Derivation of Nonlinear Schr\"{o}dinger Equation
  We propose some nonlinear Schr\"{o}dinger equations by adding some higher order terms to the Lagrangian density of Schr\"{o}dinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schr\"{o}dinger equation is weak. 
Xiang-Yao Wu, Bai-Jun Zhang, Xiao-Jing Liu, Li-Xiao, Yi-Heng Wu,   Yan-Wang, Qing-Cai Wang and Shuang Cheng
math-ph math.MP quant-ph