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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].

##### MSC:
 34L25 Scattering theory, inverse scattering involving ordinary differential operators 35Q53 KdV equations (Korteweg-de Vries equations) 34A55 Inverse problems involving ordinary differential equations
##### Keywords:
Cauchy problem for the KdV