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Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. (English) Zbl 1223.35288

The equation under study is \[ i\partial_t u=(-\Delta +V(x))u -|u|^2 u. \] The potential is a double symmetric well which arises among other in quantum tunneling. The nonlinear equation possesses a single bound state when \(\|\psi_{\Omega}\|_2\) is less than some critical value \(N_{cr}\) (bifurcation threshold) and three otherwise. For well-separated wells \(N_{cr}\) is small and the bound state are close to linear bound states.
The author investigate the long time behavior of the solution when the initial state is small and close to one of the linear bound state at the bifurcation threshold. They show that the dynamics is well approximated by the finite dimensional projection spanned by the linear eigenstates.
Several lightning numerical computations are given.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
35P25 Scattering theory for PDEs
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