Camassa-Holm and M-CIV equations with self-consistent sources: geometry and peakon solutions
Gulmira Yergaliyeva, Tolkynay Myrzakul, Gulgassyl Nugmanova, Kuralay Yesmakhanova, Ratbay Myrzakulov
aa r X i v : . [ n li n . S I] S e p Camassa-Holm and M-CIV equations with self-consistent sources:geometry and peakon solutions
Gulmira Yergaliyeva ∗ , Tolkynay Myrzakul † , Gulgassyl Nugmanova ‡ ,Kuralay Yesmakhanova § and Ratbay Myrzakulov ¶ Eurasian International Center for Theoretical Physics,Eurasian National University, Nur-Sultan, 010008, Kazakhstan
Abstract
In this paper, we study one of generalized Heisenberg ferromagnet equations with self-consistent sources, namely, the so-called M-CIV equation with self-consistent sources (M-CIVESCS).The Lax representation of the M-CIVESCS is presented. We have shown that the M-CIVESCSand the CH equation with self-consistent sources (CHESCS) is geometrically equivalent each toother. The gauge equivalence between these equations is proved. Soliton (peakon) and pseudo-spherical surfaces induced by these equations are considered. The one peakon solution of theM-CIVESCS is presented.
KEYWORDS:
Camassa-Holm equation with self-consistent sources; Heisenberg ferromagnetequation; Heisenberg ferromagnet equation with self-consistent sources; Heisenberg ferromagnetequation with self-consistent potentials; Lax representation; conservation laws; peakon; soliton.
Camassa-Holm equation (CHE) has the form u t + 2 ωu x − u xxt + 3 uu x = 2 u x u xx + uu xxx , (1.1)where u = u ( x, t ) is the fluid velocity in the x direction and ω = const is related to the criticalshallow water wave speed. This equation has several equivalent forms, for example, the followingones q t + 2 u x q + uq x = 0 , (1.2) q − u + u xx − ω = 0 , (1.3) ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] ¶ Email: [email protected] κ t + ( uκ ) x = 0 , (1.4) u − u xx + ω − νκ = 0 . (1.5)The CHE was implicitly contained in the class of multi-Hamiltonian system introduced by Fuchssteinerand Fokas [1]. It explicitly derived as a shallow water wave equation by Camassa and Holm [2] andafter their works, the CHE and its different modifications have been studied from many kinds ofviews [3]-[18]. The CHE shares most of the properties of the integrable system of KdV type andpossesses Lax representation (LR), the bi-Hamiltonian structure, smooth solitary wave solutions(as ω > ω →
0. When ω = 0, these kind of solutions are weak solutions and are called”peakons”.Integrable Heisenberg ferromagnet type equations play important role in modern physics andmathematics (see, e.g., Refs. [20]-[61]). Recently, integrable generalized Heisenberg ferromagnetequations which are equivalent to Camassa-Holm type equations were presented (see, e.g., refs.[20]-[25]). In particular, it is shown that the CHE is (geometrically and gauge) equivalent to thefollowing Myrzakulov-CIV (M-CIV) equation [20]-[27] A xt + ( uA x ) x + ( uA x + 0 . { A x , A t } ) A − β [ A, A x ] = 0 , (1.6) tr ( A x ) + 8 β ( u − u xx ) = 0 (1.7)or ( A t + uA x ) x + ( u xx − u x − u ) A − β [ A, A x ] = 0 , (1.8) tr ( A x ) + 8 β ( u − u xx ) = 0 . (1.9)Here A = (cid:18) A A − A + − A (cid:19) , A ± = A ± iA , A = I, A = ( A , A , A ) , A = 1 , (1.10) A x = − β qI, { A t , A x } = [(8 β u − q − u x + u xx )] I. (1.11)There are many integrable generalizations of the CHE. One of such generalizations is the CHE withself-consistent sources (CHESCS) [3]. Such type integrable equations with self-consistent sourceshave attracted much attention in recent years [15]. They have important applications in manybranches of physics. For example, the nonlinear Schr¨ o dinger equation with self-consistent sourcesdescribes the nonlinear interaction of an electrostatic high-frequency wave with the ion acousticwave