10. Positiveness of the Hamiltonian for the Clifford valued system

An interesting property of the Hamiltonian H of the Clifford coupled system (73) is its a priori positiveness. In fact,

$$\|\hat{H}\_3 + \hat{H}\_5 = \|(u, \xi)\|\_{H\_1}^2 + \int\_{-\infty}^{+\infty} \left(-\frac{1}{3}\mu^3 - \frac{1}{2}\mu \mathcal{P}(\xi \overline{\xi})\right) d\mathbf{x} \tag{88}$$

where the Sobolev norm ∥∥<sup>H</sup><sup>1</sup> is defined by

$$\|\|(u,\xi)\|\|\_{H\_1}^2 := \int\_{-\infty}^{+\infty} [u^2 + \mathcal{P}(\xi \overline{\xi}) + u\_x^{-2} + \mathcal{P}(\xi\_x \overline{\xi\_x})] dx. \tag{89}$$

We also noticed that

$$\hat{H}\_3 = \| (\mu, \xi) \|\_{L^2}^2 \tag{90}$$

where ∥∥L<sup>2</sup> is the L<sup>2</sup> norm.

We then have

$$\|\hat{H}\_3 + \hat{H}\_5 \ge \|(u, \xi)\|\_{H\_1}^2 - \frac{1}{2} \int\_{-\infty}^{+\infty} |u| \left( u^2 + \mathcal{P}(\xi \overline{\xi}) \right) d\mathbf{x}.\tag{91}$$

We now use the bound

$$\sup |u| \le \frac{\|u\|\_{H\_1}}{\sqrt{2}} \le \frac{\|(u, \xi)\|\_{H\_1}}{\sqrt{2}},\tag{92}$$

to obtain

$$\|\hat{H}\_3 + \hat{H}\_5 \ge \|(u, \,\xi)\|\_{H\_1}^2 - \frac{1}{2\sqrt{2}} \|(u, \,\xi)\|\_{H\_1} \|(u, \,\xi)\|\_{L^2} \,. \tag{93}$$

Consequently,

$$
\hat{H}\_3 + \hat{H}\_5 + \left(\frac{1}{4\sqrt{2}}\right)^2 \hat{H}\_3 \ge \left( \|(u,\xi)\|\_{H\_1} - \frac{1}{4\sqrt{2}} \|(u,\xi)\|\_{L^2} \right)^2 \gtrsim 0. \tag{94}
$$

Finally,

(86)

(87)

dx (88)

<sup>2</sup> <sup>þ</sup> <sup>P</sup>ðξxξxÞ�dx: (89)

<sup>L</sup><sup>2</sup> (90)

<sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>P</sup>ðξξÞÞdx: (91)

p , (92)

p ∥ðu; ξÞ∥<sup>H</sup>1∥ðu; ξÞ∥L<sup>2</sup> : (93)

fuðxÞ,uðyÞgDB ¼ ∂xδðx; yÞ,

ðyÞgDB ¼ 0:

1 2

where H is given by the last conserved quantity in Eq. (74) and can be directly expressed in

An interesting property of the Hamiltonian H of the Clifford coupled system (73) is its a priori

ðþ<sup>∞</sup> −∞ − 1 3 u3 − 1 2 uPðξξÞ � �

<sup>½</sup>u<sup>2</sup> <sup>þ</sup> <sup>P</sup>ðξξÞ þ ux

<sup>H</sup>^ <sup>3</sup> <sup>¼</sup> <sup>∥</sup>ðu; <sup>ξ</sup>Þ∥<sup>2</sup>

H<sup>1</sup> − 1 2 ðþ<sup>∞</sup> −∞ juj �

2 p ≤

H<sup>1</sup> <sup>−</sup> <sup>1</sup> 2 ffiffiffi 2

<sup>∥</sup>ðu; <sup>ξ</sup>Þ∥<sup>H</sup><sup>1</sup> ffiffiffi 2

supju<sup>j</sup> <sup>≤</sup> <sup>∥</sup>u∥<sup>H</sup><sup>1</sup> ffiffiffi

<sup>H</sup><sup>1</sup> þ

; HgDB ¼ −ϕixxx−

10. Positiveness of the Hamiltonian for the Clifford valued system

<sup>ð</sup>u<sup>2</sup>Þx−uxxx<sup>−</sup>

λ <sup>2</sup> <sup>ð</sup>uϕ<sup>i</sup> Þx,

λ <sup>4</sup> <sup>ð</sup>ϕ<sup>2</sup> <sup>i</sup> Þ<sup>x</sup>

<sup>ð</sup>yÞgDB <sup>¼</sup> <sup>δ</sup>ij∂xδðx; <sup>y</sup>Þ,

fϕ<sup>i</sup> ðxÞ,ϕ<sup>j</sup>

∂tϕ<sup>i</sup> ¼ fϕ<sup>i</sup>

<sup>H</sup>^ <sup>3</sup> <sup>þ</sup> <sup>H</sup>^ <sup>5</sup> <sup>¼</sup> <sup>∥</sup>ðu; <sup>ξ</sup>Þ∥<sup>2</sup>

<sup>H</sup><sup>1</sup> :¼

<sup>H</sup>^ <sup>3</sup> <sup>þ</sup> <sup>H</sup>^ <sup>5</sup> <sup>≥</sup> <sup>∥</sup>ðu; <sup>ξ</sup>Þ∥<sup>2</sup>

<sup>H</sup>^ <sup>3</sup> <sup>þ</sup> <sup>H</sup>^ <sup>5</sup> <sup>≥</sup> <sup>∥</sup>ðu; <sup>ξ</sup>Þ∥<sup>2</sup>

ðþ<sup>∞</sup> −∞

where the Sobolev norm ∥∥<sup>H</sup><sup>1</sup> is defined by

<sup>∥</sup>ðu; <sup>ξ</sup>Þ∥<sup>2</sup>

Consequently,

20 Lagrangian Mechanics

terms of u and ξ.

positiveness. In fact,

We also noticed that

We then have

to obtain

where ∥∥L<sup>2</sup> is the L<sup>2</sup> norm.

We now use the bound

fuðxÞ,ϕ<sup>i</sup>

∂tu ¼ fu; HgDB ¼ −

$$
\hat{H}\_5 \ge -\left(1 + \left(\frac{1}{4\sqrt{2}}\right)^2\right)\hat{H}\_3,\tag{95}
$$

Hence, for a normalized state satisfying ∥ðu; ξÞ∥L<sup>2</sup> ¼ 1, we have

$$
\hat{H}\_5 \ge -\left(1 + \left(\frac{1}{4\sqrt{2}}\right)^2\right). \tag{96}
$$

The Hamiltonian is then manifestly bounded from below in the space of normalized L<sup>2</sup> configurations and it is thus physically admissible.

The property of the Hamiltonian is relevant from the physical point of view. In particular, in showing that the soliton solution of the Clifford coupled system is Liapunov stable. The stability analysis follows ideas introduced in Ref. [35] for the KdV equation. It is based on the use of the conserved quantities of the system. It is interesting that only the first few of them, in the case of the KdV equation, are needed. In the case of the Clifford coupled system these are all the local conserved quantities of the system.
