Hardy functions and the inverse spectral method. (English) Zbl 0567.35073

Guillemin, V.; Uribe, A.

Commun. Partial Differ. Equations 8, 1455-1474 (1983).

Consider the Korteweg-de Vries equation (1) \((\partial /\partial t)q(x,t)=(\partial /\partial x)^ 3q-6q(\partial q/\partial x)\) with periodic initial data: \(q(x+\pi,0)=q(x,0)\). In this paper we will discuss the solutions of (1) for which the initial data is a function of the form \(e^{\sqrt{-1}2kx}\), \(k>0\), or a superposition of such functions (in other words Hardy functions). Our method for solving (1) will be the inverse spectral method of Gardner, Greene, Kruskal, Miura, Lax etc. This method is extremely simple for the non-periodic version of (1) but a good deal more complicated in the periodic case. We will show that for Hardy potentials the method is about as simple in the periodic case as the usual method in the non-periodic case.

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35C05	Solutions to PDEs in closed form

Keywords:

Korteweg-de Vries equation; initial data; superposition; inverse spectral method; Hardy potentials; periodic case

References:

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