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,$\mathbb{R}$) $\cong$ 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 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.

14 pages

References in corpus (4)

• Miura and Generalized Bäcklund Transformation for KdV Hierarchy
• The Static Elliptic $N$-soliton Solutions of the KdV Equation
• The Construction of the mKdV Cyclic Symmetric $N$-soliton Solution by the Bäcklund Transformation
• The Unified Soliton System as the $\mathrm{Ad}{\mathrm{S}}_2$ System

Cited by in corpus (4)

• Differential Equations of Genus Four Hyperelliptic $\wp$ Functions
• Elliptic Solutions for Higher Order KdV Equations
• Two Flows Kowalevski Top as the Full Genus Two Jacobi's Inversion Problem and Sp(4,$\mathbb{R}$) Lie Group Structure
• The Half-period Addition Formulae for Genus Two Hyperelliptic $\wp$ Functions and the Sp(4,$\mathbb{R}$) Lie Group Structure