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Large Time Asymptotics for the Fractional Order Cubic Nonlinear Schrödinger Equations

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Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equations of fractional order

$$\left\{\begin{array}{l}i\partial _{t}u-2\left( -\partial _{x}^{2} \right)^{\frac{1}{4}} \, u=F\left( u\right) \\ u\left( 0,x\right) =u_{0} \left( x\right),\end{array}\right.$$

where \({F\left( u\right) }\) is the cubic nonlinearity

$$F\left( u\right) =\lambda \left| u\right| ^{2}u$$

with \({\lambda \in \mathbf{R}}\). We find the large time asymptotics of solutions to the Cauchy problem. We use the factorization technique similar to that developed for the Schrödinger equation.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Nakao Hayashi

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

    Pavel I. Naumkin

Authors
  1. Nakao Hayashi
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  2. Pavel I. Naumkin
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Correspondence to Nakao Hayashi.

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Communicated by Nader Masmoudi.

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Hayashi, N., Naumkin, P.I. Large Time Asymptotics for the Fractional Order Cubic Nonlinear Schrödinger Equations. Ann. Henri Poincaré 18, 1025–1054 (2017). https://doi.org/10.1007/s00023-016-0502-9

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  • DOI: https://doi.org/10.1007/s00023-016-0502-9

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