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


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

A  iut+uxx=N(u,u¯,uxux¯), t,x;u(x,0)=u0(x), x,


N(u,u¯,ux,ux¯)=K1|u|2u+K2|u|2ux+K3u2ux¯+K4|ux|2u+K5u¯ux2+K6|ux|2ux, the functions Kj=Kj(|u|2) satisfy KjC([0,+);).

This equation has been derived from physics. For example if the nonlinear term is N(u)=u¯ux21+|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 ux and ux¯, and satisfies the so called Gauge condition : N(eiθu)=eiθN(u). We prove that if the initial data u03,l for any l, then there exists a positive time T> 0 and a unique solution uC([T,T]\{0};C()) 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 m,s={φ2();φm,s<+} and φm,s=(1+x2)s/2(1x2)m/2φ2(),m,=s1m,s.


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


Download Citation

Patrick-Nicolas Pipolo. "Smoothing effects for some derivative nonlinear Schrödinger equations without smallness condition." SUT J. Math. 35 (1) 81 - 112, January 1999.


Received: 1 March 1999; Published: January 1999
First available in Project Euclid: 18 June 2022

Digital Object Identifier: 10.55937/sut/991985404

Primary: 35Q55

Keywords: Schrödinger equation , smoothing effects

Rights: Copyright © 1999 Tokyo University of Science

Vol.35 • No. 1 • January 1999
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