On dipolar quantum gases in the unstable regime (Q2814472)

The authors study the Gross-Pitaevskii equation NEWLINE\[NEWLINEi\partial_t\psi=-\frac{1}{2}\Delta_x \psi+\frac{a^2}{2}|x|^2\psi+\lambda_1 |\psi|^2\psi+\lambda_2(K*|\psi|^2)\psiNEWLINE\]NEWLINE for \(x\in\mathbb{R}^3\), \(t>0\). The equation describes a Bose-Einstein condensate made of particles possessing a permanent electric or magnetic dipole moment, a so-called dipolar BEC. The dipole interaction potential \(K\) is given by \(K(x)=(1-3\cos^2 \theta)/|x|^3\) where \(\theta\) is the angle between \(x\) and the dipole axis. The physical parameters \(\lambda_1, \lambda_2\in \mathbb{R}\) lie in the unstable regime [\textit{R. Carles} et al., Nonlinearity 21, No. 11, 2569--2590 (2008; Zbl 1157.35102)]. The paper is concerned with standing wave solutions \(\psi(x,t)=e^{-i\mu t} u(x)\) of \((1)\) with prescribed \(L^2\)-norm \(||\psi(\cdot , t)||^2_2=c\).NEWLINENEWLINEThe first result states the existence of such a standing wave for \(a=0\), any \(c<0\). It is obtained via variational methods for the associated functional NEWLINE\[NEWLINE E(u)=\frac{1}{2}\int_{\mathbb{R}^3} |\nabla u|^2+\frac{\lambda_1}{2}\int_{\mathbb{R}^3} |u|^4+\frac{\lambda_2}{2}\int_{\mathbb{R}^3}(K*|u|^2)|u|^2 NEWLINE\]NEWLINE constrained to \(S(c)=\{u\in H^1(\mathbb{R}^3):||u||^2_2=c\}\). In fact, \(u\) is a mountain pass critical point of \(E\) on \(S(c)\). It is also proved that the corresponding solution \(\psi\) is unstable.NEWLINENEWLINEThe second part of the paper deals with the case \(a>0\). Here it is proved that for \(c>0\) fixed and \(a>0\) small equation (1) has two standing wave solutions, one being orbitally stable.