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Travelling waves for the Gross-Pitaevskii equation in dimension larger than two. (English) Zbl 1054.35091
The article deals with the existence of travelling vortex ring solutions
$$\psi(x,t)=U(x_1-Ct, x_2,\dots,x_N)$$ of the Gross-Pitaevskii equation $i\;\frac{\partial \psi}{\partial t}+\Delta \psi+(1-|\psi|^2)=0\quad (\psi:\mathbb{R}^N\times \mathbb{R}\to\mathbb{C},\quad N\geq 3).$ The equation on $$U$$ reads $$iC\partial U/\partial x_1 =\Delta U + (1- | H|^2)U$$ and the result is that there exists $$C_0>0$$ such that a nontrivial solution $$U_C$$ exists if $$0<C<C_0$$ with $P(U_C)/2\pi \varepsilon^{1-N}\to |\mathbb{B}^{N-1}|,\quad E(U_C)/\pi\varepsilon^{2-N}|\log\varepsilon|\to\| S^{N-2}\|$ if $$C\to 0$$, where $$\varepsilon$$ is defined by $$C = (N-2)\varepsilon|\log\varepsilon|$$. Here $E(\psi)=\tfrac12 \int|\nabla\psi|^2 +\tfrac12(1-|\psi|^2)\,dx,\quad P(\psi)=\text{Im}\int\psi\overline{\nabla\psi}\,dx$ are the energy and the momentum. The functional $$F(U) = E(U) - CP(U)$$ with his mountain pass critical point are employed.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 76B47 Vortex flows for incompressible inviscid fluids 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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