in a two component homogeneous plasma. Another example is the KdV equation with self-consistent sources. It describes the interaction of long and short capillary-gravity waves. The famousKP equation with self-consistent sources represents the nonlinear interaction of a long wave witha short wave packet propagating on the x - y plane at some angle to each other. In this paper, wewould like to study the M-CIV equation with self-consistent sources (M-CIVESCS) and its relationwith the CHESCS.This paper is organized as follows. In Section 2, we present the M-CIVESCS and its Laxrepresentation. In Section 3, the integrable motion of curves induced by the M-CIVESCS are2onstructed. The gauge equivalence between the M-CIVESCS and CHESCS is established in Section4. The peakon (soliton) and pseudo-spherical surfaces induced by the M-CIVESCS and the CHESCSare presented in Section 5 and in Section 6, respectively. In Section 7, the formulas of the one peakonsolution of the M-CIVESCS is presented. The M-CIVESCS and the CHESCS with N self-consistentsources is given in Section 8. In Section 9, the conclusion is presented. One of examples of the peakon spin systems is the following Myrzakulov-CIV equation with self-consistent sources (M-CIVESCS)[
A, A xt ] + ( u [ A, A x ]) x − β A x − α ( ω [ A, A x ]) x + 2 α χβ + α A x + 2 αβ χ x β + α A = 0 , (2.12) ψ x − ( α β −
14 )[( A + x A − − A + A − x ) ψ + 2( A − x A − A − A x ) ψ ] = 0 , (2.13) ψ x − ( α β −
14 )[2( A + A x − A + x A ) ψ + ( A + A − x − A + x A − ) ψ ] = 0 . (2.14)This equation can be written in the following equivalent form A xt + uA xx + ( uA x + 0 . { A x , A t } ) A + u A x + u I + u [ A, A x ] = 0 , (2.15) ψ x − ( α β −
14 )[( A + x A − − A + A − x ) ψ + 2( A − x A − A − A x ) ψ ] = 0 , (2.16) ψ x − ( α β −
14 )[2( A + A x − A + x A ) ψ + ( A + A − x − A + x A − ) ψ ] = 0 , (2.17)where ω = φ , χ = ω x + ω , α, β = consts and u = u − αω, u = α χβ + α − β , u = αβ χ x β + α , (2.18) A = (cid:18) A A − A + − A (cid:19) , A ± = A ± iA , A = I, A = ( A , A , A ) , A = 1 . (2.19)The LR of the M-CIVESCS reads as Ψ x = U Ψ , (2.20)Ψ t = V Ψ . (2.21)Here U = (cid:18) λ β − (cid:19) [ A, A x ] , (2.22) V = (cid:18) β − λ (cid:19) A + (cid:20) βλ − β − (cid:18) λ β − (cid:19) u (cid:21) [ A, A x ] + (cid:18) λ − β (cid:19) Z, (2.23)where Z = 12 (cid:20) u x + u xx β − αβχ x β + α (cid:21) − (cid:20) A, A t − (cid:18) β − u + αβωβ + α (cid:19) A x (cid:21) . (2.24)3 Integrable motion of space curves induced by the M-CIVESCS
In this section, we consider the integrable motion of space curves induced by the M-CIVESCS.As usual, let us consider a smooth space curve γ ( x, t ) : [0 , X ] × [0 , T ] → R in R . Let x is thearc length of the curve at each time t . The corresponding Frenet-Serret equation and its temporalcounterpart look like e e e x = C e e e , e e e t = G e e e , (3.25)where e j are the unit tangent vector ( j = 1), principal normal vector ( j = 2) and binormal vector( j = 3) which given by e = γ x , e = γ xx | γ xx | , e = e ∧ e , respectively. Here C = κ κ − κ τ − κ − τ , G = ω ω − ω ω − ω − ω , (3.26)where τ , κ , κ are the ”torsion”, ”geodesic curvature” and ”normal curvature” of the curve, re-spectively; ω j are some functions. The compatibility condition of the equations (3.25) reads as C t − G x + [ C, G ] = 0 (3.27)or in elements κ t − ω x − κ ω + τ ω = 0 , (3.28) κ t − ω x + κ ω − τ ω = 0 , (3.29) τ t − ω x − κ ω + κ ω = 0 . (3.30)We now assume A ≡ e . Let take place the following expressions κ = i, κ = λ ( q − , τ = − iλ ( q + 1) , (3.31)where q = 0 . λ ( κ + iτ ). Then we have ω = i [(0 . λ − − λu )( q + 1) + 0 . λ − ( u x + u xx )] , (3.32) ω = [(0 . λ − − λu )( q + 1) + 0 . λ − ( u x + u xx )] , (3.33) ω = i [0 . λ − − u − u x ] . (3.34)Now Eqs.(3.28)-(3.30) give us the following equations for q, u, φ : q t + 2 qu x + uq x − ω x + ω xxx = 0 , (3.35) φ xx − ( α q + 14 ) φ = 0 (3.36)or q t + 2 qu x + uq x − [( φ ) x − ( φ ) xxx ] = 0 , (3.37) φ xx − ( α q + 14 ) φ = 0 . (3.38)It is nothing but the CHESCS [3]-[4]. So, we have proved that the M-CIVESCS is the Lakshmanan(geometrical) equivalent to the CHESCS. 4 Gauge equivalence between the M-CIVESCS and the CHESCS
