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First order asymptotics for the travelling waves in Gross-Pitaevskii equation. (English) Zbl 1102.35029
The Gross-Pitaevskii equation (GP) is a relevant model in many domains of physics, e.g. Bose-Einstein condensation, superconductivity, superfluidity, nonlinear optics etc. The author gives a derivation of the asymptotic expansion for the solution of the GP in any dimension. The first-order asymptotics at infinity is computed in the general case. The $$\Gamma$$ functions which enter in many formulas come from the formula for the area of $$\mathcal{S}^{N-1}$$ [G. B. Folland, Real analysis. Modern techniques and their applications. Pure and Applied Mathematics, New York etc.: Wiley (1984; Zbl 0549.28001)]. The general asymptotic formula in theorem 2 contains the $$N$$-dimensional Ginzburg-Landau energy and the $$N$$-dimensional momentum. The proof uses Fourier transforms of tempered distributions.

##### MSC:
 35C20 Asymptotic expansions of solutions to PDEs 35Q40 PDEs in connection with quantum mechanics 35Q51 Soliton equations 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs