One-soliton Korteweg—de Vries solutions with nonzero vacuum parameters obtainable from the generalized inverse scattering method

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.