Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints
The paper studies forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small negative or oscillatory bump on a rigid flat bottom. Such wave motions are determined by a non-dimensional wave speed F, called Froude number, and F = 1 is a critical value of F. If F = 1 + lambda epsilon with a small parameter epsilon > 0, then a forced Korteweg-deVries (FKdV) equation can be derived to model the wave motion on the free surface. In this paper, the case lambda > 0 (or F > 1, called supercritical case) is considered. The steady and unsteady solutions of the FKdV equation with a negative bump function independent of time are first studied both theoretically and numerically. It is shown that there are five steady solutions and only one of them, which exists for all lambda > 0, is stable. Then, solutions of the FKdV equation with an oscillatory bump function posed on R or a finite interval are considered. The corresponding linear problems are solved explicitly and the solutions are rigorously shown to be eventually periodic as time goes to infinity, while a similar result holds for the nonlinear problem posed on a finite interval with small initial data and forcing functions. The nonlinear solutions with zero initial data for any forcing functions in the real line R or large forcing functions in a finite interval are obtained numerically. It is shown numerically that the solutions will become eventually periodic in time for a small forcing function. The behavior of the solutions becomes quite irregular as time goes to infinity, if the forcing function is large.