The Modified KdV Equation

For the modified KdV equation,

$$u_t={\partial\over\partial x}(u^3-2\sigma^2 u_{xx})\ ,\eqno (5.1)$$

it is found that

$$u=\phi^{-1}\sum_{j=0}^{\infty} u_j\phi^j\ ,\eqno (5.2)$$

where

$$u_0=2\sigma\phi_x\ .$$

The resonances occur at $j=-1,3$, and 4. The compatability conditions at $j=3,4$ are satisfied identically and the modified KdV equations possess the Painlevé property. For reference, the recursion relations are

$$u_{j-3,t}+(j-3)\phi_t u_{j-2}=\phi_{j-1,x}+(j-3)\phi_x\psi_j\ ,\eqno (5.3)$$

where

$$\psi_j=\sum_{m=0}^j\sum_{l=0}^m u_{j-m}u_{m-l}u_l-2\sigma^2 \bigl(\theta_{j-1,x}+(j-2)\phi_x\theta_j\bigr)\eqno (5.4)$$

and

$$\theta_j=u_{j-1,x}+(j-1)\phi_x u_j\ .$$

We find that

truecm $j=0\ ,\qquad u_0=2\sigma\phi_x\ ,$ (5.5)

truecm $j=1\ ,\qquad u_1=-\sigma\phi_{xx}/\Phi_x \ ,$(5.6)

truecm $j=2\ ,\qquad \phi_t=6\sigma\phi_x^2 u_2+3 \sigma^2\phi_{xx}^2/\phi_x-2\sigma^2\phi_{xxx}\ .$(5.7)

To find a Bäcklund transform, the ``resonance" functions $u_3$ and $u_4$ are set equal to zero. To obtain a finite expansion, it is round that, in general, $u_2$ must vanish as well. Thus, when

$$u_2=u_3=u_4=0\ ,\eqno (5.8)$$

(5.3) implies that

$$u_j=0\ ,\qquad j\ge 2\ ,$$

and

$$u=2\sigma\phi_x/\phi+u_1\ .\eqno (5.9)$$

Furthermore, (5.6) and (5.7) with $u_2=0$ lead to

$$u_{1t}={\partial\over\partial x}(u_1^3-2\sigma^2 u_{1xx})\ ,\eqno (5.10)$$

where

$$u_1=-\sigma\phi_{xx}/\phi_x\ .\eqno (5.11)$$

Equations (5.9), (5.10), and (5.11) constitute an explicit Bäcklund transform for the modified KdV equation.