This book highlights the dynamical behavior of self-similar waves in asymmetric dual-core waveguides. The proposed dual-core waveguide consists of two closely spaced adjoining fibers in which one fiber is active and the other is passive. Due to the linear coupling between them, the dynamics of the wave propagating through the passive core can be controlled by manipulating the dynamics of the wave propagating in the active core.  The optimal pulse compression or amplification of these waves as the length of the fiber tends to infinity is presented. The exact Mobius transform self-similar solutions that propagate through these waveguides self-similarly are subject to simple scaling rules. The book includes experiments conducted to corroborate the analytical predictions.