Previously the inverse scattering method has been applied by various authors (Gardner, Greene, Kruskal and Miura, and Lax) to obtain solutions $u(x,t)\mathrm{}$ of certain nonlinear equations [e.g., the Korteweg—de Vries (KdV) equation] under the boundary condition $u(x,t)\mathrm{}\to 0 \mathrm{as} x\to \pm \infty$. Recently via Bäcklund transformation, Au and Fung have obtained the KdV one-soliton solution which contains the vacuum parameter $b\ne 0$, and $b$ has been shown to be of physical significance. In fact, $b$ is the boundary value of $u(x,t):u(x,t)\mathrm{}\to b \mathrm{as} x\to \pm \infty$. In this investigation we provide the generalized inverse scattering theory under the more general boundary condition $u(x,t)\mathrm{}\to b\ne 0 \mathrm{as} x\to \pm \infty$. 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 $b$ 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