The zero dispersion limit of the korteweg‐de vries equation for initial potentials with non‐trivial reflection coefficient

The inverse scattering method is used to determine the distribution limit as ϵ → 0 of the solution u ( x , t , ϵ) of the initial value problem. U t − 6uu x + ϵ 2 u xxx = 0, u(x, 0) = v(x), where v ( x ) is a positive bump which decays sufficiently fast as x x→±α. The case v ( x ) ≪ 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u ( x , t , ϵ) as ϵ → 0 is reduced to a quadratic maximization problem, which is then solved.