Multiple Supercritical Solitary Wave Solutions of the Stationary Forced Korteweg-de Vries Equation and Their Stability

Abstract

The first-order approximation of long nonlinear surface waves in a channel flow of an inviscid, incompressible fluid over a bump results in a forced Korteweg–de Vries equation (fKdV): \[ \eta _t + \lambda _{\eta x} + 2\alpha \eta \eta _x + \beta \eta _{xxx} = f_x ( x ),\quad - \infty < x < \infty , t > 0. \]. The forcing represented by the function $f( x )$ in the fKdV equation is due to the bump on the bottom of the channel. In this paper, the solitary wave solutions of the stationary fKdV equation (sfKdV) are studied. The supercritical solitary wave solutions of the sfKdV equation exist only when the upstream flow velocity $c^* $ is greater than a crucial value $u_c > \sqrt {gH} $, or equivalently, $\lambda > \lambda _c > 0$. The existence of supercritical positive solitary wave solutions (SPSWS) of the sfKdV equation is proved. Some ordered properties and extreme properties of SPSWS are discussed. There may exist more than two SPSWS for a nonlocal forcing. An analytic expression of the SPSWS is found when the forcing is a rectangular bump or dent (called the well-shape forcing). Analytic solutions explicitly reveal the multiplicity of solutions and make the complicated sfKdV bifurcation behavior more transparent. 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 SPSWS for a well-shape forcing is stable.