## A class of localized solutions of the linear and nonlinear wave equations.(English)Zbl 1271.78043

The paper revisits the possibility of constructing localized exact solutions to $$(3+1)$$-dimensional linear wave equations (in optics, such solutions may represent linear version of “light bullets”, i.e., three-dimensional solitons). The solutions are constructed by means of the Fourier transform, which is a natural method to apply to linear equations with constant coefficients. In particular, solutions in the form of expanding Gaussians and several types of expanding algebraically localized solutions are produced. The solutions are characterized by finite energies, but none of them is stationary, as linear equations cannot support stationary localized modes. In addition to that, spherically symmetric localized solutions are also given for a three-dimensional wave equation with a cubic nonlinearity.

### MSC:

 78A60 Lasers, masers, optical bistability, nonlinear optics 35C07 Traveling wave solutions 35Q51 Soliton equations

### Keywords:

Gaussian; Fourier transform; spherical harmonics; light bullets