In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of type :
the functions satisfy .
This equation has been derived from physics. For example if the nonlinear term is then equation (A) appears in the classical pseudospin magnet model . The aim of this paper is to study the case : the nonlinearity depends on and , and satisfies the so called Gauge condition : . We prove that if the initial data for any , then there exists a positive time and a unique solution of the Cauchy problem (A). The result in this paper improves the previous one in  because we do not assume any size restriction on the data. Here and .
The author would like to thank the referee for his comments and Pr. Nakao Hayashi for his advices.
"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