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.