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 Mat. Sb. (N.S.), 1983, Volume 121(163), Number 4(8), Pages 469–498 (Mi msb2219)

Some new nonlinear evolution equations integrable by the inverse problem method

V. K. Mel'nikov

Abstract: Several new nonlinear evolution equations integrable by the inverse problem method are obtained. The method applied in finding these equations is believed to be essentially new. The comparison of that method with other methods for finding nonlinear evolution equations integrable by the inverse problem method is given. In particular, it is shown that the methods using the Heisenberg equation (the so-called Lax representation) are not suitable to obtain the equations studied here.
Bibliography: 23 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 49:2, 461–489

Bibliographic databases:

UDC: 517.9
MSC: 35K22, 35K55

Citation: V. K. Mel'nikov, “Some new nonlinear evolution equations integrable by the inverse problem method”, Mat. Sb. (N.S.), 121(163):4(8) (1983), 469–498; Math. USSR-Sb., 49:2 (1984), 461–489

Citation in format AMSBIB
\Bibitem{Mel83} \by V.~K.~Mel'nikov \paper Some new nonlinear evolution equations integrable by the inverse problem method \jour Mat. Sb. (N.S.) \yr 1983 \vol 121(163) \issue 4(8) \pages 469--498 \mathnet{http://mi.mathnet.ru/msb2219} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=716108} \zmath{https://zbmath.org/?q=an:0558.35063} \transl \jour Math. USSR-Sb. \yr 1984 \vol 49 \issue 2 \pages 461--489 \crossref{https://doi.org/10.1070/SM1984v049n02ABEH002721} 

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This publication is cited in the following articles:
1. I. M. Krichever, “Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model”, Funct. Anal. Appl., 20:3 (1986), 203–214
2. Zakharov V., Kuznetsov E., “Multiscale Expansions in the Theory of Systems Integrable by the Inverse Scattering Transform”, Physica D, 18:1-3 (1986), 455–463
3. N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, “Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics”, Theoret. and Math. Phys., 75:1 (1988), 329–339
4. V.K. Mel’nikov, “On equations solvable by the inverse scattering method for the Dirac operator”, Communications in Nonlinear Science and Numerical Simulation, 8:1 (2003), 9
5. I. A. Taimanov, “Singular spectral curves in finite-gap integration”, Russian Math. Surveys, 66:1 (2011), 107–144
6. Runliang Lin, Xiaojun Liu, Yunbo Zeng, “Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy”, Journal of Nonlinear Mathematical Physics, 20:2 (2013), 214
7. Adam Doliwa, Runliang Lin, “Discrete KP equation with self-consistent sources”, Physics Letters A, 2014
8. Runliang Lin, Xiaojun Liu, Yunbo Zeng, “The KP hierarchy with self–consistent sources: construction, Wronskian solutions and bilinear identities”, J. Phys.: Conf. Ser, 538 (2014), 012014
9. Q. Li, J. B. Zhang, D. Y. Chen, “The Eigenfunctions and Exact Solutions of Discrete mKdV Hierarchy with Self-Consistent Sources via the Inverse Scattering Transform”, Adv. Appl. Math. Mech, 7:05 (2015), 663
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