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 u₀ ∈ X, where \(X \in \{M_{2,q}^s(\mathbb{R}),\,{H^\sigma}(\mathbb{T}),\,{H^{{s_1}}}(\mathbb{R}) + {H^{{s_2}}}(\mathbb{T})\}\) and q ∈ [1, 2], s ⩾ 0, or σ ⩾ 0, or s₂ ⩾ s₁ ⩾ 0. Moreover, if M ^s_2,q (ℝ) ↪ L³(ℝ), or if \(\sigma \geqslant {1 \over 6}\), or if \({s_1} \geqslant {1 \over 6}\) and \({s_2} > {1 \over 2}\) we how 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.

我们在初始数据u ₀ ∈ X的三次四阶非线性薛定谔方程的柯西问题的扩展意义上证明了弱解的存在，其中\(X \in \{M_{2,q}^s(\ mathbb{R}),\,{H^\sigma}(\mathbb{T}),\,{H^{{s_1}}}(\mathbb{R}) + {H^{{s_2}}} (\mathbb{T})\}\)和q ∈ [1, 2], s ⩾ 0, or σ ⩾ 0, or s ₂ ⩾ s ₁ ⩾ 0. 此外，如果M ^s _{2, q} (ℝ) ↪ L ³ (ℝ)，或者如果\(\sigma \geqslant {1 \over 6}\)，或者如果\({s_1} \geqslant {1 \over 6}\) 和\({s_2} > {1 \over 2}\)我们如何证明柯西问题在X 中无条件地适定。由于相应相位因子的分解特性，类似的结果适用于所有高阶非线性薛定谔方程和混合阶 NLS。为了证明，我们通过分部微分技术使用范式归约，并建立在我们以前的工作基础上。