## 02.30.Ik Integrable systems

### Refine

#### Document Type

- ZIB-Report (4)

#### Language

- English (4)

#### Has Fulltext

- yes (4)

#### Is part of the Bibliography

- no (4)

#### Keywords

- computer algebra (2)
- discrete equations (2)
- integrable systems (2)
- large polynomial systems (2)
- Form (1)
- Integrable super-systems (1)
- REDUCE (1)
- Reduce (1)
- Singular (1)
- classifiaction (1)

A classification problem is proposed for supersymmetric %scaling\/-\/in\-va\-ri\-ant evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson\/-\/fermion systems are discovered under the weighting assumption $|f|=|b|=|D_t|=\oh$. The syntax of the \Reduce\ package \SsTools, which was used for intermediate computations, and the applicability of its procedures to the calculus of super\/-\/PDE are described.

We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A.~Cayley in 1845. We prove integrability (understood as $4d$-consistency) of a nonlinear difference equation defined by the $2 \times 2 \times 2$ - hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the $2\times 2\times 2\times 2$ - hyperdeterminant.

The paper describes a method for solution of very large overdetermined algebraic polynomial systems on an example that appears from a classification of all integrable 3-dimensional scalar discrete quasilinear equations $Q_3=0$ on an elementary cubic cell of the lattice ${\mathbb Z}^3$. The overdetermined polynomial algebraic system that has to be solved is far too large to be formulated. A probing' technique which replaces independent variables by random integers or zero allows to formulate subsets of this system. An automatic alteration of equation formulating steps and equation solving steps leads to an iteration process that solves the computational problem.

Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
(2005)

We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov--Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of $2^{\mbox{\scriptsize nd }}$order systems with a $3^{\mbox{\scriptsize rd }}$order or a $4^{\mbox{\scriptsize th }}$order symmetry and $3^{\mbox{\scriptsize rd }}$order systems with a $5^{\mbox{\scriptsize th }}$order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.