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) $\cong$ GL(2,$\mathbb R$) $\cong$ M\"{o}bius 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 B\"{a}cklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the B\"{a}cklund transformations of the standard form into the same common Hirota form and the same common B\"{a}cklund 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 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 B\"{a}cklund 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 B\"{a}cklund transformation, the well-known mKdV/sinh-Gordon type B\"{a}cklund 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.