A Centre-Stable Manifold for the Focussing Cubic NLS in $${\mathbb{R}}^{1+3}$$

  title={A Centre-Stable Manifold for the Focussing Cubic NLS in \$\$\{\mathbb\{R\}\}^\{1+3\}\$\$},
  author={Marius Beceanu},
  journal={Communications in Mathematical Physics},
  • M. Beceanu
  • Published 26 January 2007
  • Mathematics
  • Communications in Mathematical Physics
Consider the focussing cubic nonlinear Schrödinger equation in $${\mathbb{R}}^3$$ :$$i\psi_t+\Delta\psi = -|\psi|^2 \psi. \quad (0.1) $$It admits special solutions of the form eitαϕ, where $$\phi \in {\mathcal{S}}({\mathbb{R}}^3)$$ is a positive (ϕ > 0) solution of$$-\Delta \phi + \alpha\phi = \phi^3. \quad (0.2)$$The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional… 
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