Derivation of Nonlinear Schrödinger Equation

We propose some nonlinear Schrodinger equations by adding some higher order terms to the Lagrangian density of Schrodinger 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 Schrodinger equation is weak.