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[edit] Sockpuppetry case

You have been accused of sockpuppetry. Please refer to Wikipedia:Suspected sock puppets/R physicist for evidence. Please make sure you make yourself familiar with notes for the suspect before editing the evidence page. Cheeser1 (talk) 21:34, 23 March 2008 (UTC)

[edit] Thank you

Here I want present some Examples 1-30 regarding Myrzakulov equations. With personal regards, Sincerely Yours Ngn 92.46.69.162 (talk) 12:56, 23 March 2008 (UTC) So these Examples:

  • Please keep these Examples 1-30 until the second AfD discussion closed.


  • Example 1. Estevez P.G., Hernaez G.A. "Lax pair, Darboux transformations and solitonic solutions for a (2+1) Dimensional Non–Linear Schrodinger Equation". E–preprint: solv–int/991005, (Universidad de Salamanca, Spain), Oct.14, (1999) [7 ].

Quotation: “ … . Equation (1.1) is the Lakshmanan equivalent of the Myrzakulov-I (M-I) equation

                                                                    (1.2)

proposed in [16] as an extension to (2+1) dimensions of Heisenberg’s 1-dimensional spin model [12], [13]. The equivalence between (1.1) and (1.2) is proved in [21] and [22]. … ”


  • Example 2. Lakshmanan M. "Geometrical interpretation of (2+1)–dimensional integrable nonlinear evolution equations and localized solutions". Meeting Nonlinear systems, solitions and geometry, Oberwolfach, 19.10–25.10. (1997). p. 9.

Quotation: “ … In this lecture, it will be pointed out that by extending the formalism an important class of 2+1 dimensional integrable nonlinear evolution equations can also be interpreted as equations of motion of moving space curves but endowed with an extra spatial variable or equivalently in terms of moving surfaces (in orthogonal coordinates). Topological conserved quantities naturally follow as geometrical invariants. Underlying evolution equations are shown to be equivalent to a triad of linear equations. Geometrical equivalence between a class of 2+1 dimensional spin equations such as Myrzakulov equations and Ishimori equation with Zakharov-Strachan and Davey-Stewartson equations, respectively, will be brought out. Special localized solutions of some of these systems will also be reported. …”


  • Example 3. Chou K.S. and Qu C.Z. "Geometlic Motion of Surfaces and (2+1)–Dimensional Integrable Equations", Journal of the Physical Society of Japan, v71, №4, 1039-1043 (2002) [2]. doi:10.1143/JPSJ.71.1039

Quotation: “… In the meanwhile, other approaches to inducing motions of surfaces from (2+1)—integrable equations have been proposed [14-17]. We shall compare these results with ours in the end of this paper. … On the other hand, in ref. 16 the authors use a special parametrization (u,v,t) in which v = constant is always in arc-length. The motion not only depends on the coordinates but also on the curvature and torsion of the space curves given by v = constant, which foliate the surface. So it is not in the form (1.1). However, this approach is quite flexible and it provides ( 2 + 1 ) - integrable systems including the breaking soliton, Ishimori, Myrzakulov, and Davey-Stewartson equations. … ”


  • Example 4. Zhao W.Z., Wu K. et al. "Integrable inhomogeneous Lakshmanan-Myrzakulov equation", E-preprint: nlin.SI/0604034 (19 April 2006) [9]

Quotation: " … Eq. (1) reduces to the Myrzakulov-I (M-I) equation (see, e.g. Refs. [5-10]),

                                                           (2)

In Ref. [12], it was shown that Lakshmanan - Myrzakulov equation (LME) (1) is L-equivalent (about our … .”


  • Example 5. Gutshabash E.Sh. "Some notes on the Ishimori`s magnet model", E-preprint: nlin.SI/0302002, 2 February, 2003. [1]

Quotation: "... On the other hahd, a series of integrable models Myrzakulov`s magnets that are some modifications or generalizations of Ishimori`s model have been proposed in papers [13]-[15]...".


  • Example 6. McClain W.M., Shi Y. Hearst J.E. "A Lax pair for the dynamics of DNA modeled as a shearable and extensible elastic rod", E–preprint: nlin.SI/0108042 (University of California, Berkeley, August 23, 2001) [3].

