On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension
The authors show that the nonlinear Schrödinger equation \(iu_ t+u_{xx}=\lambda | u|^{p-1} u\), (t,x)\(\in {\mathbb{R}}\times {\mathbb{R}}\), \(\lambda\in {\mathbb{R}}\), \(p>1\), with the initial condition \(u(0,x)=\phi (x)\), \(x\in {\mathbb{R}}\) has a regularizing effect on its solution. More precisely, if the initial function \(\phi\) is in \(H^ 1({\mathbb{R}})\) (or in \(L^ 2({\mathbb{R}}))\) and decays sufficiently rapidly as \(| x| \to +\infty\), then u(t,x) is smooth for \(t\neq 0\).
scientific article; zbMATH DE number 4000493
scientific article