Unconditional uniqueness of higher order nonlinear Schrödinger equations

Friedrich Klaus; Peer Kunstmann; Nikolaos Pattakos

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 709-742
  • ISSN: 0011-4642

Abstract

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data u 0 ∈ X , where X ∈ { M 2 , q s ( ℝ ) , H σ ( 𝕋 ) , H s 1 ( ℝ ) + H s 2 ( 𝕋 ) } and q ∈ [ 1 , 2 ] , s ≥ 0 , or σ ≥ 0 , or s 2 ≥ s 1 ≥ 0 . Moreover, if M 2 , q s ( ℝ ) ↪ L 3 ( ℝ ) , or if σ ≥ 1 6 , or if s 1 ≥ 1 6 and s 2 > 1 2 we show that the Cauchy problem is unconditionally wellposed in X . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work.

How to cite

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Klaus, Friedrich, Kunstmann, Peer, and Pattakos, Nikolaos. "Unconditional uniqueness of higher order nonlinear Schrödinger equations." Czechoslovak Mathematical Journal 71.3 (2021): 709-742. <http://eudml.org/doc/298054>.

@article{Klaus2021,
abstract = {We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_\{0\}\in X$, where $X\in \lbrace M_\{2,q\}^\{s\}(\mathbb \{R\}), H^\{\sigma \}(\mathbb \{T\}), H^\{s_\{1\}\}(\mathbb \{R\})+H^\{s_\{2\}\}(\mathbb \{T\})\rbrace $ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_\{2\}\ge s_\{1\}\ge 0$. Moreover, if $M_\{2,q\}^\{s\}(\mathbb \{R\})\hookrightarrow L^\{3\}(\mathbb \{R\})$, or if $\sigma \ge \frac\{1\}\{6\}$, or if $s_\{1\}\ge \frac\{1\}\{6\}$ and $s_\{2\}>\frac\{1\}\{2\}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work.},
author = {Klaus, Friedrich, Kunstmann, Peer, Pattakos, Nikolaos},
journal = {Czechoslovak Mathematical Journal},
keywords = {normal form method; modulation space; unconditional uniqueness; higher order nonlinear Schrödinger},
language = {eng},
number = {3},
pages = {709-742},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Unconditional uniqueness of higher order nonlinear Schrödinger equations},
url = {http://eudml.org/doc/298054},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Klaus, Friedrich
AU - Kunstmann, Peer
AU - Pattakos, Nikolaos
TI - Unconditional uniqueness of higher order nonlinear Schrödinger equations
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 709
EP - 742
AB - We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_{0}\in X$, where $X\in \lbrace M_{2,q}^{s}(\mathbb {R}), H^{\sigma }(\mathbb {T}), H^{s_{1}}(\mathbb {R})+H^{s_{2}}(\mathbb {T})\rbrace $ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_{2}\ge s_{1}\ge 0$. Moreover, if $M_{2,q}^{s}(\mathbb {R})\hookrightarrow L^{3}(\mathbb {R})$, or if $\sigma \ge \frac{1}{6}$, or if $s_{1}\ge \frac{1}{6}$ and $s_{2}>\frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work.
LA - eng
KW - normal form method; modulation space; unconditional uniqueness; higher order nonlinear Schrödinger
UR - http://eudml.org/doc/298054
ER -

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