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We demonstrate theoretically and experimentally that the nonlinear interaction between excitations whose harmonic energies coincide gives rise to a strong coupling that opens a new coherent ultrafast energy relaxation path. Instead of an incoherent decay of excitations, that should take a finite time that depends on the energy difference between the initial state and the final state, the nonlinear interaction allows their coherent superposition and thus an instantaneous transfer of the excitation over energies as large as the electron-volt. Such a situation should be encountered in many systems. We demonstrate that such a model applies also for excitons in single-wall carbon nanotubes (SWCNT) where a strong nonlinear Coulomb interaction occurs between $\rm E_{11}$ and $\rm E_{22}$ states. This explains a wide panel of observations about optoelectronic properties of the SWCNT and gives a coherent picture of their features like the exciton-line spectral positions, exciton collisions and their ultrafast relaxation, as well as the low light-emission efficiency of the nanotubes., Comment: 17 pages, 6 figures