Wave turbulence description of interacti

This study investigates the wave turbulence of interacting particles within the framework of the complex Klein-Gordon equation augmented by a Mexican-hat potential. The objective is to derive wave kinetic equations that govern the behavior of two types of excitations, namely massive particles and massless Goldstone bosons, which emerge from the fundamental state. We utilized wave turbulence theory to analyze the coupled evolution of their spectra, first addressing thermodynamic solutions and examining the wave condensation transition as an analogue to Bose-Einstein condensation. Moreover, we explore nonlocal interactions in wavenumber space to comprehend the decay process of massive particles into massless counterparts. Significantly, under typical conditions, the massless particles concentrate at low wavenumber while the dynamics of waves in coexistence with a strong condensate are scrutinized. We derive a nonlocal Kolmogorov-Zakharov solution that illustrates how particles are transferred non-locally to the condensate, while concurrently, energy cascades towards larger wavenumbers through local interactions. Our findings reveal the nonlocal cascading state as a critical intermediate asymptotic between the initial wave distribution and the long-term thermodynamic state.