Spontaneously generated walking X-shaped light bullets

In quadratic nonlinear media with normal dispersion and nonvanishing group velocity mismatch between fundamental wave and second-harmonic pulses, the wave packets of two harmonics can be locked together in propagation in the form of walking X-shaped light bullets. The output wave packets are developed into X-shaped light bullets with significant group delay time due to mutual dragging and significant shifting in spatiotemporal spectrum due to delayed nonlinear phase shift. We also show that spontaneously generated phase-front tilting can balance the dragging force induced by group velocity mismatch and lead to zero-velocity walking X-shaped light bullets.