13. The master Lagrangian for the KdV equation valued on the octonion algebra

We may now use the Helmholtz procedure to obtain a Lagrangian density for the generalized Gardner equation. The master Lagrangian formulated in terms of the Casimir potential sðx; tÞ,

$$r(\mathbf{x}, t) = \mathbf{s}\_{\mathbf{x}}(\mathbf{x}, t), \tag{107}$$

is

$$L\_{\varepsilon}(\mathbf{s}) = \int\_{t\_i}^{t\_f} dt \int\_{-\infty}^{+\infty} \mathcal{L}\_{\varepsilon}(\mathbf{s}) d\mathbf{x} \tag{108}$$

where the Lagrangian density is given by

$$\mathcal{L}\_{\varepsilon}(\mathbf{s}) = \mathbb{E}\varepsilon \left[ -\frac{1}{2}\mathbf{s}\_{\mathbf{x}}\mathbf{s}\_{\mathbf{t}} - \frac{1}{6}\left(\mathbf{s}\_{\mathbf{x}}\right)^{3} + \frac{1}{2}\left(\mathbf{s}\_{\mathbf{x}\mathbf{x}}\right)^{2} + \frac{1}{72}\varepsilon^{2}\left(\mathbf{s}\_{\mathbf{x}}\right)^{4} \right]. \tag{109}$$

The Lagrangian density LεðsÞ is invariant under the action of the exceptional Lie group G2. Independent variations with respect to s yields

$$\begin{split} \delta \mathcal{L}\_{\varepsilon}(\mathbf{s}) &= \mathbb{E}e \left[ -\frac{1}{2} (\delta \mathbf{s})\_{\mathbf{x}} \mathbf{s}\_{\mathbf{t}} - \frac{1}{2} \mathbf{s}\_{\mathbf{x}} (\delta \mathbf{s})\_{\mathbf{t}} \frac{1}{6} \left( (\delta \mathbf{s})\_{\mathbf{x}} (\mathbf{s}\_{\mathbf{x}})^2 + \mathbf{s}\_{\mathbf{x}} (\delta \mathbf{s})\_{\mathbf{x}} \mathbf{s}\_{\mathbf{t}} + (\mathbf{s}\_{\mathbf{x}})^2 (\delta \mathbf{s})\_{\mathbf{x}} \right) \right] \\ &+ \mathbb{E}e \left[ \frac{1}{2} \left( (\delta \mathbf{s})\_{\mathbf{x}} \mathbf{s}\_{\mathbf{x}\mathbf{x}} + (\delta \mathbf{s})\_{\mathbf{x}} \mathbf{s}\_{\mathbf{x}\mathbf{}} \right) + \frac{1}{72} \varepsilon^{2} \Big( (\delta \mathbf{s})\_{\mathbf{x}} (\mathbf{s}\_{\mathbf{x}})^3 + \mathbf{s}\_{\mathbf{x}} (\delta \mathbf{s})\_{\mathbf{x}} (\mathbf{s}\_{\mathbf{x}})^2 + (\mathbf{s}\_{\mathbf{x}})^2 (\delta \mathbf{s})\_{\mathbf{x}} \mathbf{s}\_{\mathbf{x}} + (\mathbf{s}\_{\mathbf{x}})^3 (\delta \mathbf{s})\_{\mathbf{x}} \Big) \right]. \end{split} \tag{110}$$

Using properties of the octonion algebra we obtain from the stationary requirement δLεðsÞ ¼ 0 the generalized Gardner equation (106).

In the calculation the property to be a division algebra of the octonions is explicitly used.

If we take the limit e ! 0, we obtain a first Lagrangian for the KdV equation valued on the octonion algebra,

$$L(w) = \int\_{t\_i}^{t\_f} dt \int\_{-\infty}^{+\infty} dx \mathbb{R} e \left[ -\frac{1}{2} w\_x w\_t - \frac{1}{6} (w\_x)^3 + \frac{1}{2} (w\_{xx})^2 \right]. \tag{111}$$

Independent variations with respect to w yields, using u ¼ wx, the octonionic KdV equation (98). If we consider the following redefinition

$$\begin{aligned} \mathbf{s} & \rightarrow \hat{\mathbf{s}} = \varepsilon \mathbf{s} \\ \mathcal{L}\_{\varepsilon}(\mathbf{s}) & \rightarrow \varepsilon^2 \mathcal{L}\_{\varepsilon}(\hat{\mathbf{s}}) \end{aligned} \tag{112}$$

and take the limit e ! ∞ we obtain

$$\lim\_{\mathfrak{e}\to\infty}\mathfrak{e}^{2}\mathcal{L}\_{\mathfrak{e}}(\hat{\mathfrak{s}})=\mathcal{L}^{\mathcal{M}}(\hat{\mathfrak{s}}),\tag{113}$$

where

$$\mathcal{L}^{M}(\hat{\mathbf{s}}) = \mathbb{R}e \left[ -\frac{1}{2}\hat{s}\_{\mathbf{x}}\hat{s}\_{t} + \frac{1}{2}(\hat{s}\_{\mathbf{x}\mathbf{x}})^{2} + \frac{1}{72}(\hat{s}\_{\mathbf{x}})^{4} \right]. \tag{114}$$

We get in this limit the generalized Miura Lagrangian

$$L^{M}(\hat{\mathbf{s}}) = \int\_{t\_{\hat{l}}}^{t\_{\hat{f}}} dt \int\_{-\infty}^{+\infty} d\mathbf{x} \mathcal{L}^{M}(\hat{\mathbf{s}}).\tag{115}$$

The Miura equation is then obtained by taking variations with respect to ^s, we get

$$
\hat{r}\_t + \hat{r}\_{\text{xxx}} - \frac{1}{18} (\hat{r})\_\text{x}^3 = 0, \quad \hat{r} = \hat{s}\_\text{x}, \tag{116}
$$

while the Miura transformation arises after the redefinition process, it is <sup>u</sup> <sup>¼</sup> ^rx<sup>−</sup> <sup>1</sup> <sup>6</sup> ^r<sup>2</sup>:

Any solution of the Miura equation, through the Miura transformation, yields a solution of the KdV equation valued on the octonion algebra. Since LεðsÞ is invariant under G2, the same occurs for <sup>L</sup>ðw<sup>Þ</sup> and LMð^sÞ, and consequently for the equations arising from variations of them.

The Lagrangian formulation of the octonionic KdV equation may be used as the starting step to obtain the Hamiltonian structure of the octonion algebra valued KdV equation.