Quotation: “ … At the end, we also present a connection between our system of PDEs with Myrzakulov’s reсent unit spin description of soliton equations in (1+1) dimension [4]. … ”


  • Example 7. Lucjan Sapa. "Existence and uniqueness of the classical solution of Fourier's first problem for nonlinear parabolic-elliptic systems", UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIV, pp.83-95 (2006) [1]

Quotation: " ... a model of evolution of water waves (the systems of Davey-Stewartson) [16] and in the theory of magnetism (the Myrzakulov equations) [9] ... ".


  • Example 8. Ding Q. "The gauge equivalence of the NLS and the Schrodinger flow of maps in 2+1 dimensions", J. Phys. A: Math. Gen, v32, 5087-5096 (1999) [2] doi:10.1088/0305-4470/32/27/308

Quotation: “ … In 1998, Myrzakulov et al demonstrated in [10, 15] that the (2+1) NLS+ is gauge equivalent to the following HF model in 2+1 dimensions obtained in the same way as the (2+1) NLS+:

                                                             (7)
                                                                                                                   (8)

where with |S|2=1 and denotes the cross product. …”


  • Example 9. Гутшабаш Е.Ш. "Замечания о модели магнетика Ишимори", Журнал "Записки научных семинаров ПОМИ" , Россия, Т.291, №17. c.155-168 (2002) [1]

Quotation: " ... что в данной работе основной акцент был сделан на специфике применения ПД к модели (1) без предъявления явных решений, поскольку достаточно широкий набор и классификация этих решений приведена в [5-7]. С другой стороны, в работах [13]-[15] была предложена серия интегрируемых моделей (магнетики Мырзакулова), которые являются ...".


  • Example 10. Shi Y, MeClain W.M., Hearst J. E. A. "Lax pair for the Dynamics Models as Extensible Elastic Rod II. Discretization of the arc length", University of California, Berkeley and Wayne State University, Detroit, USA. E-preprint Los Alamos National Laboratory. November 27, 2001.

Quotation: "... We may rewrite the first component equation in (8.2b) as

                                 (9.1)

This is discrete version of the basic equation in Myrzakulov`s unit spin description of the integrable and nonintegrable PDEs [8]. … "


  • Example 11. Гутшабаш Е.Ш. "Обобщенное преобразование Дарбу в модели магнетика Ишимори на фоне спиральных структур", Письма в ЖЭТФ, T.78, в.11. С.1257-1262 (2003) [1]

Quotation: "...В заключение отметим, что развитый выше подход легко переносится на серию моделей магнетиков Мырзакулова [23,24], являющихся ..."


  • Example 12. Murugesh S. et al. "Nonlinear dynamics of moving curves and surfaces: application to physical systems", International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, v15, №1, 51-63 (2005) [2]

Quotation: "...integrable (2+1)-D nonlinear evolution equations by the above procedure. Some examples are as follows (Myrzakulov et al., 1998). (i) Myrzakulov I equation [ Myrzakulov, 1987]: This equation reads

                                                                (52)
                                                                                                              (53)

Identifying again the spin vector with the unit tangent vector and following a procedure ... (ii) Ishimori II equation [Ishimori, 1984]: The equation has the form

                                               (54)
                                                                                                 (55)

The procedure adapted for (i) ...".


  • Example 13. Bracken P. and Grundland A.M. "Solutions of the generalized Weierstrass representation in four-dimensional Euclidean space", Journal of Nonlinear Mathematical Physics, v9, №3, 357-381 (2002)[1] doi:10.2991/jnmp.2002.9.3.8

Quotation: "... A more general (2+1)-dimensional integrable spin model is described by the pair of equations [17],

                                   (54a)
                                                                                                            (54b)

where ..."


