The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with non-trivial reflection coefficient. (English) Zbl 0544.35081

Venakides, Stephanos

Commun. Pure Appl. Math. (to appear)

The inverse scattering method is used to determine the distribution limit as \(\epsilon \to 0\) of the solution u(x,t,\(\epsilon)\) of the initial value problem: \(u_ t-6uu_ x+\epsilon^ 2u_{xxx}=0,\quad u(x,0)=v(x)\) where v(x) is a positive bump which decays sufficiently fast as \(x\to \pm \alpha\). The case v(x)\(\leq 0\) has been solved by P. D. Lax and C. D. Levermore. The computation of the distribution limit of u(x,t,\(\epsilon)\) as \(\epsilon \to 0\) is reduced to a quadratic maximization problem, which is then solved.

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35R30	Inverse problems for PDEs
35P25	Scattering theory for PDEs

Keywords:

zero dispersion limit; Korteweg-de Vries equation; non-trivial reflection coefficient; inverse scattering method; distribution limit; initial value problem