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The Cauchy problem for the KdV equation with non-decreasing initial data. (English) Zbl 0810.34090
What is integrability, Springer Ser. Nonlinear Dyn., 273-318 (1991).
Using the approach of integral equations to solve the Korteweg-de Vries (KdV) equation [see the author, Nonlinear equations and operator algebras, English translation, Reidel, Dordrecht (1988; Zbl 0644.47053)], i.e. the solution of the integral equation \[ y(z)+ e^{-2z(x- 4z^ 2 t)} p(z)\left[\nu(z)y(-z)+ \int {y(z')\over z'+ z} d\mu(z')- 1\right]=0\tag{I} \] yields a solution of the KdV equation by the formula \[ u(x,t)= 2(d/dx) \int y(z) d\mu(z), \] the author discusses how to obtain a solution of the KdV equation which satisfies the Cauchy problem \(u(x,0)= q(x)\). See a paper by D. Sh. Lundina [Teor. Funkts. Funkts. Anal. Prilozh. 44, 57-66 (1985; Zbl 0583.47046)] for a class of initial data \(q\) which yields solutions of the KdV equation.
In the present paper, first the case of reflectionless potentials is considered: the real locally integrable potential \(q\) is reflectionless if and only if the Weyl function \(n(\lambda)\) associated to \(-y''+ qy= \lambda^ 2 y\) is a rational fraction. Let \(B(-\mu^ 2)\) be the set of reflectionless potentials \(q\) for which \(H+ \mu^ 2 I\geq 0\), where \(H= d^ 2/dx^ 2+ q(x)\). Characterizations of \(B(- \mu^ 2)\) are given and the closure of \(B(-\mu^ 2)\) in the topology of uniform convergence is investigated. Weyl functions of \(-y''+ qy= \lambda^ 2 y\) are characterized for potentials in the closure of \(B(-\mu^ 2)\). Let \(\widetilde B= \bigcup_{\mu>0} \overline{B(-\mu^ 2)}\). It is proved that if \(q= c+\widetilde q\), where \(c\in \mathbb{R}\) and \(\widetilde q\in \widetilde B\), then the Cauchy problem for the KdV equation with the initial function \(u(x,0)= q(x)\) has a bounded solution. Then the author relates the initial data \(q\) to the parameters \(p\), \(\nu\) and \(\mu\) of the integral equation (I).
For the entire collection see [Zbl 0808.35001].

34L25 Scattering theory, inverse scattering involving ordinary differential operators
35Q53 KdV equations (Korteweg-de Vries equations)
34A55 Inverse problems involving ordinary differential equations