Gamow vectors explain the shock profile

The description of shock waves beyond the shock point is a challenge in nonlinear physics and optics. Finding solutions to the global dynamics of dispersive shock waves is not always possible due to the lack of integrability. Here we propose a new method based on the eigenstates (Gamow vectors) of a reversed harmonic oscillator in a rigged Hilbert space. These vectors allow analytical formulation for the development of undular bores of shock waves in a nonlinear nonlocal medium. Experiments by a photothermal induced nonlinearity confirm theoretical predictions: the undulation period as a function of power and the characteristic quantized decays of Gamow vectors. Our results demonstrate that Gamow vectors are a novel and effective paradigm for describing extreme nonlinear phenomena.