Geodesic
Complete integrability, gauge equivalence and Lax representation of inhomogeneous nonlinear evolution equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 92 (1992) no. 3, pp. 374-386 Cet article a éte moissonné depuis la source Math-Net.Ru

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The gauge equivalence between the inhomogeneous versions of the nonlinear Schrödinger and the Heisenberg ferromagnet equations is studied. An unexplicit criterion for integrability is proposed. Examples of gauge equivalent inhomogeneous nonlinear evolution equations are presented. It is shown that in the nonintegrable cases the $M$-operators in their Lax representations possess unremovable pole singularities lying on the spectrum of the $L$-operators. 
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     author = {V. S. Gerdjikov},
     title = {Complete integrability, gauge equivalence and {Lax} representation of inhomogeneous nonlinear evolution equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {374--386},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TMF_1992_92_3_a1/}
}
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V. S. Gerdjikov. Complete integrability, gauge equivalence and Lax representation of inhomogeneous nonlinear evolution equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 92 (1992) no. 3, pp. 374-386. http://geodesic.mathdoc.fr/item/TMF_1992_92_3_a1/

[1] Manakov S. V., Novikov S. P., Pitaevskii L. I., Zakharov V. E., The Theory of Solitons: The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultant Bureau, New-York, 1984 | MR | Zbl

[2] Ablowitz M. J., Seegur H., Solitons and the Inverse Scattering Transform, Studies in Appplied Mathematics 4, SIAM, Philadelphia, PA, 1981 | MR | Zbl

[3] Calogero F., Degasperis A., Spectral Transform and Solitons, v. I. Studies in Mathematics ans its Applications, North Holland, Amsterdam, 1982 | MR | Zbl

[4] Faddeev L. D., Takhtadjan L. A., Hamiltonian Method in the Theory of Solitons, Springer-Verlag, Berlin, 1986 | MR

[5] Arkadi'ev V. A., Pogrebkov A. K., Polivanov M. K., TMF, 53:2 (1982), 163–180 | MR

[6] Manakov S. V., Zakharov V. E., Bordag L. A., Its A. R., Matveev V. B., Phys. Lett. A, 63 (1977), 205–206 | DOI

[7] Fokas A. S., Physica D, 35 (1989), 167 | DOI | MR | Zbl

[8] Khristov E. Kh., Kliment Ohridsky'76, v. 2. Mecanique, 1987, 167, 191

[9] Kaup D. J., Newell A., Proc. Roy. Soc. London A, 361 (1978), 413 | DOI

[10] Leon J., Phys. Lett. A, 144 (1990), 444 | DOI | MR

[11] Fokas A. S., Ablowitz M. J., Stud. Appl. Math., 80 (1989), 253 | DOI | MR | Zbl

[12] Gerdjikov V. S., Ivanov M. I., Preprint of INRNE, Sofia, 1990

[13] Melnikov V. K., Reports on NEEDS 90, Dubna, july 1990; NEEDS 91, Gallipoli, june 1991 | Zbl

[14] Latifi A., Leon J., Phys Lett. A, 152 (1991), 171 | DOI | MR

[15] Gerdjikov V. S., Yanevski A. B., Commun. Math. Phys., 103 (1986), 549 | DOI | MR | Zbl

[16] Calogero F., Xiaodin, J. Math. Phys., 32 (1991), 875, 2703 | DOI | MR

[17] Calogero F., Nucci C., J. Math. Phys., 32 (1991), 72 | DOI | MR | Zbl

[18] Gerdjikov V. S., Lett. Math. Phys., 6 (1982), 315 | DOI | MR | Zbl

[19] Gerdjikov V. S., Khristov E. Kh., Bulgarian J. Phys., 7 (1980), 28, 119 (Russian) | MR

[20] Gerdjikov V. S., Ivanov M. I., Bulgarian J. Phys., 10 (1983), 13, 130 (Russian) | MR

[21] Gerdjikov V. S., Ivanov M. I., Vaklev Y. S., Inverse Problems, 2:4 (1986), 413–432 | DOI | MR | Zbl

[22] Gerdjikov V. S., Inverse Problems, 2 (1986), 51 | DOI | MR | Zbl

[23] Ablowitz M. J., Kaup D. J., Newell A. C., J. Math. Phys., 15 (1974), 1851 | DOI | MR

[24] Claude C., Latifi A., Leon J., J. Math. Phys., 32 (1991), 3321 | DOI | MR | Zbl

[25] Gerdjikov V. S., Lax Representation Does not Mean Complete Integrability, Internal Report of ICTP IC/91/274, 1991 | MR