Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system

Xue Geng; Liang Guan; Dianlou Du; Xue Geng; Liang Guan; Dianlou Du

doi:10.3934/math.2022557

AIMS Mathematics

2022, Volume 7, Issue 6: 9989-10008. doi: 10.3934/math.2022557

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Research article

Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system

1.
School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, Henan, China
2.
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, Henan, China

Received: 28 December 2021 Revised: 22 February 2022 Accepted: 03 March 2022 Published: 21 March 2022
MSC : 35Q53, 37J15, 37J35

The $ \mathfrak{gl}_3(\mathbb{C}) $ rational Gaudin model governed by $ 3\times 3 $ Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.
- three-wave resonant interaction system,
- non-hyperelliptic algebraic curve,
- separated variables,
- action-angle variables
Citation: Xue Geng, Liang Guan, Dianlou Du. Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system[J]. AIMS Mathematics, 2022, 7(6): 9989-10008. doi: 10.3934/math.2022557

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Abstract

The $ \mathfrak{gl}_3(\mathbb{C}) $ rational Gaudin model governed by $ 3\times 3 $ Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.

References

[1]	M. Gaudin, Diagonalisation d$'$une classe d$'$Hamiltoniens de spin, J. Phys. France., 37 (1976), 1089–1098. https://doi.org/10.1051/jphys:0197600370100108700 doi: 10.1051/jphys:0197600370100108700
[2]	M. Gaudin, The bethe wavefunction (Français), Paris: Masson, 1983. https://doi.org/10.1017/CBO9781107053885
[3]	B. Jur${\rm\check{c}}$o, Classical Yang-Baxter equations and quantum integrable systems, J. Math. Phys., 30 (1989), 1289. https://doi.org/10.1063/1.528305 doi: 10.1063/1.528305
[4]	A. Reyman, M. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable systems, In: Dynamical systems vii, Berlin: Springer, 1994,116–225. https://doi.org/10.1007/978-3-662-06796-3_7
[5]	B. Feigin, E. Frenkel, N. Reshetikhin, Gaudin model, Bethe ansatz and critical level, Commun. Math. Phys., 166 (1994), 27–62. https://doi.org/10.1007/BF02099300 doi: 10.1007/BF02099300
[6]	A. Hone, V. Kuznetsov, O. Ragnisco, B$ \rm \ddot{a} $cklund transformations for the $\mathfrak{sl}(2)$ Gaudin magnet, J. Phys. A: Math. Gen., 34 (2001), 2477. https://doi.org/10.1088/0305-4470/34/11/336 doi: 10.1088/0305-4470/34/11/336
[7]	E. Sklyanin, Separation of variables in the Gaudin model, J. Soviet. Math., 47 (1989), 2473–2488. https://doi.org/10.1007/BF01840429 doi: 10.1007/BF01840429
[8]	E. Sklyanin, Separation of variables in the classical integrable $SL(3)$ magnetic chain, Commun. Math. Phys., 150 (1992), 181–191. https://doi.org/10.1007/BF02096572 doi: 10.1007/BF02096572
[9]	V. Kuznetsov, Quadrics on real Riemannian spaces of constant curvature: separation of variables and connection with Gaudin magnet, J. Math. Phys., 33 (1992), 3240. https://doi.org/10.1063/1.529542 doi: 10.1063/1.529542
[10]	D. Du, X. Geng, On the relationship between the classical Dicke-Jaynes-Cummings-Gaudin model and the nonlinear Schrödinger equation, J. Math. Phys., 54 (2013), 053510. https://doi.org/10.1063/1.4804943 doi: 10.1063/1.4804943
[11]	E. Kalnins, V. Kuznetsov, W. Miller, Quadrics on complex Riemannian spaces of constant curvature, separation of variables, and the Gaudin magnet, J. Math. Phys., 35 (1994), 1710. https://doi.org/10.1063/1.530566 doi: 10.1063/1.530566
[12]	J. Eilbeck, V. Enol'skii, V. Kuznetsov, A. Tsiganov, Linear $r$-matrix algebra for classical separable systems, J. Phys. A: Math. Gen., 27 (1994), 567. https://doi.org/10.1088/0305-4470/27/2/038 doi: 10.1088/0305-4470/27/2/038