Above, we have shown that the M-CIVESCS and the CHESCS are the geometrical (Lakshmanan)equivalent each to other. Now we consider the possible gauge equivalence between these equations[26]. First, we note that from the results of the previous section and from the isomorphism so (3) ≈ su (2) follow the LR for the CHESCS of the form [3]Φ x = U Φ , (4.39)Φ t = V Φ , (4.40)where Φ = (cid:18) φ φ (cid:19) , U = (cid:18) − . λλq . (cid:19) , V = V + V ′ . (4.41)Here V = (cid:18) u x + u xx − λ λ − uλ u x + u xx + q λ − uqλ λ − u x + u xx (cid:19) , (4.42) V ′ = − αλ χ λ + α ) σ + αλ ωλ + α (cid:18) q (cid:19) − αλχ x λ + α ) Σ , Σ = (cid:18) (cid:19) . (4.43)The compatibility condition Φ xt = Φ tx given by U t − V x + [ U , V ] = 0 (4.44)is equivalent to the CHESCS (3.35)-(3.36). Consider the transformation Ψ = g − Φ, where Ψ is thesolution of the equations (2.20)-(2.21) and g = Φ | λ = β . Then the Lax pairs of the M-CIVESCS andCHESCS is related by the following equations U = g − U g − g − g x , V = g − V g − g − g t . (4.45)Note that these LR can be rewritten in the equivalent scalar forms. For example, the equivalentscalar form of the LR for the CHESCS is given by [3] φ xx = ( λ q + 14 ) φ , (4.46) φ t = (cid:18) λ − u + αλ ωλ + α (cid:19) φ x + (cid:18) u x − αλ ω x λ + α ) (cid:19) φ . (4.47)The compatibility condition φ xxt = φ txx is equivalent to the CHESCS (3.35)-(3.36). Finally wepresent the following important relation between the solutions of the M-CIVESCS and the CHESCS: tr ( A x ) = − β q = − β ( u − u xx ) (4.48)or A x = − β q = − β ( u − u xx ) . (4.49)5 Peakon (soliton) surfaces
As well-known, the Sym-Tafel formula gives an interesting connection between the classical geom-etry of manifolds immersed in R n and the integrable systems. Using the Sym-Tafel formula, herewe want to construct the soliton (peakon) surface induced by the CHESCS and the M-CIVESCS.To this end, let us consider a λ -family of parametric surfaces given by the matrix r = r ( x, t, λ ).According to the Sym-Tafel formula, this matrix defines as r = Φ − Φ λ , (5.50)where Φ is the solution of the equations (4.39)-(4.40). We have r x = Φ − U λ Φ , r t = Φ − V λ Φ . (5.51)Then the components of the metric tensor define as g ij = r ,i · r ,j , (5.52)where r = ( r , r , r ) is the position vector, r , ≡ r x , r , ≡ r t . Then for example, the firstfundamental form of the soliton surface is given by I = g ij dx i dx j = r x dx + 2 r x r t dxdt + r t dt , (5.53)where r x = 12 tr ( r x ) , r x r t = 12 tr ( r x r t ) , r t = 12 tr ( r t ) . (5.54)For example, g = r x = q. (5.55)Similarly, we can construct the second fundamental form in the standard way (see, e.g., [19]). Let us now we present the main elements of the soliton (peakon) surfaces induced by the M-CIVESCS. In this case, the matrix r is given by r = Ψ − Ψ λ . (5.56)Here Ψ is the solution of the equations (2.20)-(2.21). These equations give r x = Ψ − U λ Ψ , r t = Ψ − V λ Ψ . (5.57)Then the components of the metric tensor is given by the formula (5.52). For example, g = − β tr ( A x ) . (5.58)Now it is not difficult to construct the first and second fundamental forms of the peakon (soliton)surfaces that we can do in the standard way (see, e.g., [19]).6 Pseudo-spherical surfaces
In this section, we briefly present the main facts on the pseudo-spherical surfaces induced by theM-CIVESCS and the CHESCS.
