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