7. A duality relation among the Lagrangians of the parametric coupled KdV system

We consider a generalization of the Gardner construction for the KdV equation. The Gardner transformation for the system (6) and (7) is given by

$$
\mu = r + \varepsilon r\_{\pi} - \frac{1}{6} \varepsilon^2 (r^2 + \lambda s^2) \tag{53}
$$

$$
\sigma = \mathbf{s} + \varepsilon \mathbf{s}\_{\mathbf{x}} - \frac{1}{3} \varepsilon^2 r \mathbf{s},
\tag{54}
$$

where ε is a real parameter and rðx; tÞ,sðx; tÞ are the fields which describe the Gardner ε-deformation. The Gardner equations are

$$r\_t + r\_{\text{xxx}} + r r\_x + \lambda s s\_x - \frac{1}{6} \varepsilon^2 [(r^2 + \lambda s^2) r\_x + 2\lambda r s s\_x] = 0 \tag{55}$$

$$\mathbf{s}\_t + \mathbf{s}\_{xxx} + r\mathbf{s}\_x + sr\_x - \frac{1}{6}\epsilon^2[(r^2 + \lambda \mathbf{s}^2)\mathbf{s}\_x + 2rs\_x] = \mathbf{0}.\tag{56}$$

Any solution of Eqs. (55) and (56) define through Eqs. (53) and (54) a solution of the system (6), (7).

ðþ<sup>∞</sup> −∞ dx rðx; <sup>t</sup><sup>Þ</sup> and <sup>ð</sup>þ<sup>∞</sup> −∞ dx sðx; tÞ are conserved quantities of the system (55) and (56). Assuming a formal power series on ε of the solutions of Eqs. (55) and (56) and inverting Eqs. (53) and (54), one obtains an infinite sequence of conserved quantities for the system (6), (7). It is an integrable system in this sense.

If we consider the ε ! 0 limit for the Gardner transformation Eqs. (53), (54) and Gardner Eqs. (55) and (56), we get the original system (6) and (7). On the other side, if we redefine

$$
\mu = \varepsilon r \tag{57}
$$

$$\mathbf{v} = \mathbf{\varepsilon}\mathbf{s} \tag{58}$$

and rewrite Eqs. (53) and (54), we get

L<sup>M</sup> <sup>1</sup> ¼ − 1 2 σtσx− λ 2 ρt ρx− 1 2 σxσxxx−

Eq. (48) being valid only for λ≠0.

L<sup>M</sup> <sup>2</sup> ¼ − 1 2 σtρx− 1 2 σxρ<sup>t</sup>

<sup>f</sup>vðxÞ,vðx^ÞgDB <sup>¼</sup> <sup>1</sup>

which is the Poisson structure associated with L<sup>M</sup>

the Poisson structure associated with L<sup>M</sup>

The corresponding Hamiltonian densities H<sup>M</sup>

and

14 Lagrangian Mechanics

v by

Eqs. (6) and (7).

<sup>1</sup> and L<sup>M</sup>

that L<sup>M</sup>

λ

−σxxxρ<sup>x</sup> þ

Each of them has a Poisson structure that follows from the Dirac approach. The Dirac

3

vxδðx−x^Þ þ <sup>2</sup>

<sup>1</sup> and

<sup>3</sup> vxδðx−x^Þ þ <sup>2</sup><sup>λ</sup>

vxδðx−x^Þ þ <sup>2</sup>

3

<sup>1</sup> and H<sup>M</sup>

<sup>1</sup> <sup>¼</sup> <sup>v</sup><sup>2</sup>−u<sup>2</sup>

The Hamilton equations obtained from these Hamiltonian structures coincide, of course, with

From these two Poisson structures, we may construct a pencil of Poisson structures as we described in the previous section, see Ref. [22] for the details of the construction. We notice

construct a hierarchy of higher order Hamiltonians from them. The same occurs with L<sup>1</sup> and L2. However, the two pencils are of different dimensions and we may obtain from them a hierarchy of higher order Hamiltonians which extends the hierarchy of the KdV equation.

<sup>2</sup> in the construction are of the same dimension. It is then not possible to

uxδðx−x^Þ þ <sup>2</sup>

<sup>3</sup><sup>λ</sup> uxδðx−x^Þ þ <sup>2</sup>

3

3

uxδðx−x^Þ þ <sup>2</sup>

3

v∂xδðx−x^Þ,

<sup>3</sup> <sup>v</sup>∂xδðx−x^<sup>Þ</sup>

v∂xδðx−x^Þ

3

u∂xδðx−x^Þ,

<sup>1</sup> <sup>¼</sup> <sup>−</sup>uv: (52)

<sup>2</sup> are given in terms of the fields u and

u∂xδðx−x^Þ

<sup>3</sup><sup>λ</sup> <sup>u</sup>∂xδðx−x^<sup>Þ</sup>

brackets, for the original fields u; v in the coupled system (6) and (7) are given by

<sup>λ</sup> <sup>∂</sup>xxxδðx−x^Þ þ <sup>1</sup>

3

3

2 .

