8. Hamiltonian structure for a KdV system valued on a Clifford algebra

In this section, we continue the discussion of the Lagrangian and Hamiltonian structures for the coupled KdV systems. We discuss a coupled system arising from the breaking of the supersymmetry on the N ¼ 1 supersymmetric KdV equation. The details of this system may be found in Ref. [15]. The system is formulated in terms of a real valued field uðx; tÞ and a Clifford algebra valued field ξðx; tÞ. The field ξðx; tÞ is expressed in terms of an odd number of generators ei; i ¼ 1, … of the Clifford algebra

$$\mathcal{L} = \sum\_{i=1}^{\infty} \varphi\_i \mathbf{e}\_i + \sum\_{ijk} \varphi\_{ijk} \mathbf{e}\_i \mathbf{e}\_j \mathbf{e}\_k + \dotsb \tag{69}$$

where

u^ ¼ μx−

tion. A direct relation of these two systems arises from the present construction.

and using the Helmholtz approach we obtain the Lagrangian densities

<sup>12</sup> <sup>ð</sup>wx<sup>Þ</sup> 4 − λ <sup>2</sup> <sup>ð</sup>wx<sup>Þ</sup> 2 ðyxÞ 2

> 1 2 wxyt − 1 2 wtyx− 1 2 ðwxÞ 2

þ 1 <sup>18</sup> <sup>ε</sup><sup>2</sup> ðwxÞ 3 yx þ 1 <sup>18</sup> <sup>ε</sup><sup>2</sup>

lim ε!0

L<sup>M</sup>

lim<sup>ε</sup>!<sup>∞</sup> <sup>L</sup><sup>M</sup>

lim<sup>ε</sup>!<sup>∞</sup> <sup>L</sup><sup>M</sup>

LG<sup>1</sup> ¼ L<sup>1</sup> ; lim

σ ¼ εw ; ρ ¼ εy

<sup>G</sup>1ðσ; <sup>ρ</sup>Þ ¼ <sup>L</sup><sup>M</sup>

<sup>G</sup>2ðσ; <sup>ρ</sup>Þ ¼ <sup>L</sup><sup>M</sup>

<sup>G</sup><sup>1</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup>LG<sup>1</sup> ; <sup>L</sup><sup>M</sup>

ε!0

þ ε2 <sup>72</sup> <sup>λ</sup><sup>2</sup> ðyxÞ 4 ;

yx−yxwxxx−

λðyxÞ 3 wx:

<sup>G</sup><sup>2</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup>LG<sup>2</sup>

<sup>1</sup> ðσ; ρÞ

Miura equations given by Eq. (47).

16 Lagrangian Mechanics

We introduce the Casimir potentials

LG<sup>1</sup> ¼ −

1 2 wxwt− 1 6 ðwxÞ 3 þ 1 2 ðwxxÞ 2 − λ <sup>2</sup> wxðyx<sup>Þ</sup> 2 − λ <sup>2</sup> yxyt <sup>þ</sup>

<sup>ε</sup><sup>2</sup> <sup>−</sup> <sup>1</sup>

LG<sup>2</sup> ¼ −

− 1 6

If we take the weak coupling limit ε ! 0 we obtain

where L<sup>1</sup> and L<sup>2</sup> were defined in Section 3.

and take the strong coupling limit ε ! ∞, we get

If we redefine

1 6 μ2 − 1 6

> 1 3

v^ ¼ νx−

which is exactly the Miura transformation. In the same limit, we obtain from Eqs. (55), (56) the

We now construct using the Helmholtz approach a master Lagrangian for the Gardner equations. The master Lagrangians, there are two of them, are ε dependent and following the above limits we obtain all the Lagrangian structures we discussed previously. The KdV coupled system and the modified KdV coupled system are then dual constructions corresponding to the weak coupling limit ε ! 0 and to the strong coupling limit ε ! ∞ respectively, of the master construc-

λμ<sup>2</sup> (61)

μν (62)

r ¼ wx; s ¼ yx (63)

λ <sup>6</sup> <sup>ð</sup>yx<sup>Þ</sup> 3

λ <sup>2</sup> <sup>ð</sup>yxx<sup>Þ</sup> 2

LG<sup>1</sup> ¼ L<sup>2</sup> (66)

<sup>2</sup> <sup>ð</sup>σ; <sup>ρ</sup>Þ, (68)

(64)

(65)

(67)

$$
\varepsilon\_i \varepsilon\_\rangle + \varepsilon\_\not p\_i = -2\delta\_{\not j},\tag{70}
$$

and ϕ<sup>i</sup> ; ϕijk; … are real valued fields. We define by ξ the conjugate of ξ,

$$\overline{\xi} = \sum\_{i=1}^{\infty} \varphi\_i \overline{e}\_i + \sum\_{ijk} \varphi\_{ijk} \overline{e}\_k \overline{e}\_j \overline{e}\_i + \dotsb \tag{71}$$

where ei ¼ −ei. We denote by PðξξÞ the projector of the product ξξ to the identity element of the algebra

$$\mathcal{P}(\xi \overline{\xi}) = \sum\_{i=1}^{\infty} \rho\_i^2 + \sum\_{ijk} \rho\_{ijk}^2 + \cdots \tag{72}$$

We proposed in Ref. [15] the following coupled KdV system arising from the breaking of the supersymmetry in the N ¼ 1 supersymmetric equation [9]:

$$\begin{aligned} u\_t &= -u\_{\text{xxx}} - \mu u\_x - \frac{1}{4} (\mathcal{P}(\xi \overline{\xi}))\_x \\ \xi\_t &= -\xi\_{\text{xxx}} - \frac{1}{2} (\xi u)\_x. \end{aligned} \tag{73}$$

In distinction to the N ¼ 1 supersymmetric KdV equation the coupled system (73) has only a finite number of local conserved quantities,

$$\begin{aligned} \hat{H}\_{\frac{1}{2}} &= \int\_{-\infty}^{+\infty} \xi dx, \\ \hat{H}\_{1} &= \int\_{-\infty}^{+\infty} u dx, \\ V \triangleq \hat{H}\_{3} &= \int\_{-\infty}^{+\infty} \left( u^{2} + \mathcal{P}(\xi \overline{\xi}) \right) dx, \\ M \triangleq \hat{H}\_{5} &= \int\_{-\infty}^{+\infty} \left( -\frac{1}{3} u^{3} - \frac{1}{2} u \mathcal{P}(\xi \overline{\xi}) + (u\_{x})^{2} + \mathcal{P}(\xi\_{x} \overline{\xi}\_{x}) \right) dx. \end{aligned} \tag{74}$$

It is interesting to remark that the following nonlocal conserved charge of Super KdV [32] is also a nonlocal conserved charge for the system (73), in terms of the Clifford algebra valued field ξ,

$$\int\_{-\infty}^{\infty} \xi(\mathbf{x}) \int\_{-\infty}^{\mathbf{x}} \xi(\mathbf{s}) d\mathbf{s} d\mathbf{x}.\tag{75}$$

However, the nonlocal conserved charges of Super KdV in Ref. [33] are not conserved by the system (73). For example,

$$\int\_{-\infty}^{\infty} \mu(\mathbf{x}) \int\_{-\infty}^{\mathbf{x}} \xi(\mathbf{s}) d\mathbf{s} d\mathbf{x}.\tag{76}$$

is not conserved by Eq. (73).

The system (73) has multisolitonic solutions. In Ref. [34], we showed that the soliton solution is Liapunov stable under perturbation of the initial data.
