Damping of periodic waves in physically significant wave systems

Damping of periodic waves in the classically important nonlinear wave systems - nonlinear Schrödinger, Korteweg-deVries (KdV), and modified KdV - is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m = m(t). The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m = 0 to soliton waves when m = 1.