Shallow water cnoidal wave interactions

The nonlinear dynamics of cnoidal waves, within the context of the general N-cnoidal wave solutions of the periodic Korteweg-deVries (KdV) and Kadomtsev-Petvishvilli (KP) equations, are considered. These equations are important for describing the propagation of small-but-finite amplitude waves in shallow water; the solutions to KdV are unidirectional while those of KP are directionally spread. Herein solutions are constructed from the theta-function representation of their appropriate inverse scattering transform formulations. To this end a general theorem is employed in the construction process: All solutions to the KdV and KP equations can be written as the linear superposition of cnoidal waves plus their nonlinear interactions. The approach presented here is viewed as significant because it allows the exact construction of N degree-of-freedom cnoidal wave trains under rather general conditions.