A solution of the Korteweg–de Vries equation in a half-space bounded by a wall

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Abstract
We give a solution of the Korteweg–de Vries equation in the half‐space 0<r<∞ with the boundary condition V (0) =0. The boundary condition may be interpreted as the requirement that the plane which bounds the half‐space be a rigid wall. Aside from possible physical interest, this solution, which is obtained from one of the potentials for the radial Schrödinger equation which do not scatter, appears to indicate that the radial Schrödinger equation and the corresponding Gel’fand–Levitan equation play a role in the case of the half‐space bounded by a wall similar to that of the one‐dimensional Schrödinger equation (−∞<x<∞) and its corresponding Gel’fand–Levitan equation in the more usual full space treatment of the KdV equation. A possible interpretation of the solution presented in this paper is that it corresponds to the reflection of a wave by a wall, in which the incident wave is singular and the reflected wave is nonsingular but highly dispersive.

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