Elliptic Solutions for Higher Order KdV Equations

We study higher order KdV equations from the GL(2,$mathbb{R}$) $cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/elliptic $N$-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find $N$-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.

Common Hirota Form B{a}cklund Transformation for the Unified Soliton System

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.

The Unified Soliton System as the ${rm AdS_2}$ System

We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon equations.

The Static Elliptic $N$-soliton Solutions of the KdV Equation

Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative B{a}cklund transformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $wp$-function type differential equation.

The Construction of the mKdV Cyclic Symmetric $N$-soliton Solution by the B{a}cklund Transformation

We study group theoretical structures of the mKdV equation. The Schwarzian type mKdV equation has the global M{o}bius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local M{o}bius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV B{a}cklund transformation can generate the mKdV cyclic symmetric $N$-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.