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Asymptotics for the travelling waves in the Gross-Pitaevskii equation. (English) Zbl 1092.35103
The author gives a rigorous derivation of the asymptotic expansion for the solution of the Gross-Pitaevskii equation (GP) in dimensions 2 and 3. Then a generalization to all dimensions is given. Only subsonic travelling waves of finite energy, with $$0<c<\sqrt{2}$$, are considered. The proof uses the behaviour at infinity of the kernel for the sequence of convolution kernels, which is calculated by Fourier transform. The uniform convergence is proved by the Ascoli-Arzela theorem. The second part is devoted to an improved estimate for the travelling waves in GP. In the third part the existence of first-order asymptotics for the subsonic waves of finite energy is established.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 76G25 General aerodynamics and subsonic flows 35C20 Asymptotic expansions of solutions to PDEs