We discuss an integrable partial differential equation arising from the hyperdeterminant.
It is also well known (see for example [5,6]) that the Bateman equation can be linearized by a Legendre transformation and the general implicit solution is given by
where f and g are smooth functions. This partial differential equation plays a central role in studying the integrability of partial differential equation using the Painlevé test [5,7]. For example the differential equation appears in the Painleve analysis of the inviscid Burgers equation, double sine-Gordan, discrete Boltzmann equation.
Here we generalize the condition given above from the determinant to the 2 × 2 × 2 hyperdeterminant of a 2 × 2 × 2 hypermatrix and derive the nonlinear partial differential equation and discuss its properties. The extension to 2 × 2 × 2 × 2 hyperdeterminants will be straightforward.
Cayley [8] in 1845 introduced the hyperdeterminant. Gelfand et al [9] give an in debt discussion of the hyperdeterminant. The hyperdeterminant arises as entanglement measure for three qubits [10,11,12], in black hole entropy [13]. The Nambu-Goto action in string theory can be expressed in terms of the hyperdeterminant [14]. A computer algebra program for the hyperdeterminant is given by Steeb and Hardy [11] Let ǫ 00 = ǫ 11 = 0, ǫ 01 = 1, ǫ 10 = -1, i.e. we consider the 2 × 2 matrix
Then the determinant of a 2 × 2 matrix A 2 = (a ij ) with i, j = 0, 1 can be defined as
Thus det A 2 = a 00 a 11 -a 01 a 10 . In analogy the hyperdeterminant of the 2 × 2 × 2 array A 3 = (a ijk ) with i, j, k = 0, 1 is defined as
There are 2 If only one of the coefficients a ijk is nonzero we find that the hyperdeterminant of A 3 is 0.
Given a 2 × 2 × 2 hypermatrix A 3 = (a jkℓ ), j, k, ℓ = 0, 1 and the 2 × 2 matrix S = s 00 s 01 s 10 s 11 .
The multiplication A 3 S which is again a 2 × 2 hypermatrix is defined by
a jkr s rℓ .
If det(S) = 1, i.e. S ∈ SL(2, C), then Det(A 3 S) = Det(A 3 ).
In analogy with the Bateman equation we set
and obtain the nonlinear partial differential equation
The partial differential equation is invariant under the permutations of x 1 , x 2 , x 3 . The group of symmetries is SL(3, R). The equation can also be linearized by a Legendre transformation.
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[2] Fairlie D. B., Integrable systems in higher dimensions, Prog. Theor. Phys. Supp. 1995, N 118, 309-327.
[3] Derjagin V. and Leznov A., Geometrical symmetries of the universal equation, Nonlinear Mathematical Physics, 2 (1995), 46-50.
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