Note on Backlund transformations

The method of obtaining Backlund transformations proposed by Chern and Tenenblat (1986) fits completely the approach of obtaining Backlund transformations by prolongation techniques. For KdV, MKdV and sine-Gordon equations the only difference consists in the application of a non-linear representation of the prolongation algebra other than the usual one. This representation can be obtained by a coordinate transformation of the prolongation variabl

Prolongation structure and lax representation of the boomeron equation

An Estabrook-Wahlquist prolongation structure and a Lax representation for the Boomeron equation introduced by Calogero and Degasperis is determined

Polynomial solutions to the WDVV equations in four dimensions

All polynomial solutions of the WDVV equations for the case n = 4 are determined. We find all five solutions predicted by Dubrovin, namely those corresponding to Frobenius structures on orbit spaces of finite Coxeter groups. Moreover we find two additional series of polynomial solutions of which one series is of semi-simple type (massive). This result supports Dubrovin's conjecture if modified appropriately

Prolongation structure of the KdV equation in the bilinear form of Hirota

The prolongation structure of the KdV equation in the bilinear form of Hirota is determined, the resulting Lie algebra is realised and the Backlund transformation obtained from the prolongation structure is derived. The results are compared with those found by Wahlquist and Estabrook (1975) and by Hirota (1980)

Triviality of the prolongation algebra of the Kuramoto-Sivashinsky equation

The authors apply the well known Wahlquist-Estabrook prolongation technique to the Kuramoto-Sivashinsky equation. The prolongation algebra turns out to be trivial in the sense that it is commutative. This supports nonintegrability of the equation

WDVV equations for pure Super-Yang-Mills theory

In the literature, there are two proofs that the prepotential of $N=2$ pure Super-Yang-Mills theory satisfies the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. We show that these two methods are in fact equivalent. \u

The explicit structure of the prolongation algebra of the hirota-satsuma system

For a coupled system of KdV equations the prolongation Lie algebra is explicitly determined. It turns out to be of Kac-Moody type

Quasi-periodic solutions of the Boomeron equation

The quasi-periodic solutions for the Boomeron equation are determined by means of function-theoretical methods related to Riemann surfaces and theta functions. Also determined are the so-called Boomerons as degenerations of the quasi-periodic solutions. Moreover it is indicated that there are no higher-order Boomeron equations than the second-order one

Symmetries of the WDVV equations and Chazy-type equations

We investigate the symmetry structure of the WDVV equations. We obtain an $r$-parameter group of symmetries, where $r = (n^2 + 7n + 2)/2 + \lfloor n/2 \rfloor$. Moreover it is proved that for $n=3$ and $n=4$ these comprise all symmetries. We determine a subgroup, which defines an $SL_2$-action on the space of solutions. For the special case $n=3$ this action is compared to the $SL_2$-symmetry of the Chazy equation. For $n=4$ and $n=5$ we construct new, Chazy-type, solutions

Second order reductions of the WDVV Equations related to classical Lie algebras

We construct second order reductions of the generalized Witten-Dijkgraaf-Verlinde-Verlinde system based on simple Lie algebras. We discuss to what extent some of the symmetries of the WDVV system are preserved by the reduction.Comment: 6 pages, 1 tabl