zbMATH — the first resource for mathematics

Generalized solutions to the Korteweg-de Vries and the regularized long- wave equations. (English) Zbl 0757.35068
The purpose of this paper is to study generalized solutions to the KdV equation and the regularized long-wave (Benjamin-Bona-Mahony) equation: \[ u_ t+u_ x+uu_ x-u_{xxt}=0 \] in the framework of generalized functions introduced by Colombeau. The solutions are found in certain algebras of new generalized functions containing spaces of distributions. On one hand, this allows to handle initial data with strong singularities. On the other hand, a suitable scaling allows one to introduce an infinitesimally small coefficient \(\nu\): \(u_ t+uu_ x+\nu u_{xxx}=0\). Thereby the authors produce generalized solutions in the sense of Colombeau to the inviscid Burgers equation.

35Q53 KdV equations (Korteweg-de Vries equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
46F10 Operations with distributions and generalized functions
35L65 Hyperbolic conservation laws
Full Text: DOI