Abstract
Previously the inverse scattering method has been applied by various authors (Gardner, Greene, Kruskal and Miura, and Lax) to obtain solutions of certain nonlinear equations [e.g., the Korteweg—de Vries (KdV) equation] under the boundary condition . Recently via Bäcklund transformation, Au and Fung have obtained the KdV one-soliton solution which contains the vacuum parameter , and has been shown to be of physical significance. In fact, is the boundary value of . In this investigation we provide the generalized inverse scattering theory under the more general boundary condition . The one-soliton solution obtainable from this inverse scattering method is identical to the new solution just found by Au and Fung [Phys Rev. B 25, 6460 (1982)]. The solution containing nonzero is outside the square-integrable class. This extension of the class of functions has a crucial feature in attempting to understand physical observables.
- Received 5 July 1983
DOI:https://doi.org/10.1103/PhysRevA.29.2370
©1984 American Physical Society