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تسجيل مستخدم جديدGeneralized Adler-Moser Polynomials and Multiple vortex rings for the Gross-Pitaevskii equation
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New finite energy traveling wave solutions with small speed are constructed for the three dimensional Gross-Pitaevskii equation begin{equation*} iPsi_t= Delta Psi+(1-|Psi|^2)Psi, end{equation*} where $Psi$ is a complex valued function defined on ${mathbb R}^3times{mathbb R}$. These solutions have the shape of $2n+1$ vortex rings, far away from each other. Among these vortex rings, $n+1$ of them have positive orientation and the other $n$ of them have negative orientation. The location of these rings are described by the roots of a sequence of polynomials with rational coefficients. The polynomials found here can be regarded as a generalization of the classical Adler-Moser polynomials and can be expressed as the Wronskian of certain very special functions. The techniques used in the derivation of these polynomials should have independent interest.
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