Attractive Bose-Einstein condensates are investigated with numerical continuation methods capturing stationary solutions of the Gross-Pitaevskii equation. The branches of stable (elliptic) and unstable (hyperbolic) solutions are found to meet at a critical particle number through a generic Hamiltonian saddle node bifurcation. The condensate decay rates corresponding to macroscopic quantum tunneling, two and three body inelastic collisions, and thermally induced collapse are computed from the exact numerical solutions. These rates show experimentally significant differences with previously published rates. Universal scaling laws stemming from the bifurcation are derived. [S0031-9007(99)08550-6] PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 47.20.Ky Experimental Bose-Einstein condensation (BEC) in ultracold vapors of 7 Li atoms [1] opened a new field in the study of macroscopic quantum phenomena. Condensates with attractive interactions are known to be metastable in spatially localized systems, provided that the number of condensed particles is below a critical value N c [2]. Various physical processes compete to determine the lifetime of attractive condensates. Among them one can distinguish macroscopic quantum tunneling (MQT) [3,4], inelastic two and three body collisions (ICO) [5][6][7], and thermally induced collapse (TIC) [4,8]. The MQT and TIC contributions were evaluated in the literature using a variational Gaussian approximation to the condensate density. However, this approximation is known to be in substantial quantitative error-e.g., as high as 17% on N c [3,9]-when compared to the exact solution of the Gross-Pitaevskii (GP) equation. Experimentally, the recent observations of Feshbach resonances in BEC of sodium atoms offer new possibilities to investigate the dynamics of condensates with negative scattering lengths close to zero temperature (in the nK range) [10]. Reliable theoretical evaluations of the lifetime of metastable condensates are thus needed for quantitative comparisons with experiments.The basic goal of the present Letter is to numerically compute the bifurcation diagram of the stationary solutions of the GP equation. Both the stable (elliptic) and unstable (hyperbolic) branches of solutions will then be used to obtain decay rates and compare them to the known (Gaussian approximation) ones. At low enough temperature, neglecting the thermal and quantum fluctuations, a Bose condensate can be represented by a complex wave function c͑x, t͒ that obeys the dynamics of the GP equation [11,12]. Specifically, we consider a condensate of N particles of mass m and (negative) effective scattering length a in a radial confining harmonic potential V ͑r͒ mv 2 r 2 ͞2. Using variables rescaled by the natural quantum harmonic oscillator units of time t 0 1͞v and length L 0 ph ͞mv:t t͞t 0 ,x x͞L 0 , andã 4pa͞L 0 , the condensate is described by the action

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We report the experimental observation of an endogeneous antisymmetric wave source in a quasione-dimensional chemical system. Substantiated by numerical simulations, a theoretical interpretation relying on the interaction between Turing and Hopf modes is proposed.