Geodesic
Quantizing the KdV Equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 333-344

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We consider the quantization procedure for the Gardner–Zakharov–Faddeev and Magri brackets using the fermionic representation for the KdV field. In both cases, the corresponding Hamiltonians are sums of two well-defined operators. Each operator is bilinear and diagonal with respect to either fermion or boson (current) creation/annihilation operators. As a result, the quantization procedure needs no space cutoff and can be performed on the entire axis. In this approach, solitonic states appear in the Hilbert space, and soliton parameters become quantized. We also demonstrate that the dispersionless KdV equation is uniquely and explicitly solvable in the quantum case. 
@article{TMF_2001_129_2_a12,
     author = {A. K. Pogrebkov},
     title = {Quantizing the {KdV} {Equation}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {333--344},
     publisher = {mathdoc},
     volume = {129},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a12/}
}
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A. K. Pogrebkov. Quantizing the KdV Equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 333-344. http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a12/