1. The original idea is due toC. S. Gardner, J. M. Greene, M. D. Kruskal andR. M. Miura:Phys. Rev. Lett.,19, 1095 (1967). For the approach used below see:F. Calogero:Lett. Nuovo Cimento,14, 443 (1975);F. Calogero andA. Degasperis:Nuovo Cimento,32 B, 201 (1976);39 B, 1 (1977).
2. See also eqs. (3.23)-(3.25) ofF. Calogero:Nonlinear evolution equations solvable by the inverse spectral transform, to appear in theProceedings of the Conference on the Mathematical Problems in Theoretical Physics, Rome, June 6–15, 1977, Lecture Notes in Physics (Berlin, 1978).
3. A direct derivation of (27) obtains setting φ(x)=x in eq. (2.13) of the paper byF. Calogero: inStudies in Mathematical Physics, edited byE. H. Lieb, B. Simon andA. S. Wightman (Princeton, N. J., 1976), p. 107.
4. F. Calogero andA. Degasperis:Lett. Nuovo Cimento,22, 138 (1978).