The presented research proposes the $α$-modification of the Beaulieu-Xie shadowed fading channel for wireless communications. For the assumed model the closed-form analytical description of the basic statistical characteristics is carried out (i.e., probability density function, cumulative distribution function, and their asymptotics). The derived statistical description is exemplified on the problems of the average bit error rate and ergodic capacity calculation, for which the exact analytic and high signal-to-noise ratio asymptotic expressions are derived. The performed extensive numerical analysis demonstrates high correspondence with the analytical work and helps to study the dependence of the channel nonlinearity effects on the bit error probability.