From Ginzburg-Landau to Gross-Pitaevskii

In this note we consider the Gross-Pitaevskii equation iphi(t) + Deltaphi + phi(1 - \phi\(2)) = 0, where phi is a complex-valued function defined on R-N x R, and study the following 2-parameters family of solitary waves: phi(x, t) = e(iomegat)(x(1) - ct, x'), where (omega, c) is an element of R-2 nu is an element of L-loc(3) (R-N, C) amd x' denotes the vector of the last N = 1 variables in R-N. We prove that every distribution solution phi, of the considered form, satisfies the following universal (and sharp) L-infinity-bound: parallel tophiparallel to(Linfinity)(2)(R-N x R) less than or equal to max {0,1 - omega + c(2)/4}. This bound has two consequences. The first one is that phi is smooth and the second one is that a solution phi not equivalent to 0 exists, if and only if 1 - omega + c(2)/4 > 0 We also prove a non-existence result for some solitary waves having finite energy. Some more general nonlinear Schrodinger equations are considered in the third and last section. The proof of our theorems is based on previous results of the author ([7]) concerning the Ginzburg-Landau system of equations in R-N.