Consider the following linear problem d Φ = Y Φ , (6.59)where Y = 12 (cid:18) ω ω − ω ω + ω − ω (cid:19) = U dx + V dt. (6.60)The integrable condition of the 1-form Y is given by d Y = ( U t − V x + [ U , V ]) dt ∧ dx = 0 . (6.61)This means that the one-forms ω j satisfy the following structure equations dω = ω ∧ ω , (6.62) dω = ω ∧ ω , (6.63) dω = ω ∧ ω . (6.64)These equations are equivalent to the CHESCS. We now return to the M-CIVESCS. To construct the pseudo-spherical surfaces induced by thisequation, let us consider the linear problem d Ψ = Y Ψ , (6.65)where the 1-form Y reads as Y = 12 (cid:18) σ σ − σ σ + σ − σ (cid:19) = U dx + V dt. (6.66)As in the previous subsection, we consider the integrable condition of the 1-form Y : d Y = ( U t − V x + [ U , V ]) dt ∧ dx = 0 (6.67)which in components takes the form dσ = σ ∧ σ , (6.68) dσ = σ ∧ σ , (6.69) dσ = σ ∧ σ . (6.70)These equations is equivalent to the M-CIVESCS. Thus the M-CIVESCS and the CHESCS describesome kind pseudo-spherical surfaces as their analogies without sources (see, e.g., refs. [16]-[17]).7 One peakon solution of the M-CIV ESCS
As the integrable equation, the M-CIVESCS has all ingredients of integrable systems like LR,conservation laws, bi-Hamiltonian structure, soliton solutions and so on. In particular, it admitsthe peakon solutions. Here let us present a one peakon solution of the M-CIVESCS. To constructthis 1-peakon solution, we use the corresponding 1-peakon solution of the CHESCS [3]. The 1-peakon solution of the M-CIVESCS has the form A + = 2 g g | g | + | g | , A = | g | − | g | | g | + | g | , (7.71) ψ = ¯ g φ + ¯ g φ | g | + | g | , ψ = − g φ + g φ | g | + | g | . (7.72)Here φ = λ − ( φ x + 0 . φ ) , g j = φ j | λ = β , (7.73)where φ = p σ ′ ( t ) ce − | x − ct + σ ( t ) | , u = ce −| x − ct + σ ( t ) | (7.74)is the 1-peakon solution of the CHESCS [3]. N sources case In the previous sections, we have considered the M-CIVESCS and the CHESCS with the one self-consistent source. In this section, we present, in short form, integrable generalizations of theseequations with N self-consistent sources. N -sources The M-CIVE with N self-consistent sources has the form[ A, A xt + ( uA x ) x ] − β A x − N X j =1 λ j ( ω j [ A, A x ]) x + 2 λ j χ j β + λ j A x + 2 λ j β χ jx β + λ j A ! = 0 , (8.75) ψ jx − ( λ j β −
14 )[( A + x A − − A + A − x ) ψ j + 2( A − x A − A − A x ) ψ j ] = 0 , (8.76) ψ jx − ( λ j β −
14 )[2( A + A x − A + x A ) ψ j + ( A + A − x − A + x A − ) ψ j ] = 0 , (8.77)where ω j = φ j , χ j = ω jx + ω j , α, β = consts and A = (cid:18) A A − A + − A (cid:19) , A ± = A ± iA , A = I, A = ( A , A , A ) , A = 1 . (8.78)8he LR of the M-CIVESCS reads as Ψ x = U Ψ , (8.79)Ψ t = V Ψ . (8.80)Here U = (cid:18) λ β − (cid:19) [ A, A x ] , (8.81) V = (cid:18) β − λ (cid:19) A + (cid:20) βλ − β − (cid:18) λ β − (cid:19) u (cid:21) [ A, A x ] + (cid:18) λ − β (cid:19) N X j =1 Z j , (8.82)where Z j = 12 (cid:20) u x + u xx β − λ j βχ jx β + λ j (cid:21) − (cid:20) A, A t − (cid:18) β − u + λ j βω j β + λ j (cid:19) A x (cid:21) . (8.83) N -sources The CHE with N self-consistent sources has the form [3] q t + 2 qu x + uq x − N X j =1 [( ϕ j ) x − ( ϕ j ) xxx ] = 0 , (8.84) ϕ j,xx − ( λ j q + 14 ) ϕ j = 0 . (8.85)Its LR reads as [3] φ xx = ( 14 + λq ) φ, (8.86) φ t = u x φ + ( 12 λ − u ) φ x + 2 N X j =1 λλ j φ j λ − λ j ( φ jx φ − φ j φ x ) , (8.87)which means that the CHESCS is Lax integrable [3]. The integrable generalized Heisenberg ferromagnet equation with self-consistent sources, namely,the M-CIVESCS is investigated. The integrable motion of space curves induced by the M-CIVESCSis constructed. Using this result, the geometrical equivalence between the M-CIVESCS and theCHESCS is established. It is shown that the M-CIVESCS and the CHESCS is gauge equivalenteach to other. The simplest conservation law and the one peakon solution are constructed. Thepeakon (soliton) surfaces induced by the M-CIVESCS and the CHESCS are presented.9 cknowledgements
This work was supported by the Ministry of Edication and Science of Kazakhstan under grants011800935 and 011800693.
