On the well-posedness of the periodic fractional Schrödinger equation

Beckett Sanchez (), Oscar Riaño () and Svetlana Roudenko ()
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Beckett Sanchez: Brown University
Oscar Riaño: Universidad Nacional de Colombia, Ciudad Universitaria
Svetlana Roudenko: Florida International University

Partial Differential Equations and Applications, 2025, vol. 6, issue 3, 1-28

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

Keywords: Periodic fractional nonlinear Schrödinger equation; Well-posedness; Weighted spaces; Combined nonlinearities; Logarithmic potential; 35A01; 35B10; 35Q40; 35Q55 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00327-0

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