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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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