From Ginzburg-Landau to Gross-Pitaevskii

Abstract

In this note we consider the Gross-Pitaevskii equation iϕ t +Δϕ+ϕ(1−∣ϕ∣2)=0, where ϕ is a complex-valued function defined on ℝN × ℝ, and study the following 2-parameters family of solitary waves: ϕ(x, t)=e iωt v(x 1−ct, x′), where $\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}(\omega, c)\in {\Op R}^2, {\upsilon} \in L_{\rm loc}^3 ({\Op R}^N, {\bf C})$ and x′ denotes the vector of the last N−1 variables in ℝ N . We prove that every distribution solution ϕ, of the considered form, satisfies the following universal (and sharp) L ∞-bound: $$\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}\Vert\varphi\Vert_{L^\infty({\Op R}^N\times{\Op R})}^2 \le \max\bigg\{0, 1-\omega+{c^2\over 4}\bigg\}.$$ This bound has two consequences. The first one is that ϕ is smooth and the second one is that a solution φ≢0 exists, if and only if $1-\omega+{c^2\over 4} > 0$ . 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 .