<p>Nonlinear structures are an important phenomenon observed in the channels of field-effect transistors (FETs). This study investigates the propagation characteristics of nonlinear waves in the channels of nano-FETs under external magnetic fields and quantum effects. A KdV Burgers equation governing the evolution of shock waves and a KdV equation describing the features of solitary waves are derived by the reductive perturbation method. Both numerical results and theoretical analysis reveal that there are two types of nonlinear waves, shock waves and solitary waves, in the channels of FETs, as well as transitions not only between shock waves and solitary waves but also between oscillatory shock waves and monotone shock waves. The external magnetic field weakens the oscillations of shock waves. These findings significantly enhance our understanding of plasma dynamics and nonlinear phenomena, offering crucial insights for controlling THz radiation and advancing THz transistor radiation sources.</p>