Asymptotics for the fractional nonlinear Schrödinger equation with 2<α<52

• Published in: Journal of pseudo-differential operators and applications Vol. 13; no. 3
• Main Authors: Hayashi, Nakao, Mendez-Navarro, Jesus A., Naumkin, Pavel I.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2022; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Asymptotic properties; Cauchy problems; Defocusing; Functional Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Partial Differential Equations; Schrodinger equation; Asymptotics of solutions; Fractional nonlinear Schrödinger equation; Modified scattering; 35B40; Factorization technique; 35Q35
• ISSN: 1662-9981; 1662-999X
• DOI: 10.1007/s11868-022-00460-z

We study the Cauchy problem for the fractional nonlinear Schrödinger equation i ∂ t u - 1 α ∂ x α u = λ u α u , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where λ ∈ R , the fractional derivative ∂ x α = F - 1 ξ α F , the order α ∈ 2 , 5 2 . Our aim is to find the asymptotics of solutions to the fractional nonlinear Schrödinger equation in the defocusing case λ > 0 . We show that the asymptotics differs from that in the case of the usual cubic nonlinear Schrödinger equation. To prove our main result, we develop the Factorization Techniques which was proposed in our previous works.