Higher-order nonlinear Schrödinger equations with singular data

Abstract

We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{c} i\partial _{t}u-\frac{1}{2k}\left( -\partial _{x}^{2}\right) ^{k}u=\lambda \left| u\right| ^{2p}u,\text { }\left( t,x\right) \in \left[ 0,T\right] \times \mathbb {R}\mathbf {,} \\ u\left( 0,x\right) =u_{0}\left( x\right) , x \in \mathbb {R}\mathbf {,} \end{array} \right. \end{aligned}$$

where $k,p\in \mathbb {N}\mathbf {,}$ $k\ge 2,$ $\lambda \in \mathbb {C}$. We prove local existence of solutions for the case of singular initial data $u_{0}\left( x\right)$ including the Dirac delta function.