Modified Scattering for the Derivative Fractional Cubic Nonlinear Schrödinger Equation with Dispersion Order $\alpha\in(4,5)$

Source abstract

We consider the Cauchy problem for the derivative fractional cubic nonlinear Schrödinger equation $$\begin{cases} \partial_{t}u-\dfrac{1}{\alpha}\left\vert \partial_{x}\right\vert ^{\alpha}u=i\lambda\partial_{x}(\left\vert u\right\vert ^{2}u), & t\gt 0,~ x\in\mathbb{R}, \\ u\left( 0,x\right) =u_{0}\left( x\right) , & x\in\mathbb{R}, \end{cases}$$ where $\alpha\in\left( 4,5\right) ,$ and $\lambda\in\mathbb{R}.$ We show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case. We develop the method based on the factorization techniques. Also we apply the known results on the $\mathbf{L}^{2}$ - boundedness of pseudodifferential operators.