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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.

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