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Fractional nonlinear Schrödinger equation

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Abstract

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} i\partial _{t}u+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2} }u=\lambda \left| u\right| ^{2}u,\,\, t>0, &{}\quad x\in \mathbb {R},\\ u\left( 1,x\right) =u_{0}\left( x\right) ,&{}\quad x\in \mathbb {R}. \end{array}\right. \end{aligned}$$

We develop the factorization technique to obtain the large-time asymptotic behavior of solutions which has a logarithmic phase modifications for large time comparing with the linear problem.

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Acknowledgements

We are grateful to unknown referees for many useful suggestions and comments. The work of P.I.N. is partially supported by CONACYT 283698 and PAPIIT project IN100616.

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Authors and Affiliations

  1. Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia, CP 58089, Michoacán, Mexico

    Jesus A. Mendez-Navarro & Pavel I. Naumkin

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

    Isahi Sánchez-Suárez

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  1. Jesus A. Mendez-Navarro
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  2. Pavel I. Naumkin
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  3. Isahi Sánchez-Suárez
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Correspondence to Pavel I. Naumkin.

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Mendez-Navarro, J.A., Naumkin, P.I. & Sánchez-Suárez, I. Fractional nonlinear Schrödinger equation. Z. Angew. Math. Phys. 70, 168 (2019). https://doi.org/10.1007/s00033-019-1207-y

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