Semiclassical evolution of two rotating solitons for the nonlinear Schrödinger equation with electric potential
The author of this interesting paper studies the dynamics in semiclassical limit of some classes of solutions for an energy-subcritical focusing nonlinear Schrödinger (NLS) equation in the presence of an electric potential. It has the form \[ i\varepsilon \phi (t,x)= -\varepsilon^2 \Delta \phi (t,x) + E(x)\phi (t,x) - |\phi (t,x)|^{p-1}\phi (t,x), \tag{\(NLS_\varepsilon\)} \] where \((t,x)\in \mathbb R\times \mathbb R^2\), \(p>1\), \(\varepsilon \geq 0\). The electric potential is defined as follows: \(E(x)= 1\) for \(|x|\leq 1\), \(E(x)=|x|^2\) for \( |x|\geq 1+\eta \) (\(\eta << 1\)), and \(E(x)\in C^{\infty }(\mathbb R^n)\), otherwise. The equation \((NLS_{\varepsilon})\) is a model for the motion of a quantum particle in presence of an external electric potential. It appears in several applications such as nonlinear optics, water waves and Bose-Einstein condensates. An interesting case here is the semiclassical limit, namely when the Planck constant \(\varepsilon :=\hbar \) tends to zero. This limit could be interpreted as the correspondence principle between classical mechanics and quantum mechanics. The author proves the existence of a symmetric configuration of two solitons rotating around a common pole. The existence of such solitons derives from the balance between the attracting force of the pair of solitons and the centrifugal force due to the rotation and not from the trapping effect of the potential.
scientific article; zbMATH DE number 5707455