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A nonuniform condensate is usually described by the Gross-Pitaevskii (GP)equation, which is derived with the help of the c-number ansatz $\hat{\Psi}(\mathbf{r},t)=\Psi (\mathbf{r},t)$. Proceeding from a more accurateoperator ansatz $\hat{\Psi}(\mathbf{r},t)=\hat{a}_{0}\Psi (\mathbf{r},t)\sqrt{N}$, we find the equation $i\hbar \frac{\partial \Psi(\mathbf{r},t)}{\partial t}=-\frac{\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi(\mathbf{r},t)}{\partial \mathbf{r}^{2}}+\left( 1-\frac{1}{N}\right) 2c\Psi(\mathbf{r},t)|\Psi(\mathbf{r},t)|^{2}$ (the GP$_{N}$ equation). It differsfrom the GP equation by the factor $(1-1/N)$, where $N$ is the number of Boseparticles. We compare the accuracy of the GP and GP$_{N}$ equations byanalyzing the ground state of a one-dimensional system of point bosons withrepulsive interaction ($c>0$) and zero boundary conditions. Both equations aresolved numerically, and the system energy $E$ and the particle density profile$\rho (x)$ are determined for various values of~$N$, the mean particle density$\bar{\rho}$, and the coupling constant $\gamma =c/\bar{\rho}$. The solutionsare compared with the exact ones obtained by the Bethe ansatz. The results showthat in the weak coupling limit ($N^{-2}\ll \gamma \lesssim 0.1$), the GP andGP$_{N}$ equations describe the system equally well if $N\gtrsim 100$. Forfew-boson systems ($N\lesssim 10$) with $\gamma \lesssim N^{-2}$ the solutionsof the GP$_{N}$ equation are in excellent agreement with the exact ones. Thatis, the multiplier $(1-1/N)$ allows one to describe few-boson systems with highaccuracy. This means that it is reasonable to extend the notion ofBose-Einstein condensation to few-particle systems.

Nonuniform Bose-Einstein condensate. I. An improvement of the Gross-Pitaevskii method