Common Hirota form Bäcklund transformation for the unified Soliton system

Abstract

We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) ≅ GL(2, ) ≅ Möbius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the N-soliton solutions through the KdV type Bäcklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the Bäcklund transformations of the standard form into the same common Hirota form and the same common Bäcklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the N-soliton solutions has the same common form for KdV/mKdV/sinh-Gordon systems. Then the N-soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV N-soliton solutions. We also give general addition formulae coming from the KdV type Bäcklund transformation which plays not only an important role to construct the trigonometric/hyperbolic N-soliton solutions but also an essential role to construct the elliptic N-soliton solutions. In contrast to the KdV type Bäcklund transformation, the well-known mKdV/sinh-Gordon type Bäcklund transformation gives the non-cyclic symmetric N-soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.