of the above equations when initial data satisfy a non-vanishing condition .

Paper Structure

This paper contains 8 sections, 4 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Well-posedness results for nonlinearities of class $C^J$
  4. Nonlinear estimates
  5. Proof of Theorem \ref{['mainTHM2']}
  6. Well-posedness results for series nonlinearities
  7. Nonlinear estimates
  8. Proof of Theorem \ref{['mainTHM1']}

Key Result

Theorem 1.1

Let $s>0$ and $J\in \mathbb{Z}^{+}$ be such that $J>\frac{N}{2}+s$. Consider the Cauchy problem NLS, where $\mathcal{N}(\cdot)$ satisfies the following conditions: Let $u_0\in H^J(\mathbb T^N)$ be such that there exists an $\eta>0$, for which nonvanish holds. Then, there exist a time $T>0$ and a unique solution $u\in C([-T,T];H^J(\mathbb{T}^N))$ of NLS with initial condition $u_0$ such that More

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Remark 4.1
  • proof : Proof of Lemma \ref{['nonestimates']}