Modified scattering for the nonlinear nonlocal Schrödinger equation in two space dimensions

Abstract

We consider the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity$ \begin{equation*} \left\{ \begin{array}{c} i\partial _{t}u-\mathbf{\Lambda }u = \lambda \left\vert u\right\vert u,\text{ } t>0,{\ }x\in \mathbb{R}^{2}\mathbf{,} \\ u\left( 0,x\right) = u_{0}\left( x\right) ,{\ }x\in \mathbb{R}^{2} \mathbf{,} \end{array} \right. \end{equation*} $where the linear pseudodifferential operator $ \mathbf{\Lambda }u = \left(1-\Delta \right) ^{-1}\left( -\Delta +a\Delta ^{2}\right) u $ is characterized by its symbol $ \Lambda \left( \xi \right) = \frac{\left\vert\xi \right\vert ^{2}+a\left\vert \xi \right\vert ^{4}}{1+\left\vert \xi\right\vert ^{2}}. $ We will show below the modified scattering for solutions of equation (1). We continue to develop the factorization techniques which was started in our previous papers. The crucial points of our approach presented here are the $ \mathbf{L}^{2} $ - boundedness of the pseudodifferential operators.

Locations

• Discrete and Continuous Dynamical Systems - View - PDF