论文标题
3D偶极bose-Einstein冷凝物具有吸引人的三体相互作用的归一化基态
Normalized ground states for 3D dipolar Bose-Einstein condensate with attractive three-body interactions
论文作者
论文摘要
我们研究了具有有吸引力的三体相互作用的3D偶性玻璃纤维凝结方程的归一化基态的存在：\ begin {align} \ label {1}-ΔU+βU+βU+βU+βU+λ_1| U |^2 |^2 U+λ_2（k*| U |^2） $λ_2= 0 $或$ u $是径向，（\ ref {1}）还原为立方 - Quintic nls \ begin {align} \ label {2}-ΔU+βU+βU+βU+βU+βu+βU+λ_1| [31]。特别是，显示出任何$λ_1<0 $和$ c> 0 $，（\ ref {2}）具有带有质量$ c $的径向对称基础状态解决方案，对于$λ_1\ geq 0 $，（\ ref Ref {2}），没有非整体解决方案。我们表明，通过将偶极 - 偶极相互作用添加到（\ ref {2}）中，（\ ref {2}）的几何性质发生了巨大变化，并且技术不再使用[31]的技术来获得相似的结果。更准确地说，由于偶极 - 偶极相互作用电位的轴对称性质，与（\ ref {1}）相对应的能量在对称重排下不稳定，因此基于溶液的径向对称性的常规参数是不可应用的。我们将通过吸引微妙的变分和扰动方法来克服这一困难，并证明以下内容： （i）如果对$（λ_1，λ_2）$不稳定且$λ_1<0 $，则对于任何$ c> 0 $，（\ ref {1}）具有带有质量$ c $的基态解决方案。 （ii）如果对$（λ_1，λ_2）$不稳定，$λ_1\ geq 0 $，则存在一些$ c^*= c^*（λ_1，λ_2）\ geq 0 $，因此对于所有$ c> c> c^*$，（\ c^$ c $ c $ c $ co $ c $ co $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c。此外，在这种情况下，（\ ref {1}）的任何非平凡解决方案都必须是非主题的。 （iii）如果对$（λ_1，λ_2）$是\ textit {stable}，则（\ ref {1}）没有非平凡的解决方案。
We study the existence of normalized ground states for the 3D dipolar Bose-Einstein condensate equation with attractive three-body interactions: \begin{align}\label{1} -Δu+βu+λ_1|u|^2 u+λ_2 (K*|u|^2)u-|u|^4u=0.\tag{DBEC} \end{align} When $λ_2=0$ or $u$ is radial, (\ref{1}) reduces to the cubic-quintic NLS \begin{align}\label{2} -Δu+βu+λ_1|u|^2 u-|u|^4u=0\tag{CQNLS}, \end{align} which has been recently studied by Soave in [31]. In particular, it was shown that for any $λ_1<0$ and $c>0$, (\ref{2}) possesses a radially symmetric ground state solution with mass $c$ and for $λ_1\geq 0$, (\ref{2}) has no non-trivial solution. We show that by adding a dipole-dipole interaction to (\ref{2}), the geometric nature of (\ref{2}) changes dramatically and techniques as the ones from [31] cannot be used anymore to obtain similar results. More precisely, due to the axisymmetric nature of the dipole-dipole interaction potential, the energy corresponding to (\ref{1}) is not stable under symmetric rearrangements, hence conventional arguments based on the radial symmetry of solutions are inapplicable. We will overcome this difficulty by appealing to subtle variational and perturbative methods and prove the following: (i) If the pair $(λ_1,λ_2)$ is unstable and $λ_1<0$, then for any $c>0$, (\ref{1}) has a ground state solution with mass $c$. (ii) If the pair $(λ_1,λ_2)$ is unstable and $λ_1\geq 0$, then there exists some $c^*=c^*(λ_1,λ_2)\geq 0$ such that for all $c>c^*$, (\ref{1}) has a ground state solution with mass $c$. Moreover, any non-trivial solution of (\ref{1}) in this case must be non-radial. (iii) If the pair $(λ_1,λ_2)$ is \textit{stable}, then (\ref{1}) has no non-trivial solutions.
