MULTIPLE SUPERCRITICAL SOLITARY WAVE SOLUTIONS OF THE STATIONARY FORCED KORTEWEG-DE VRIES EQUATION AND THEIR STABILITY

The first-order approximation of long nonlinear surface waves in a cha nnel flow of an inviscid, incompressible fluid over a bump results in a forced Korteweg-de Vries equation (fKdV): eta(t) + lambda eta(x) + 2 alpha eta eta(x) + beta eta(xxx) = f(x)(x), -infinity < x < infinity, t > 0. The forcing represented by the function f(x) in the fKdV equat ion is due to the bump on the bottom of the channel. In this paper, th e solitary wave solutions of the stationary fKdV equation (sfKdV) are studied. The supercritical solitary wave solutions of the sfKdV equati on exist only when the upstream flow velocity c is greater than a cru cial value u(c) > root gH, or equivalently, lambda > lambda(c) > 0. Th e existence of supercritical positive solitary wave solutions (SPSWS) of the sfKdV equation is proved. Some ordered properties and extreme p roperties of SPSWS are discussed. There may exist more than two SPSWS for a nonlocal forcing. An analytic expression of the SPSWS is found w hen the forcing is a rectangular bump or dent (called the well-shape f orcing). Analytic solutions explicitly reveal the multiplicity of solu tions and make the complicated sfKdV bifurcation behavior more transpa rent. Multiple SPSWS are also found numerically when the forcing is a partly negative and partly positive bump, and two semi-elliptic bumps, respectively, Numerical simulations show that only one of the four SP SWS for a well-shape forcing is stable.