12. The Gardner formulation for the octonion valued algebra KdV equation

Associated with the real KdV equation, there is a Gardner ε-transformation and a Gardner equation which allows to obtain in a direct way the corresponding infinite sequence of conserved quantities. There exists a generalization of this approach for the KdV valued on the octonion algebra. The generalized Gardner transformation, expressed in terms of a new field rðx; tÞ valued on the octonion is given by

$$
\mu = r + \varepsilon r\_x - \frac{1}{6} \varepsilon^2 r^2. \tag{105}
$$

The generalized Gardner equation is then

$$(r\_t + r\_{xxx} + \frac{1}{2}(r r\_x + r\_x r) - \frac{1}{12}\left((r^2)r\_x + r\_x(r^2)\right)\varepsilon^2 = 0\tag{106}$$

where ε is a real parameter.

If rðx; tÞ is a solution of the generalized Gardner equation (106), then uðx; tÞ is a solution of the octonion algebra valued KdV equation (98). It has been shown in Ref. [23] that <sup>ð</sup>þ<sup>∞</sup> −∞ Re½rðx; tÞ�dx is a conserved quantity of Eq. (106). We can then invert Eq. (105), assuming a formal εexpansion of the solution rðx; tÞ, to obtain an infinite sequence of conserved quantities for the KdV equation valued on the octonion algebra.
