Stability of forced steady solitary waves

This paper explores the basic mechanism underlying the remarkable phenomenon that a forcing excitation stationary in character and sustained at near resonance in a shallow channel of uniform water depth generates a non-stationary response in the form of a sequential upstream emission of solitary waves. Adopting the forced Korteweg-de Vries (fKdV) model and using two of its steady forced solitary wave solutions as primary flows, the stability of these two transcritical steady motions is investigated, and their bifurcation diagrams relating these solutions to other stationary solutions determined, with the forcing held fixed. The corresponding forcing functions are characterized by a velocity parameter for one, and an amplitude parameter for the other of the steadily moving excitations. The linear stability analysis is first pursued for small arbitrary perturbations of the primary flow, leading to a singular, non-self-adjoint eigenvalue problem, which is solved by applying techniques of matched asymptotic expansions with suitable multiscales for singular perturbations, about the isolated bifurcation points of the parametric space pertaining to the stationary perturbations. The eigenvalues and eigenfunctions are then obtained for the full range of the parameters by numerical continuation of the eigenvalues branching off from the stationary-perturbation solutions that were determined by the local analysis. A highly accurate numerical scheme is developed as required for this purpose. The linear stability analysis identifies three categories of evolution of infinitesimal disturbances superimposed to the steady state; they occur in three different parametric regimes. The first, called periodical bifurcating regime, is characterized by complex eigenvalues, with a real part much smaller than the imaginary part, signifying that small departures from the steady state will oscillate with an amplitude growing at a slow exponential rate. In the second regime, called the aperiodical bifurcating regime, the eigenvalues are purely real, implying that small departures from the steady state grow exponentially. For the third regime, linear stability theory is unable to find any eigenvalue (including zero) to exist. In this last case, however, a nonlinear analysis based on the functional hamiltonian formulation is possible, with the hamiltonian conserved for forcings of constant velocity, and the steady state is shown to be stable. For this reason, this regime will be called the stable supercritical regime. Finally, extensive numerical simulations using various finite difference schemes are carried out to find how the solution evolves once the instability of the solution manifests, with results fully confirming the predictions obtained analytically for the various regimes. The numerical simulations show that the instability in the periodical bifurcating regime, for the type of forcings considered, causes the steady solutions to evolve into the phenomenon of periodical production of upstream-advancing solitary waves.