Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation (Q2788608)

The defocusing Gross-Pitaevskii equation (GPE) in one dimension NEWLINE\[NEWLINEi\partial_t \Psi +\partial_{xx}\Psi+\Psi(1-|\Psi|^2)=0NEWLINE\]NEWLINE supports dark solitons in the form of hyperbolic tangent functions. The soliton family is parametrized by the velocity \(c\in(-\sqrt{2},\sqrt{2})\). This paper proves an asymptotic stability result for dark solitons of nonzero velocity. The selected phase space is the energy space \(\chi(\mathbb{R}):=\{\Psi\in H^1_{\text{loc}}(\mathbb{R}): \Psi'\in L^2(\mathbb{R}), 1-|\Psi|^2\in L^2(\mathbb{R})\}\), which is a complete metric space when equipped with the distance NEWLINE\[NEWLINEd(\Psi_1,\Psi_2):=\|\Psi_1-\Psi_2\|_{L^\infty([-1,1])}+\|\Psi'_1-\Psi'_2\|_{L^2(\mathbb{R})}+\||\Psi_1|-|\Psi_2|\|_{L^2(\mathbb{R})}.NEWLINE\]NEWLINE Building on their previous result on the orbital stability of the dark solitons, the authors prove asymptotic stability, i.e. for initial conditions sufficiently close to a soliton with velocity \(c\) the solution converges in time to a soliton with a possibly different nonzero velocity. The convergence is in \(L^\infty_{\text{loc}}(\mathbb{R})\) for \(\Psi\) and weakly in \(L^2(\mathbb{R})\) for \(\partial_x\Psi\).NEWLINENEWLINEThe proof makes no use of integrability of the equation or of the explicit form of the dark solitons. The main steps are, firstly, rewriting the GPE in the, so called, hydrodynamical formulation, for which orbital stability is known. This is a formulation in terms of the variables \(1-\rho^2\) and \(-\partial_x\varphi\), where \(\rho\) and \(\varphi\) are the modulus and phase of \(\Psi\). Next, the authors prove asymptotic stability of the hydrodynamical form. Here the main work dwells in replacing the uniform estimates on the deviation from the soliton and on the spatial shift and velocity, provided in the orbital stability result, by convergence results for \(t\to \infty\). Finally, the asymptotic stability result is transferred to the original formulation of the GPE.