References [1] Fuchssteiner B., Fokas A.S. Physica D, , 47 (1981)[2] Camassa R., Holm D. Phys. Rev. Lett., , 1661 (1993)[3] Huang Y., Yao Y., Zeng Y. On Camassa-Holm equation with self-consistent sources and itssolutions , [arXiv:0811.2552][4] Baltaeva I.I., Urazboev G.U.
About the Camassa-Holm equation with a self-consistent source ,Ufa mathematical journal, , N2, 10-18 (2011)[5] Yehui Huang. On Multi-Soliton Solution of Degasperis-Procesi Equation with Self-ConsistentSources , International Journal of Nonlinear Science, , No.3, 217-223 (2014)[6] Yehui Huang, Yunbo Zeng, Orlando Ragnisco. The Degasperis-Procesi equation with self-consistent sources , [arXiv:0807.0085][7] Qi Li, Wen Zhang, Qiu-Yuan Duan, Deng-Yuan Chen.
The transformation between the AKNShierarchy and the KN hierarchy with self-consistent sources , Journal of Nonlinear MathematicalPhysics, , No.4, 483490 (2011)[8] Runliang Lin, Haishen Yao, Yunbo Zeng. Restricted Flows and the Soliton Equation with Self-Consistent Sources , SIGMA, , 096 (2006)[9] Yuqin Yao, Yehui Huang, Yunbo Zeng. The two-component Camassa-Holm equation with self-consistent sources and its multisoliton solutions , Theoretical and Mathematical Physics, ,No1, 63-73 (2010)[10] Wen-Xiu Ma, Ruguang Zhou.
A coupled AKNS-Kaup-Newell soliton hierarchy , Journal ofMathematical Physics, (1999).[arXiv:solv-int/9908005][11] Yuqin Yao, Yunbo Zeng.
Integrable Rosochatius deformations of higher-order constrained flowsand the soliton hierarchy with self-consistent sources , [arXiv:0806.2251][12] Zhihua Yang, Ting Xiao, Yunbo Zeng.
Integrable dispersionless KdV hierarchy with sources ,[arXiv:0606047][13] Yuqin Yao, Yehui Huang, Guixiang Dong, Yunbo Zeng.
The new integrable deformations ofshort pulse equation and sine-Gordon equation, and their solutions , [arXiv:1012.0462][14] Yuqin Yao, Yunbo Zeng.
The generalized Kupershmidt deformation for constructing new inte-grable systems from integrable bi-Hamiltonian systems , [arXiv:1005.0281][15] Myrzakulov R.
Soliton Equations with Self-Consistent Sources , [arXiv:1409.0486]1016] Reyes E.G.
Geometric integrability of the CamassaHolm Equation , Letters in MathematicalPhysics, , No2, 117131 (2002)[17] Reyes E.G. Pseudo-potentials, nonlocal symmetries and integrability of some shallow waterequations , [arXiv:nlin/0212045][18] Ivanov R., Lyons T., Orr N.
A dressing method for soliton solutions of the Camassa-Holmequation , AIP Conference Proceedings 1895, 040003 (2017): https://doi.org/10.1063/1.5007370[19] Cieslinski J.
