## Abstract

In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of derivative type: iu_{t} + u_{xx} = N(u, ū, u_{x}, ū_{x}), t ∈ R, x ∈ R; u(0, x) = u_{0}(x), x ∈ R, (A) where N(u, ū, u_{x}, ū_{x}) = K_{1}\u\^{2}u + K_{2}\u\^{2}u_{x} + K_{3}u^{2}ū_{x} + K_{4}\u_{x}\^{2}u + K_{5}ūu^{2}_{x} + K_{6}\u_{x}\^{2}u_{x}, the functions K_{j} = K_{j}(\u\^{2}), K_{j}(z) ∈ C^{∞}([0,∞)). If the nonlinear terms N = ūu^{2}_{x}/1+|u|^{2} then equation (A) appears in the classical pseudospin magnet model [16]. Our purpose in this paper is to consider the case when the nonlinearity N depends both on u_{x} and ū_{x}. We prove that if the initial data u_{0} ∈ H^{3,∞} and the norms ||u_{0}||3,l are sufficiently small for any l ∈ N, (when N depends on ū_{x}), then for some time T > 0 there exists a unique solution u ∈ C^{∞}([-T,T]\{0};C^{∞}(R)) of the Cauchy problem (A). Here H^{m,s} = {φ ∈ L^{2}; ||φ||_{m,s} < ∞}, ||φ||_{m,s} = ||(1 + x^{2})^{s/2}(1 - ∂^{2}_{x})^{m/2}φ||_{L2}, H^{m,∞} = ∩_{s≥1}H^{m,s}.

Original language | English |
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Pages (from-to) | 685-695 |

Number of pages | 11 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 5 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1999 Jul |

## Keywords

- Derivative type
- Nonlinear Schrödinger
- Smoothing effects