## Abstract

In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of type :

$$\begin{array}{c}\left(A\right)i{u}_{t}+{u}_{xx}=\mathcal{N}(u,\overline{u},{u}_{x}\overline{{u}_{x}}),t\in \mathrm{\mathbb{R}},x\in \mathrm{\mathbb{R}};\\ u(x,0)={u}_{0}(x),x\in \mathrm{\mathbb{R}},\end{array}$$

where

$$\mathcal{N}(u,\overline{u},{u}_{x},\overline{{u}_{x}})={K}_{1}|u{|}^{2}u+{K}_{2}|u{|}^{2}{u}_{x}+{K}_{3}{u}^{2}\overline{{u}_{x}}+{K}_{4}|{u}_{x}{|}^{2}u+{K}_{5}\overline{u}{u}_{x}^{2}+{K}_{6}|{u}_{x}{|}^{2}{u}_{x},$$ the functions ${K}_{j}={K}_{j}(|u{|}^{2})$ satisfy ${K}_{j}\in {\mathcal{C}}^{\infty}([0,+\infty );\u2102)$.

This equation has been derived from physics. For example if the nonlinear term is $\mathcal{N}(u)=\frac{\overline{u}{u}_{x}^{2}}{1+|u{|}^{2}}$ then equation (A) appears in the classical pseudospin magnet model [18]. The aim of this paper is to study the case : the nonlinearity depends on ${u}_{x}$ and $\overline{{u}_{x}}$, and satisfies the so called Gauge condition : $\mathcal{N}({e}^{i\theta}u)={e}^{i\theta}\mathcal{N}(u)$. We prove that if the initial data ${u}_{0}\in {\mathscr{H}}^{3,l}$ for any $l\in \mathbb{N}$, then there exists a positive time $T>\text{}0$ and a unique solution $u\in {\mathcal{C}}^{\infty}([-T,T]\backslash \{0\};{\mathcal{C}}^{\infty}(\mathbb{R}))$ of the Cauchy problem (A). The result in this paper improves the previous one in [11] because we do not assume any size restriction on the data. Here and ${\Vert \phi \Vert}_{m,s}{=\Vert {(1+{x}^{2})}^{s/2}{(1-{\partial}_{x}^{2})}^{m/2}\phi \Vert}_{{\mathcal{L}}^{2}(\mathbb{R})},{\mathscr{H}}^{m,\infty}={\cap}_{s\ge 1}{\mathscr{H}}^{m,s}$.

## Acknowledgement

The author would like to thank the referee for his comments and Pr. Nakao Hayashi for his advices.

## Citation

Patrick-Nicolas Pipolo. "Smoothing effects for some derivative nonlinear Schrödinger equations without smallness condition." SUT J. Math. 35 (1) 81 - 112, January 1999. https://doi.org/10.55937/sut/991985404

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