I am interested in finding a solution of the NLS equation (defocusing): $$i \partial_t u = -\frac 12 u_{xx} + |u|^2 u$$ I hope to find a solution that is somewhat similar to that of the NLS equation (focusing): $$i \partial_t u = -\frac 12 u_{xx} - |u|^2 u$$ I'll outline the procedure for this focusing case here--
- We may consider a solution of the form $u = v(x) e^{it}$ where $v(x) \in \mathbb R$. Plugging this in, we arrive at the ODE for $v$: $$-v = -\frac 12 v'' - v^3$$ Which we can rewrite as: $$\frac 12 \left((v')^2\right)' = v'\partial_v (v^2 - \frac 12 v^4)$$ Integrating both sides (and using B.C. vanishing at $\pm \infty$), we arrive at: $$\frac{dv}{dx} = \pm\sqrt{2 v^2 - v^4} $$ Rewriting, we have: $$\int \frac{dv}{\sqrt{2v^2 - v^4}}= -\frac{\tanh ^{-1}\left(\sqrt{1-2 v^{2}}\right)}{\sqrt{2}} = \pm x + c_0$$ For a positive soliton centered at the origin, we then have: $$v(x) = \sqrt{2}\operatorname{sech}(\sqrt{2}\cdot x)$$
Now, going back to the defocusing case, if we likewise propose an ansatz of the form $u = v(x) e^{it}$ where $v(x)$ vanishes at $\pm \infty$, we have $v(x)$ solves the ODE: $$v + v^3= \frac 12 v''$$ But multiplying both sides by $v$ and integrating on $\mathbb R$, we have: $$\int_\mathbb{R} v^2 + v^4 + \frac 12 v'^2 dx = 0 \implies v(x) = 0$$ Hence, for this case, there is no such non-trivial solution of this ODE if $v$ is real.
Now, my question is this: is there an alternative ansatz I can use here for the defocusing case that would yield an analogous result to the focusing case? Indeed, if we try: $v(x) = i \sqrt{2} \operatorname{sech}(\sqrt{2} \cdot x)$ this solves the ODE, but it fails for the ansatz (it is not real, so the $ | \cdot |^2$ term causes issues).