6. The Miura transformation for the parametric coupled KdV system

It is well known that the KdV equation admits two Hamiltonian structures, one of them is a particular case of our previous construction. It is obtained by considering only the uðx; tÞ field, imposing vðx; tÞ ¼ 0: In this case, the two previous Hamiltonians structures reduce to only one and there is no pencil of Poisson structures. The second Hamiltonian structure for the KdV equation arises from a Miura transformation, which is also a particular case of the following construction. The corresponding Miura transformation for our coupled system becomes

$$\begin{aligned} \mu &= \mu\_x - \frac{1}{6}\mu^2 - \frac{\lambda}{6}\nu^2\\ \upsilon &= \nu\_x - \frac{1}{3}\mu\nu. \end{aligned} \tag{46}$$

and the modified KdV system (MKdVS)

$$\begin{aligned} \mu\_t + \mu\_{\text{xxx}} - \frac{1}{6}\mu^2\mu\_x - \frac{\lambda}{6}\nu^2\mu\_x - \frac{\lambda}{3}\mu\nu\nu\_x &= 0 \\\\ \nu\_t + \nu\_{\text{xxx}} - \frac{1}{6}\mu^2\nu\_x - \frac{\lambda}{6}\nu^2\nu\_x - \frac{1}{3}\mu\nu\mu\_x &= 0. \end{aligned} \tag{47}$$

It is interesting that from Eq. (47), following the Helmholtz procedure, which is also valid for the MKdVS system, we obtain two singular Lagrangians densities L<sup>M</sup> <sup>i</sup> ; i ¼ 1; 2, expressed in terms of the Casimir potentials σ; ρ where μ ¼ σx; ν ¼ ρ<sup>x</sup> :

$$\mathcal{L}\_1^M = -\frac{1}{2}\sigma\_t \sigma\_x - \frac{\lambda}{2}\rho\_t \rho\_x - \frac{1}{2}\sigma\_x \sigma\_{\text{xxx}} - \frac{\lambda}{2}\rho\_x \rho\_{\text{xxx}} + \frac{1}{72}\sigma\_x^4 + \frac{\lambda^2}{72}\rho\_x^{\*4} + \frac{\lambda}{12}\rho\_x^2 \sigma\_x^2 \tag{48}$$

and

$$\mathcal{L}\_2^M = -\frac{1}{2}\sigma\_t \rho\_x - \frac{1}{2}\sigma\_x \rho\_t - \sigma\_{\text{xxx}}\rho\_x + \frac{1}{18}\sigma\_x^{~3}\rho\_x + \frac{\lambda}{18}\rho\_x^{~3}\sigma\_x,\tag{49}$$

Eq. (48) being valid only for λ≠0.

Each of them has a Poisson structure that follows from the Dirac approach. The Dirac brackets, for the original fields u; v in the coupled system (6) and (7) are given by

$$\begin{aligned} \{u(\mathbf{x}), u(\hat{\mathbf{x}})\}\_{\mathrm{DB}} &= \mathfrak{d}\_{\mathrm{xxx}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{1}{3} \mathfrak{u}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{2}{3} \mathfrak{u} \mathfrak{d}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{v(\mathbf{x}), v(\hat{\mathbf{x}})\}\_{\mathrm{DB}} &= \frac{1}{\lambda} \mathfrak{d}\_{\mathrm{xxx}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{1}{3\lambda} \mathfrak{u}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{2}{3\lambda} \mathfrak{u} \mathfrak{d}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{u(\mathbf{x}), v(\hat{\mathbf{x}})\}\_{\mathrm{DB}} &= \frac{1}{3} v\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{2}{3} \mathfrak{u} \mathfrak{d}\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}), \end{aligned} \tag{50}$$

which is the Poisson structure associated with L<sup>M</sup> <sup>1</sup> and

$$\begin{aligned} \{u(\mathbf{x}), u(\hat{\mathbf{x}})\}\_{DB} &= \frac{\lambda}{3} \upsilon\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{2\lambda}{3} \upsilon \partial\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{\upsilon(\mathbf{x}), \upsilon(\hat{\mathbf{x}})\}\_{DB} &= \frac{1}{3} \upsilon\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{2}{3} \upsilon \partial\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}})\\ \{u(\mathbf{x}), \upsilon(\hat{\mathbf{x}})\}\_{DB} &= \partial\_{\mathbf{x}\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{1}{3} \mu\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}) + \frac{2}{3} \mu \partial\_{\mathbf{x}} \delta(\mathbf{x} - \hat{\mathbf{x}}), \end{aligned} \tag{51}$$

the Poisson structure associated with L<sup>M</sup> 2 .

The corresponding Hamiltonian densities H<sup>M</sup> <sup>1</sup> and H<sup>M</sup> <sup>2</sup> are given in terms of the fields u and v by

$$\begin{aligned} \mathcal{H}\_1^M &= v^2 - u^2\\ \mathcal{H}\_1^M &= -uv. \end{aligned} \tag{52}$$

The Hamilton equations obtained from these Hamiltonian structures coincide, of course, with Eqs. (6) and (7).

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 that L<sup>M</sup> <sup>1</sup> and L<sup>M</sup> <sup>2</sup> in the construction are of the same dimension. It is then not possible to 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.