The Darboux-Bianchi-Backlund transformation and soliton surfaces ,[arXiv:1303.5472][20] Mussatayeva A., Myrzakul T., Nugmanova G., Yesmakhanova K.
Gauge equivalence betweenthe Myrzakulov-CIV equation and the Camassa-Holm Equation [21] Mussatayeva A., Myrzakul T., Nugmanova G., Yesmakhanova K., Myrzakulov R.
IntegrableMotion of Curves, Spin Equation and Camassa-Holm Equation , [arXiv:1907.10910][22] Kutum B.
Gauge equivalence between the M-CV equation and the two-component Camassa-Holm equation [23] Taishiyeva A., Myrzakul T., Nugmanova G., Yesmakhanova K.
On the Gauge Equivalencebetween the 2-component Camassa-Holm and Myrzakulov-CVI equations [24] Taishiyeva A., Myrzakul T., Nugmanova G., Yesmakhanova K., Myrzakulov R.
GeometricFlows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Fer-romagnet Equation [25] Kutum B., Nugmanova G., Myrzakul T., Yesmakhanova K., Myrzakulov R.
Integrable Defor-mation of Space Curves, Generalized Heisenberg Ferromagnet Equation and Two-ComponentModified Camassa-Holm Equation , [arXiv:1908.01371][26] Yergaliyeva G., Yesmakhanova K., Nugmanova G., Myrzakul T.
Gauge equivalence betweenthe Myrzakulov-CIV equation with self-consistent sources and Camassa-Holm equation withself-consistent sources [27] Mussatayeva A., Myrzakul T., Nugmanova G., Yesmakhanova K., Myrzakulov R.
IntegrableMotion of Curves, Spin Equation and Camassa-Holm Equation , [arXiv:1907.10910][28] G. Nugmanova, Z. Zhunussova, K. Yesmakhanova, G. Mamyrbekova, R. Myrzakulov. Interna-tional Journal of Mathematical, Computational, Statistical, Natural and Physical Engineering, , N8, 328-331 (2015).[29] U. Saleem, M. Hasan. J. Phys. A: Math. Theor., , 045204 (2010).[30] R. Myrzakulov, S. Vijayalakshmi, G. Nugmanova , M. Lakshmanan Physics Letters A, ,14-6, 391-396 (1997).[31] R. Myrzakulov, S. Vijayalakshmi, R. Syzdykova, M. Lakshmanan, J. Math. Phys., , 2122-2139 (1998). 1132] R. Myrzakulov, M. Lakshmanan, S. Vijayalakshmi, A. Danlybaeva , J. Math. Phys., , 3765-3771 (1998).[33] Myrzakulov R, Danlybaeva A.K, Nugmanova G.N. Theoretical and Mathematical Physics,V.118, 13, P. 441-451 (1999).[34] Myrzakulov R., Nugmanova G., Syzdykova R. Journal of Physics A: Mathematical & Theo-retical, V.31, 147, P.9535-9545 (1998).[35] Myrzakulov R., Daniel M., Amuda R. Physica A., V.234, 13-4, P.715-724 (1997).[36] Myrzakulov R., Makhankov V.G., Pashaev O.?. Letters in Mathematical Physics, V.16, N1,P.83-92 (1989)[37] Myrzakulov R., Makhankov V.G., Makhankov A. Physica Scripta, V.35, N3, P. 233-237 (1987)[38] Myrzakulov R., Pashaev O.K., Kholmurodov Kh. Physica Scripta, V.33, N4, P. 378-384 (1986)[39] Anco S.C., Myrzakulov R. Journal of Geometry and Physics, v.60, 1576-1603 (2010)[40] Myrzakulov R., Rahimov F.K., Myrzakul K., Serikbaev N.S. On the geometry of stationaryHeisenberg ferromagnets . In: ”Non-linear waves: Classical and Quantum Aspects”, KluwerAcademic Publishers, Dordrecht, Netherlands, P. 543-549 (2004)[41] Myrzakulov R., Serikbaev N.S., Myrzakul Kur., Rahimov F.K.
