An analytical expression for the optical field of an Airy beam partially blocked by an opaque obstacle is derived. The self-healing properties of the Airy beam are studied and discussed in detail. The study shows that the self-healing process of the Airy beam is affected by many factors such as the opaque obstacle size, propagation distance, wavelength, and parameters of the Airy beam. The self-healing process is caused by the convergence of energy from the side to the position of the opaque obstacle and is finished when the convergence of energy flow disappears. When the propagation distance is short, the main lobe of the Airy beam is affected by the obstruction located near the main lobe. When the propagation distance is long, the main lobe of the Airy beam can be affected by the obstruction located far away from the main lobe. The result agrees with the existing results and can be explained by the caustic of the Airy beam.

The effect of Kerr nonlinearity on an Airy beam is investigated by using the nonlinear Schr\"odinger equation. Based on the moments method, the evolution of the Airy beam width in the rms sense is analytically described. Numerical simulations indicate that the central parts of the major lobe of the Airy beam initially give rise to radial compression during propagation in a focusing medium, even though the rms beam width broadens. The partial collapse of the center parts of the major lobe of the beam appear below the threshold for a global collapse. The evolutions of the field distributions of the Airy beams are different during propagation in different Kerr media while the beams still travel along the parabolic trajectory just as the beam propagates in free space.

Analytical propagation expression of an Airy beam in uniaxial crystals orthogonal to the optical axis is derived. The ballistic dynamics of an Airy beam in uniaxial crystals is also investigated. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index. As an example, the propagation of an Airy beam in the positive uniaxial crystals orthogonal to the optical axis is demonstrated. The acceleration of an Airy beam in the transversal direction along the optical axis is more rapidly than that in the other transversal direction. With increasing the ratio of the extraordinary refractive index to the ordinary refractive index, the acceleration of the Airy beam in the transversal direction along the optical axis speeds up and the acceleration of the Airy beam in the other transversal direction slows down. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. The effective beam size of the Airy beam in the transversal direction along the optical axis is always larger than that in the other transversal direction.