Exact Analytic Spectra of Asymmetric Modulation Instability in Systems with Self-Steepening Effect

Nonlinear waves become asymmetric when asymmetric physical effects are present within the system. One example is the self-steepening effect. When exactly balanced with dispersion, it leads to a fully integrable system governed by the Chen-Lee-Liu equation. The latter provides a natural basis for the analysis of asymmetric wave dynamics just as nonlinear Schrödinger or Korteweg-de Vries equations provide the basis for analyzing solitons with symmetric profile. In this work, we found periodic wave trains of the Chen-Lee-Liu equation evolved from fully developed modulation instability and analyzed a highly nontrivial spectral evolution of such waves in analytic form that shows strong asymmetry of its components. We present the conceptual basis for finding such spectra that can be used in analyzing asymmetric nonlinear waves in other systems.