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Modified scattering for the fractional nonlinear Schrödinger equation with \(\alpha \in ({3 \over 2},2)\)

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Abstract

We study the Cauchy problem for the fractional nonlinear Schrödinger equation

$$\left\{{\matrix{{i{\partial _t}u + {1 \over \alpha}{{\left| {{\partial _x}} \right|}^\alpha}u = \lambda |u{|^2}u,\,\,t>0,} \hfill\;\;\;\;\;\;\;\;\;\;\;\; {x \in \mathbb{R},} \hfill \cr {u(0,x) = {u_0}(x),} \hfill\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{x \in \mathbb{R},} \hfill \cr}} \right.$$

where λ ∈ ℝ, the fractional derivative \(|{\partial _x}{|^\alpha} = {{\cal F}^{-1}}|\xi {|^\alpha}{\cal F}\), the order \(\alpha \in ({3 \over 2},2)\). Our aim is to prove the modified scattering for solutions of the fractional nonlinear Schrödinger equation. We develop the factorization techniques which we proposed in our previous works.

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Acknowledgments

We would like to thank the referee for useful suggestions on the second version of the draft. The work of N. H. is partially supported by JSPS KAKENHI Grant Numbers JP20K03680, JP19H05597. The work of P. I. N. is partially supported by CONACYT and PAPIIT project IN103221.

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Nakao Hayashi

  2. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, AP 61-3 (Xangari) Morelia, CP 58089, Michoacán, Mexico

    Pavel I. Naumkin

  3. Universidad Politécnica de Uruapan, Uruapan, CP 60210, Michoacán, Mexico

    Isahi Sánchez-Suárez

Authors
  1. Nakao Hayashi
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  2. Pavel I. Naumkin
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  3. Isahi Sánchez-Suárez
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Correspondence to Nakao Hayashi.

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Hayashi, N., Naumkin, P.I. & Sánchez-Suárez, I. Modified scattering for the fractional nonlinear Schrödinger equation with \(\alpha \in ({3 \over 2},2)\). JAMA 150, 609–644 (2023). https://doi.org/10.1007/s11854-023-0284-1

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  • DOI: https://doi.org/10.1007/s11854-023-0284-1

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