Nonlinear effects in wave scattering and generation

When a fluid flow interacts with a topographic feature, and the fluid can support wave propagation, there is the potential for waves to be generated upstream and/or downstream. In many cases when the topographic feature has a small amplitude the situation can be successfully described using a linearised theory, and any nonlinear effects are determined as a small perturbation on the linear theory. However, when the flow is critical, that is, the system supports a long wave whose group velocity is zero in the reference frame of the topographic feature, then typically the linear theory fails and it is necessary to develop an intrinsically nonlinear theory. It is now known that in many cases such a transcriticai, weakly nonlinear and weakly dispersive theory leads to a forced Korteweg-de Vries (fKdV) equation. In canonical form, this is $$ - {u_t} - \Delta {u_x} + 6u{u_x} + {u_{xxx}} + {f_x} = 0,$$ where u(x, t) is the amplitude of the critical mode, t is the time coordinate, x is the spatial coordinate, Δ is the phase speed of the critical mode, and f(x) is a representation of the topographic feature.