  • Example 14. Chou K.-S., Qu C.-Z. "Integrable equations arising from motions of plane curves. II", Journal of Nonlinear Science, v13, N7. pp. 487-517 (2003).[2] doi:10.1007/s00332-003-0570-0

Quotation: " … Surfaces (with or without motion) have also been related to integrable equations since the works of [10], [11], [33]. A recent trend is to study surfaces in classical geometries such as affine, conformal, projective, and Lie sphere geometries. A few examples include the papers [33]-[45]. So Klein geometry for surfaces has already been anticipated in the study of integrable equations. …"


  • Example 15. Gutshabash E.Sh. "Generalized Darboux transform in the Ishimori magnet model on the nackround of spiral structures", JETP Letters", v.78, №11, 740-744 (2003) [2]

Quotation: " ... Note in conclusion that the approach developed in this work can easily be extended to a series of Myrzakulov magnet models [23, 24], which are modifications of the Ishimori model; for them, the first Lax-pair equation either is close or coincides with Eq. (3a) and the main modifications concern the functional Q in Eq. (3b). ..."


  • Example 16. Qu C., Zhang S. "Motion of curves and surfaces in affine geometry", Chaos, Solitons and Fractals, V.20, N5. pp. 1013-1019 (2004).[4] doi:10.1016/j.chaos.2003.09.005

Quotation: " … Surfaces (with or without motion)...are also related to integrable equations since the works of [18-22]. A recent trend is to study surfaces in classical . geometries such as affine, conformal, projective and Lie sphere geometries. But only few examples of integrable equations are found to be related to surface motions. There have been several ways to concern with this subject. An useful approach [23-26] is to consider space curve motions by adding an extra space variable. In this way the Ishimori equation, M-I equation , (2+1) – dimensional isotropic LLE equation and 2+1 – dimensional Schrodinger equation [24, 25] have been obtained from motion of space curves in Euclidean geometry. … Similar to the approach in [23-26], we endow an additional space variable y to the motion of space curves, the y evolution of the trihedral is determined by the following system if linear equations …"


  • Example 17. Zai Y., Albeverio S., Zhao W.Z., Wu K. "Prolongation structure of the (2+1)-dimensional integrable Heisenberg ferromagnet model", Journal of Physics A: Mathematical and General. V.39, N9, pp.2117-2126 (2006). [2] doi:10.1088/0305-4470/39/9/008

Quotation: " … Many efforts have been devoted to the study of its (2+1)-dimensional extensions [7, 8]. … A simple (2+1)-dimensional integrable equation is given by [8]

.                                          (2)

We now analyse this equation by using the prolongation structure … “


  • Example 18. Dimakis A., Muller-Hoissen F. "Bicomplex formulation and Moyal deformation of (2+1)-dimensional Fordy-Kulish systems", Journal of Physics A: Mathematical and General. V.34, pp.2571-2581 (2001). [3] doi:10.1088/0305-4470/34/12/305

Quotation: " … Fordy-Kulish systems generalize the nonlinear Schrodinger equation, respectively the Heisenberg magnet or the Da Rios equation. The latter equations are associated with the simplest Hermitian symmetric space SU(2) / S(U(1) x U(1)) in the series SU(N) / S(U(n) x U(N-n)). The corresponding extended Fordy-Kulish system associated with this space reproduces the (2+1)-dimensional Myrzakulov system [7] and also the sine-Gordon equation, as shown in the previous section. There may be a way to undestant the extended Fordy-Kulish systems as generalizations of the sine-Gordon equation in a similar way as they are quite obvious generalizations of the NLS equation. The extended Fordy-Kulish systems are matrix generalization of the Myrzakulov system, of course … “


  • Example 19. Zhang Z.H., Deng M., Zhao W.Z., Wu K. On the integrable inhomogeneous Myrzakulov I equation. nlin.SI/0603069 (30 March 2006) [3]

Quotation: “ … integrable extensions have been constructed [5]. One of its important integrable extensions is given by

                                                                    (3)

that is the Myrzakulov I (M-I) equation [5-7]. The M-I equation (3) is geometrical and gauge equivalent … . … Thus the integrable inhomogeneous M-I equation is

                                                   (20)

such as M-VIII, Ishimori and M-IX equations [5-7, 11]. …”


  • Example 20. Senthil Kumar C., Lakshmanan M., Grammaticos B., Ramani A. "Nonintegrability of (2+1)-dimensional continuum isotropic Heisenberg spin system: Painleve analysis". Physics Letters A, v.356, №4-5, pp.339-345 (2006) [2]. doi:10.1016/j.physleta.2006.03.074

