Forced KdV Equation

Abstract

In this chapter we study the forced Korteweg-de Vries equation (fKdV) :

$$ {u_{t}} + \lambda {u_{x}} + 2\alpha u{u_{x}} + \beta {u_{{xxx}}} = f'(x), - \infty < x < \infty $$

where λ, α < 0 and β < 0 are constants, and f(x) is a given function (called the forcing) which is differentiate and has a compact support (i.e. it is nonzero only in a closed bounded set). This equation is an asymptotically reduced result from Euler equations of fluid motion and corresponding boundary conditions. The unknown function u(x, t) represents the first order elevation of the free surface of the fluid. The forcing function f(x) is due to the bottom topography of a fluid domain (such as a bump on the bottom of a two dimensional channel), or due to an external pressure on the free surface (such as the wind stress on the surface of an ocean). Solutions of this fKdV are characterized according to the value of λ. We will show that there exist two values of λ (λ _L < 0, λ _C > 0) such that

• (a) when λ ≥ λ _C the fKdV admits at least two stationary solitary wave solutions and λ = λ _C is the turning point of the bifurcation curve;

• (b) when λ ≤ λ _L , the fKdV admits only one downstream cnoidal wave solution and λ = λ _L is the cut-off point at which the cnoidal wave becomes a hydraulic fall;

• (c) when λ _L < λ < λ _C , the fKdV admits no steady state solutions and solitons are periodically generated at the site of forcing and radiated upstream.