Appendix A:    Recursion Relations for the KdV Equation and Conserved Densities

For the KdV equation

$$u_t+{\partial\over\partial x}\left({u^2\over 2}+u_{xx}\right)=0\eqno (A1)$$

we get

$$u=\phi^{-2}\sum_{j=0}^{\infty} u_j\phi^j\eqno (A2)$$

and find

$$u_{j-3,t}+(j-4)\phi_t u_{j-2}+\psi_{j-1,x}+(j-4)\phi_x\psi_j=0 , \eqno (A3)$$

where

$$\psi_j=\sum_{m=0}^j {u_{j-m}u_m\over 2}+\theta_{j-1,x}+(j-3)\phi_x \theta_j\eqno (A4)$$

and

$$\theta_j=u_{j-1,x}+(j-2)\phi_x u_j .\eqno (A5)$$

In (A2), $(\phi,u_4,u_6)$ are arbitrary.

In this appendix we show how the recursion relations (A3) may be used to formally define an infinite set of conserved densities for the KdV equation. By repeated application of (A3) for $j,j+1,\ldots$ it is easily demonstrated that

$${\partial\over\partial t}A_j+{\partial\over\partial x}B_j=0 ,\eqno (A6)$$

where

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...&+(j-4)(j-3)(j-2)(\phi^3/3!)u_j+\cdots ,\cr\crcr}} \eqno (A7)$$

or, for $j\ge 5$,

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...5}=\sum_{K=0}^{\infty} u_{j+2+K}\phi^K ,\cr\crcr}} \eqno (A8)$$

and

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...(j-3)(j-2)(\phi^3/3!)\psi_{j+2}+\cdots ,\cr\crcr}} \eqno (A9)$$

or, for $j\ge 5$,

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...\sum_{K=0}^{\infty}\psi_{j+4+K}\phi^K .\cr\crcr}} \eqno (A10)$$

For the conserved densities $A_j$, we find that

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...phi\phi_x)_x ,\cr A_4&=12(\phi_x)_x .\cr\crcr}} \eqno (A11)$$

Thus, the densities $A_0-A_4$ are trivial, i.e., gradients. Furthermore,

$$A_5=\sum_{K=2}^{\infty} u_K\phi^{K-2}=\sum_{K=0}^{\infty} u_{K+2} \phi^K\eqno (A12)$$

and

$$A_{5+j}={1\over j!}{\delta^j\over\delta\phi^j} A_5 ,\eqno (A13)$$

where, for constant $\epsilon$,

$${\delta\over\delta\phi} f(\phi)={d\over d\epsilon} f(\phi+\epsilon) \vert_{\epsilon=0} .$$

We note that

$$u=12{\partial^2\over\partial x^2}\ln\phi+A_5 .\eqno (A14)$$