Continuous vacua in bilinear soliton equations
Journal of Physics A: Mathematical and General
We discuss the freedom in the background field (vacuum) on top of which the solitons are built. If the Hirota bilinear form of a soliton equation is given by A(D_x⃗) GF=0, B(D_x⃗)( FF - GG)=0 where both A and B are even polynomials in their variables, then there can be a continuum of vacua, parametrized by a vacuum angle ϕ. The ramifications of this freedom on the construction of one- and two-soliton solutions are discussed. We find, e.g., that once the angle ϕ is fixed and we choose u=arctan
... F as the physical quantity, then there are four different solitons (or kinks) connecting the vacuum angles ±ϕ, ±ϕ±Π2 (defined modulo π). The most interesting result is the existence of a "ghost" soliton; it goes over to the vacuum in isolation, but interacts with "normal" solitons by giving them a finite phase shift.