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Orbital stability of the black soliton for the Gross-Pitaevskii equation. (English) Zbl 1171.35012
The authors of this interesting paper consider the one-dimensional Gross-Pitaevskii equation \[ i\Psi_t + \Psi_{xx}=\Psi (|\Psi |^2-1) , \quad (t,x)\in \mathbb R \times \mathbb R , \] which is a version of the defocusing cubic nonlinear Schrödinger equation. The boundary condition is given at infinity \(|\Psi (x,t)|\to 1\), as \(|x|\to +\infty \). The conserved Hamiltonian is a Ginzburg-Landau energy \[ E(\Psi )=(1/2)\int_{\mathbb R }|\Psi '|^2dx + (1/4)\int_{\mathbb R }(1-|\Psi |^2)^2dx . \] The authors establish the orbital stability of the black soliton, or kink solution, that is, \(v_0=\tanh{(x/\sqrt{2})}\), with respect to perturbations in the energy space.

MSC:
35B35 Stability in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
35Q51 Soliton equations
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