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 TMF, 2001, Volume 129, Number 2, Pages 333–344 (Mi tmf540)

Quantizing the KdV Equation

A. K. Pogrebkov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/tmf540

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English version:
Theoretical and Mathematical Physics, 2001, 129:2, 1586–1595

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Document Type: Article

Citation: A. K. Pogrebkov, “Quantizing the KdV Equation”, TMF, 129:2 (2001), 333–344; Theoret. and Math. Phys., 129:2 (2001), 1586–1595

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. K. Pogrebkov, “Boson-fermion correspondence and quantum integrable and dispersionless models”, Russian Math. Surveys, 58:5 (2003), 1003–1037
2. Pogrebkov A.K., “Hierarchy of quantum explicitly solvable and integrable models”, Bilinear Integrable Systems: From Classical to Quatum, Continuous to Discrete, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 201, 2006, 231–244
3. Zarmi Ya., “Quantized representation of some nonlinear integrable evolution equations on the soliton sector”, Phys Rev E, 83:5, Part 2 (2011), 056606
4. Zarmi Ya., “Nonlinear Quantum-Dynamical System Based on the Kadomtsev-Petviashvili II Equation”, J. Math. Phys., 54:6 (2013), 063515
5. Pustilnik M. Matveev K.A., “Fate of Classical Solitons in One-Dimensional Quantum Systems”, Phys. Rev. B, 92:19 (2015), 195146
6. Sotiriadis S., “Equilibration in One-Dimensional Quantum Hydrodynamic Systems”, J. Phys. A-Math. Theor., 50:42 (2017), 424004
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