11. The KdV equation valued on the octonion algebra

A famous theorem by Hurwitz establishes that the only real normalized division algebras are the reals R, the complex C, the quaternions ℍ, and the octonions O. In particular, these division algebras are directly related to the existence of super Yang-Mills in several dimensions: 3, 4, 6, and 10 dimensions [36]. The octonion algebra may be explicitly used in the formulation of superstring theory in 10 dimensions and in the supermembrane theory in 11 dimensions, relevant theories in the search for a unified theory of all the known fundamental forces in nature.

The extension of the KdV equation to a partial differential equation for a field valued on a octonion algebra is then an interesting goal [23].

We showed in the previous sections that an extension of the KdV equation to the field valued on a Clifford algebra give rise to a coupled system with Liapunov stable soliton solution but without an infinite sequence of local conserved quantities.

In the present section, we analyze the KdV extension where the field is valued on the octonion algebra. The system shares several properties of the original real KdV equation. It has soliton solutions and also has an infinite sequence of local conserved quantities derived from a Bäcklund transformation and a bi-Lagrangian and bi-Hamiltonian structure [23]. We will show in this section the construction of the bi-Lagrangian structure.

The octonion algebra contains as subalgebras all other division algebras, hence our construction may be reduced to any of them.

The KdV equation on the octonion algebra can be seen as a coupled KdV system, as we will see it has some similarities to the construction in the previous sections. However, it is invariant under the exceptional Lie group G2, the automorphisms of the octonions, and under the Galileo transformations. Those symmetries are not present in the model constructed on a Clifford algebra.

We denote u ¼ uðx; tÞ a function with domain in R · R valued on the octonionic algebra. If we denote ei; i ¼ 1, …; 7 the imaginary basis of the octonions, u can be expressed as

$$u(\mathbf{x},t) = b(\mathbf{x},t) + \overrightarrow{B}(\mathbf{x},t) \tag{97}$$

where bðx; tÞ is the real part and B ! <sup>¼</sup> <sup>∑</sup><sup>7</sup> <sup>i</sup>¼<sup>1</sup>Biðx; <sup>t</sup>Þei its imaginary part.

The KdV equation formulated on the algebra of octonions, or simply the octonion KdV equation, is given by

$$
\mu\_t + \mu\_{\text{xxx}} + \frac{1}{2} (\mu^2)\_x = 0,\tag{98}
$$

when B ! ¼ 0 ! it reduces to the scalar KdV equation. In terms of b and B ! the equation can be reexpressed as

$$b\_t + b\_{xxx} + b b\_x - \sum\_{i=1}^{7} B\_i B\_{ix} = 0,\tag{99}$$

$$(\left(\mathcal{B}\_{i}\right)\_{t} + \left(\mathcal{B}\_{i}\right)\_{\text{xxx}} + \left(b\mathcal{B}\_{i}\right)\_{x} = \mathbf{0}.\tag{100}$$

Eq. (98) is invariant under the Galileo transformation given by

$$\begin{array}{l} \tilde{\mathbf{x}} = \mathbf{x} + ct, \\ \tilde{t} = t, \\ \tilde{u} = u + c \end{array} \tag{101}$$

where c is a real constant.

Additionally, Eq. (98) is invariant under the automorphisms of the octonions, that is, under the group G2. If under an automorphism

$$
u \to \phi(u) \tag{102}$$

then

$$
\mu\_1 \mu\_2 \to \phi(\mu\_1 \mu\_2) = \phi(\mu\_1)\phi(\mu\_2) \tag{103}
$$

and consequently

Bäcklund transformation and a bi-Lagrangian and bi-Hamiltonian structure [23]. We will

The octonion algebra contains as subalgebras all other division algebras, hence our construc-

The KdV equation on the octonion algebra can be seen as a coupled KdV system, as we will see it has some similarities to the construction in the previous sections. However, it is invariant under the exceptional Lie group G2, the automorphisms of the octonions, and under the Galileo transformations. Those symmetries are not present in the model constructed on a

We denote u ¼ uðx; tÞ a function with domain in R · R valued on the octonionic algebra. If we

The KdV equation formulated on the algebra of octonions, or simply the octonion KdV

1 2 ðu2

> 7 i¼1

<sup>~</sup><sup>x</sup> <sup>¼</sup> <sup>x</sup> <sup>þ</sup> ct; <sup>~</sup><sup>t</sup> <sup>¼</sup> <sup>t</sup>; u~ ¼ u þ c

Additionally, Eq. (98) is invariant under the automorphisms of the octonions, that is, under the

!

<sup>i</sup>¼<sup>1</sup>Biðx; <sup>t</sup>Þei its imaginary part.

ðx; tÞ (97)

Þ<sup>x</sup> ¼ 0, (98)

!

BiBix ¼ 0, (99)

ðBiÞ<sup>t</sup> þ ðBiÞxxx þ ðbBiÞ<sup>x</sup> ¼ 0: (100)

u ! φðuÞ (102)

u1u<sup>2</sup> ! φðu1u2Þ ¼ φðu1Þφðu2Þ (103)

the equation can be

(101)

uðx; tÞ ¼ bðx; tÞ þ B

denote ei; i ¼ 1, …; 7 the imaginary basis of the octonions, u can be expressed as

ut þ uxxx þ

! it reduces to the scalar KdV equation. In terms of b and B

bt þ bxxx þ bbx− ∑

! <sup>¼</sup> <sup>∑</sup><sup>7</sup>

Eq. (98) is invariant under the Galileo transformation given by

show in this section the construction of the bi-Lagrangian structure.

tion may be reduced to any of them.

where bðx; tÞ is the real part and B

Clifford algebra.

22 Lagrangian Mechanics

equation, is given by

when B ! ¼ 0

then

reexpressed as

where c is a real constant.

group G2. If under an automorphism

$$\left( [\phi(\mu)]\_t + [\phi(\mu)]\_{\text{xxx}} + \frac{1}{2} \left( [\phi(\mu)]^2 \right)\_x = 0. \tag{104}$$
