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At least for an equilibrium situation, where $\dfrac{\partial n}{\partial t}=0$ and $\dfrac{\partial n}{\partial x}=0$, you would easily see that $$d_1 n - d_2 n^2=0,$$ with the two solutions you anticipated. I am not sure what you mean with a soliton solution for an equation like these, as the basis of your equation is not a wave equation. Probably you ...
Why do you say it is not a soliton solution? I did not verify your answer but assuming its true, then it seems like a soliton to me. Your vacuum consists of $\phi=0,\pm 1$. This solution obviously interpolates between the two vacua $\phi(+\infty)\rightarrow +1$ and $\phi(-\infty)\rightarrow -1$, and is therefore topologically stable. Moreover the region ...