| Abstract: |
| The 1D cubic defocusing nonlinear Schr\odinger equation (NLS) or Gross-Pitaevskii equation
$$i\partial_t u + \partial_x^2u + 2(1-|u|^2)u=0$$
supports dark solitons of the form $u(x,t) = q_\beta(x-2t\sin \beta)$, where $q_\beta(x) =\cos \beta \tanh(x\cos \beta)+i\sin \beta$. These solutions have the spatial asymptotic behavior $|u(x,t) \to 1$ as $x\to \pm \infty$. Orbital stability of dark solitons have been studied in several papers over the last twenty years. Here, we introduce a slightly different method based on the Lyapunov method that avoids working in hydrodynamical (amplitude/phase) coordinates. |
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