Modified scattering for higher-order nonlinear Schrödinger equation in one space dimension (Q2064553)

The paper under review considers the higher-order nonlinear Schrödinger equation in one space dimension: \[ \begin{cases} i\partial_tu + \frac{1}{2}\partial_x^2 u -\frac1{\alpha} |\partial_x|^\alpha u = \lambda|u|^2 u, \quad t > 0,~ x \in \mathbb R,\\ u (0, x) = u_0 (x) , x \in \mathbb R\,. \end{cases} \] Assuming \(\alpha\in\{4\}\cup [5,+\infty)\) and the initial data is small, the authors prove the global well posedness of the Cauchy problem and derive the asymptotic profile of the solution for large time, which involves, after adequate transforms, the quantity \[ \frac{W_+}{i\sqrt{\Lambda''}}\exp\Big(-\frac{i\lambda}{\Lambda''}|W_+|^2\,\log t\Big) \] where \(\Lambda(\xi)=\frac12\xi^2+\frac1\alpha |\xi|^\alpha\) and \(W_+(\xi)\) is an essentially bounded function.