On continuous limits of somegeneralized compressible Heisenberg spin chains . Journal of NATO Science Series II. Mathe-matics, Physics and Chemistry, V 153, P. 535-542 (2004)[42] R.Myrzakulov, G. K. Mamyrbekova, G. N. Nugmanova, M. Lakshmanan. Symmetry, (3),1352-1375 (2015). [arXiv:1305.0098][43] R.Myrzakulov, G. K. Mamyrbekova, G. N. Nugmanova, K. Yesmakhanova, M. Lakshmanan.Physics Letters A, , N30-31, 2118-2123 (2014). [arXiv:1404.2088][44] Myrzakulov R., Martina L., Kozhamkulov T.A., Myrzakul Kur. Integrable Heisenberg ferro-magnets and soliton geometry of curves and surfaces . In book: ”Nonlinear Physics: Theoryand Experiment. II”. World Scientific, London, P. 248-253 (2003)[45] Myrzakulov R.
Integrability of the Gauss-Codazzi-Mainardi equation in 2+1 dimensions . In”Mathematical Problems of Nonlinear Dynamics”, Proc. of the Int. Conf. ”Progress in Non-linear sciences”, Nizhny Novgorod, Russia, July 2-6, 2001, V.1, P.314-319 (2001)[46] Chen Chi, Zhou Zi-Xiang.
Darboux Tranformation and Exact Solutions of the Myrzakulov-IEquations . Chin. Phys. Lett., , N8, 080504 (2009)[47] Chen Hai, Zhou Zi-Xiang. Darboux Transformation with a Double Spectral Parameter for theMyrzakulov-I Equation . Chin. Phys. Lett., , N12, 120504 (2014)[48] Zhao-Wen Yan, Min-Ru Chen, Ke Wu, Wei-Zhong Zhao. J. Phys. Soc. Jpn., , 094006 (2012)[49] Yan Zhao-Wen, Chen Min-Ru, Wu Ke, Zhao Wei-Zhong. Commun. Theor. Phys., , 463-468(2012) 1250] K.R. Ysmakhanova, G.N. Nugmanova, Wei-Zhong Zhao, Ke Wu. Integrable inhomogeneousLakshmanan-Myrzakulov equation , [nlin/0604034][51] Zhen-Huan Zhang, Ming Deng, Wei-Zhong Zhao, Ke Wu.
On the integrable inhomogeneousMyrzakulov-I equation , [arXiv: nlin/0603069][52] Martina L, Myrzakul Kur., Myrzakulov R, Soliani G. Journal of Mathematical Physics, V.42,13, P.1397-1417 (2001).[53] Xiao-Yu Wu, Bo Tian, Hui-Ling Zhen, Wen-Rong Sun and Ya Sun. Journal of Modern Optics,2015.[54] Z.S. Yersultanova, M. Zhassybayeva, K. Yesmakhanova, G. Nugmanova, R. Myrzakulov. In-ternational Journal of Geometric Methods in Modern Physics, , N1, 1550134 (2016)..[arXiv:1404.2270][55] Myrzakul Akbota and Myrzakulov Ratbay. Integrable Motion of Two Interacting Curves andHeisenberg Ferromagnetic Equations , Abstracts of XVIII-th Intern. Conference ”Geometry,Integrability and Quantization”, June 3-8, 2016, Bulgaria.[56] Myrzakul Akbota and Myrzakulov Ratbay.
Integrable motion of two interacting curves, spinsystems and the Manakov system . International Journal of Geometric Methods in ModernPhysics, , N1, 1550134 (2016). [arXiv:1606.06598][57] Myrzakul Akbota and Myrzakulov Ratbay. Darboux transformations and exact soliton solu-tions of integrable coupled spin systems related with the Manakov system , [arXiv:1607.08151][58] Myrzakul Akbota and Myrzakulov Ratbay.
Integrable geometric flows of interactingcurves/surfaces, multilayer spin systems and the vector nonlinear Schrodinger equation . In-ternational Journal of Geometric Methods in Modern Physics, , N1, 1550134 (2016).[arXiv:1608.08553][59] Myrzakulova Zh., Myrzakul A., Nugmanova G., MyrzakulovR. Notes on Integrable Motionof Two Interacting Curves and Two-layer Generalized Heisenberg Ferromagnet Equations ,[arXiv:1811:12216][60] Hussien R.A., Mohamed S.G.
Generated Surfaces via Inextensible Flows of Curves in R .Journal of Applied Mathematics, v.2016, Article ID 6178961 (2016).[61] Bekova G., Yesmakhanova K., Shaikhova G., Nugmanova G., Myrzakulov R. Geometric for-mulation and soliton solutions of the Myrzakulov-LXXIII equation and the complex short pulseequation ,[62] C. Qu, J. Song and R. Yao.