Tohoku Mathematical Journal
2025
June
SECOND SERIES VOL. 77, NO. 2

Tohoku Math. J.
77 (2025), 239-267

Title SELF-SIMILAR CHARACTER OF THE LARGE-TIME SYMPTOTICS OF SOLUTIONS TO THE DERIVATIVE FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION

Author Nakao Hayashi and Pavel I. Naumkin

(Received November 28, 2022, revised June 6, 2023)
Abstract. We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional derivative nonlinear Schrödinger equation in one space dimension \begin{equation*} \left\{ \begin{array}{c} i\partial_{t}u-\Lambda ( -i\partial_x) u=i\lambda \partial_x F(u) , t>0, x\in \mathbb{R}, \\ u(0,x) =u_{0}(x) , x\in \mathbb{R}, \end{array} \right. \end{equation*} where \begin{equation*} \Lambda (-i\partial_x) =\frac{1}{\alpha}(-\partial_x^2)^{\frac{\alpha}{2}}=\frac{1}{\alpha} |\partial_x|^{\alpha}, \alpha>3, F(u) =|u|^{\alpha -1}u \end{equation*} and $\lambda \in \mathbb{R}$. The fractional derivative $|\partial_x| ^{\alpha}=\mathcal{F}^{-1}|\xi|^{\alpha}\mathcal{F}$, where $\mathcal{F}$ stands for the Fourier transformation and $\mathcal{F}^{-1}$ is its inverse transformation. We have the mass conservation law $\int_{\mathbb{R}}u(t,x)dx=\int_{\mathbb{R}}u_0(x)dx$. We prove that solutions are stable in the neighborhood of the self-similar solutions under the non-zero total mass condition $\int_{\mathbb{R}}u_0(x)dx\neq 0.$

Mathematics Subject Classification. Primary 35B40; Secondary 35Q92.
Key words and phrases. Fractional Schrödinger equation, large time symptotics, self-similar solutions, nonlinearity of divergence type.

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