Higher-order nonlinear Schrödinger equation in 2D case

Abstract

We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation in two dimensional case \[ \left\{\!\!\! \begin{array}{c} i\partial _{t}u+\frac{b}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\lambda \left\vert u\right\vert u,\text{ }t>0,\quad x\in \mathbb{R}^{2}\,\mathbf{,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,\quad x\in \mathbb{R}^{2} \,\mathbf{,} \end{array} \right. \] where $\lambda \in \mathbb{R}\mathbf{,}$ $b>0.$ We develop the factorization techniques for studying the large time asymptotics of solutions to the above Cauchy problem. We prove that the asymptotics has a modified character.