Quotation: " … Many other integrable generalizations have also been obtained by Myrzakulov and coworkers [11]-[14]. … where , and is a scalar field and , and the Myrzakulov I (M-I) equation [13]

,                                                           (8a)
                                         ,                                                          (8b)

where u(x,y,t) is a scalar field. … “


  • Example 21. Zhen-Huan Zhang, Ming Deng, Wei-Zhong Zhao, Ke Wu. On the (2+1)-dimensional Integrable Inhomogeneous Heisenberg Ferromagnet Equation. Journal of the Physical Society of Japan, v.75, №10, pp.104002-104006 (2006) [3]. doi:10.1143/JPSJ.75.104002

Quotation: “… By using the prolongation structure theory proposed by Morris, we give a (2+1)-dimensional integrable inhomogeneous Heisenberg Ferromagnet equation, namely, the inhomogeneous Myrzakulov I equation. Through the motion of space curves endowed with an additional spatial variable, its geometrical equivalent counterpart is also presented. … ” Ngn 13 января 2008


  • Example 22. Marian Malec, Lucjan Sapa. "A finite difference method for nonlinear parabolic-elliptic systems of second-order partial differential equations". OPUSCULA MATHEMATICA, v.27, №2, pp.259-289 (2007) [1].

Quotation: "… water wave (the Davey-Stewartson equation) [28] and in the theory of magnetism (the Myrzakulov equations) [13]. Another example … "


  • Example 23. Bracken P. "Reductions of Chern-Simons theory related to integrable systems which have geometric applications", International Journal of Modern Physics, v.18, №9, 1261-1275 (2004)[1]

Quotation: "The equations which express the change of the frame are related to the Gauss{Weingarten equations for a surface in R [3,5]"


  • Example 24. Catherine Sulem, Pierre-Louis Sulem. The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse.(Monograph). Applied Mathematical Sciences. v.139. Springer, - Page 65

Quotation: "Other extensions were recently reviewed in Myrzakulov, Vijayalakshmi, Syzdykova and Lakshmanan [1998]... "


  • Example 25. Changzheng Qu, Yanyan Li. Deformation of surfaces induced by motions of curves in higher-dimensional similarity geometries (in press)

Quotation: "... In [35-42], Myrzakulov et al extended the theory of moving space curve formalism in 1+1 dimensions to 2+1 dimensions by adding an extra space variable to the motion of curves in $R^{3}$. They showed that the 2+1-dimensional Ishimori equation, Myrzakulov I equation, Myrzakulov III equation, 2+1-dimensional isotropic Heisenberg ferromagnet model and the 2+1-dimensional Schrodinger equation arise from such motions ... "


  • Example 26. M. Daniel, K. M. Ramizhamani, R. Sahadevan. Nonlinear Dynamics: Integrability and Chaos. (book). - Science - 2000 - 482 pages

Quotation: "... A major part of this work was done in collaboration with Dr. R. Myrzakulov (see [31]) and we acknowledge gratefully his contributions. ... "


  • Example 27. Elena Ausejo, Mariano Hormigón. Paradigms and Mathematics (book). - Mathematics - 1996 - 501 pages (Page 324).

Quotation: "... Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically ... "


  • Example 28. Zentralblatt MATH - Page 253. by European Mathematical Society, Heidelberger Akademie der Wissenschaften - 1999

Quotation: "... The extensions are proven to be integrable under the meaning that they possess the Painlevé property. Myrzakulov, R.; Vijayalakshmi, S.; ... "


  • Example 29. Robert Wayne Carroll. Quantum Theory, Deformation, and Integrability. (book) - Science - 2000 - 420 pages (page 386)

Quotation: "... [625] M. Lakshmanan, R. Myrzakulov, S. Vijayalakshmi and A. Danlybaeva. solv-int 9709009. ..."


  • Example 30. Radha R. Induced explode-decay dromions in the (2 + 1) dimensional nonisospectral nonlinear Schrödinger (NLS) equation. The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics. Volume 45, Number 2 / November, 2007, Pages 317-320 (2007).

Quotation: "... integrable inhomogeneous Myrzakulov I equation ..." %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%