Nonlocal description of X waves in quadratic nonlinear materials

We study localized light bullets and X waves in quadratic media and show how the notion of nonlocality can provide an alternative simple physical picture of both types of multidimensional nonlinear waves. For X waves we show that a local cascading limit in terms of a nonlinear Schrödinger equation does not exist-one needs to use the nonlocal description, because the nonlocal response function does not converge toward a function. Also, we use the nonlocal theory to show that the coupling to the second harmonic is able to generate an X shape in the fundamental field despite having anomalous dispersion, in contrast to the predictions of the cascading limit.