H<sup>M</sup>

H<sup>M</sup>

<sup>f</sup>uðxÞ,uðx^ÞgDB <sup>¼</sup> <sup>∂</sup>xxxδðx−x^Þ þ <sup>1</sup>

<sup>f</sup>uðxÞ,vðx^ÞgDB <sup>¼</sup> <sup>1</sup>

<sup>f</sup>uðxÞ,uðx^ÞgDB <sup>¼</sup> <sup>λ</sup>

<sup>f</sup>vðxÞ,vðx^ÞgDB <sup>¼</sup> <sup>1</sup>

<sup>f</sup>uðxÞ,vðx^ÞgDB <sup>¼</sup> <sup>∂</sup>xxxδðx−x^Þ þ <sup>1</sup>

<sup>2</sup> <sup>ρ</sup>xρxxx <sup>þ</sup>

1 <sup>18</sup> <sup>σ</sup><sup>x</sup> 3 ρ<sup>x</sup> þ λ <sup>18</sup> <sup>ρ</sup><sup>x</sup> 3

1 <sup>72</sup> <sup>σ</sup><sup>x</sup> 4 þ λ2 <sup>72</sup> <sup>ρ</sup><sup>x</sup> 4 þ λ <sup>12</sup> <sup>ρ</sup><sup>2</sup> xσ2

<sup>x</sup> (48)

(50)

(51)

σx, (49)

$$
\mu = \frac{\mu}{\varepsilon} + \mu\_x - \frac{1}{6}\mu^2 - \frac{1}{6}\lambda\nu^2 \tag{59}
$$

$$
\omega = \frac{\nu}{\varepsilon} + \nu\_{\text{x}} - \frac{1}{\mathfrak{Z}} \mu \nu. \tag{60}
$$

Taking the limit ε ! ∞ we obtain

$$
\hat{\mu} = \mu\_{\text{x}} - \frac{1}{6}\mu^2 - \frac{1}{6}\lambda\mu^2 \tag{61}
$$

$$
\hat{\nu} = \nu\_x - \frac{1}{3}\mu\nu\tag{62}
$$

which is exactly the Miura transformation. In the same limit, we obtain from Eqs. (55), (56) the Miura equations given by Eq. (47).

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

We introduce the Casimir potentials

$$r = w\_x, \quad s = y\_x \tag{63}$$

and using the Helmholtz approach we obtain the Lagrangian densities

$$\begin{split} \mathcal{L}\_{G1} &= -\frac{1}{2} w\_{\text{x}} w\_{\text{t}} \cdot \frac{1}{6} (w\_{\text{x}})^3 + \frac{1}{2} (w\_{\text{xx}})^2 - \frac{\lambda}{2} w\_{\text{x}} (y\_{\text{x}})^2 - \frac{\lambda}{2} y\_{\text{x}} y\_{\text{t}} + \frac{\lambda}{2} (y\_{\text{xx}})^2 \\ &- \frac{1}{6} \varepsilon^2 \left[ -\frac{1}{12} (w\_{\text{x}})^4 - \frac{\lambda}{2} (w\_{\text{x}})^2 (y\_{\text{x}})^2 \right] + \frac{\varepsilon^2}{72} \lambda^2 (y\_{\text{x}})^4, \end{split} \tag{64}$$
 
$$\mathcal{L}\_{C2} = -\frac{1}{2} w\_{\text{y}} \mu - \frac{1}{2} \eta\_{\text{y}} \mu - \frac{1}{2} (w\_{\text{x}})^2 \mu - \nu \ \eta\_{\text{y}} \dots - \frac{\lambda}{2} (\nu\_{\text{y}})^3$$

$$\begin{split} \mathcal{L}\_{\text{G2}} &= -\frac{1}{2} w\_x y\_t - \frac{1}{2} w\_l y\_x - \frac{1}{2} (w\_x)^2 y\_x - y\_x w\_{\text{xxx}} - \frac{1}{6} (y\_x)^3 \\ &+ \frac{1}{18} \varepsilon^2 (w\_x)^3 y\_x + \frac{1}{18} \varepsilon^2 \lambda (y\_x)^3 w\_{\text{x}}. \end{split} \tag{65}$$

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

$$\lim\_{\varepsilon \to 0} \mathcal{L}\_{\text{G1}} = \mathcal{L}\_1 \quad , \quad \lim\_{\varepsilon \to 0} \mathcal{L}\_{\text{G1}} = \mathcal{L}\_2 \tag{66}$$

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

If we redefine

$$\begin{aligned} \mathcal{L}\_{\text{G1}}^{\text{M}} &= \varepsilon^2 \mathcal{L}\_{\text{G1}} \quad , \quad \begin{aligned} \sigma &= \varepsilon w \\ \mathcal{L}\_{\text{G1}}^{\text{M}} & \end{aligned} , \quad \begin{aligned} \rho &= \varepsilon y \\ \mathcal{L}\_{\text{G2}}^{\text{M}} & \end{aligned} \tag{67}$$

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

$$\begin{split} \lim\_{\ell \to \infty} \mathcal{L}\_{\text{G1}}^{\mathcal{M}}(\sigma, \rho) &= \mathcal{L}\_{1}^{\mathcal{M}}(\sigma, \rho) \\ \lim\_{\ell \to \infty} \mathcal{L}\_{\text{G2}}^{\mathcal{M}}(\sigma, \rho) &= \mathcal{L}\_{2}^{\mathcal{M}}(\sigma, \rho), \end{split} \tag{68}$$

where L<sup>M</sup> <sup>1</sup> and L<sup>M</sup> <sup>2</sup> were defined in Section 5. Consequently, all the Lagrangian structure and the associated Hamiltonian structure of the coupled system (6), (7) arises from the master Lagrangians. They can also be combined to a unique master Lagrangian depending on a parameter k as was done in Section 4. The field equations of the master Lagrangians are the Gardner equations, the spatial integral of rðx; tÞ and sðx; tÞ define an ε-deformed conserved quantity of the Gardner equations which implies an infinite sequence of conserved quantities of the original coupled KdV system (6), (7).