[13]	J. Harnad, P. Winternitz, Classical and quantum integrable systems in 263-1263-1263-1 and separation of variables, Commun. Math. Phys., 172 (1995), 263–285. https://doi.org/10.1007/BF02099428 doi: 10.1007/BF02099428
[14]	C. Cao, Y. Wu, X. Geng, Relation between the Kadometsev-Petviashvili equation and the confocal involutive system, J. Math. Phys., 40 (1999), 3948. https://doi.org/10.1063/1.532936 doi: 10.1063/1.532936
[15]	X. Yang, D. Du, From nonlinear Schr$\rm\ddot{o}$dinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferentiale quations, J. Math. Phys., 51 (2010), 083505. https://doi.org/10.1063/1.3453389 doi: 10.1063/1.3453389
[16]	D. Du, X. Yang, An alternative approach to solve the mixed AKNS equations, J. Math. Anal. Appl., 414 (2014), 850–870. https://doi.org/10.1016/j.jmaa.2014.01.041 doi: 10.1016/j.jmaa.2014.01.041
[17]	D. Scott, Classical functional Bethe ansatz for $ SL(N)$: separation of variables for the magnetic chain, J. Math. Phys., 35 (1994), 5831. https://doi.org/10.1063/1.530712 doi: 10.1063/1.530712
[18]	M. Gekhtman, Separation of variables in the classical $SL(N)$ magnetic chain, Commun. Math. Phys., 167 (1995), 593–605. https://doi.org/10.1007/BF02101537 doi: 10.1007/BF02101537
[19]	E. Sklyanin, Separation of variables: new trends, Prog. Theor. Phys. Supp., 118 (1995), 35–60. https://doi.org/10.1143/PTPS.118.35 doi: 10.1143/PTPS.118.35
[20]	B. Dubrovin, T. Skrypnyk, Separation of variables for linear Lax algebras and classical $r$-matrices, J. Math. Phys., 59 (2018), 091405. https://doi.org/10.1063/1.5031769 doi: 10.1063/1.5031769
[21]	S. Manakov, Example of a completely integrable nonlinear wave field with nontrivial dynamics (lee model), Theor. Math. Phys., 28 (1976), 709–714. https://doi.org/10.1007/BF01029027 doi: 10.1007/BF01029027
[22]	R. Beals, D. Sattinger, On the complete integrability of completely integrable systems, Commun. Math. Phys., 138 (1991), 409–436. https://doi.org/10.1007/BF02102035 doi: 10.1007/BF02102035
[23]	M. Adams, J. Harnad, J. Hurtubise, Darboux coordinates and Liouville-Arnold integration in loop algebras, Commun. Math. Phys., 155 (1993), 385–413. https://doi.org/10.1007/bf02097398 doi: 10.1007/bf02097398
[24]	L. Dickey, Integrable nonlinear equations and Liouville's theorem, I, Commun. Math. Phys., 82 (1981), 345–360. https://doi.org/10.1007/BF01237043 doi: 10.1007/BF01237043
[25]	L. Dickey, Soliton equations and Hamiltonian systems, Singapore: World Scientific Publishing, 2003. https://doi.org/10.1142/5108
[26]	Y. Wu, X. Geng, A finite-dimensional integrable system associated with the three-wave interaction equations, J. Math. Phys., 40 (1999), 3409. https://doi.org/10.1063/1.532896 doi: 10.1063/1.532896
[27]	F. Calogero, A. Degasperis, Novel solution of the system describing the resonant interaction of three waves, Physica D, 200 (2005), 242–256. https://doi.org/10.1016/j.physd.2004.11.007 doi: 10.1016/j.physd.2004.11.007
[28]	V. Zakharov, S. Manakov, Resonant interaction of wave packets in nonlinear media, JETP Lett., 18 (1973), 243–245.
[29]	V. Zakharov, S. Manakov, The theory of resonant interaction of wave packets in nonlinear media, Sov. Phys. JETP Lett., 42 (1975), 842–850.
[30]	D. Kaup, The three-wave interaction-a nondispersive phenomenon, Stud. Appl. Math., 55 (1976), 9–44. https://doi.org/10.1002/sapm19765519 doi: 10.1002/sapm19765519
[31]	R. Conte, A. Grundland, M. Musette, A reduction of the resonant three-wave interaction to the generic sixth Painlevé equation, J. Phys. A: Math. Gen., 39 (2006), 12115. https://doi.org/10.1088/0305-4470/39/39/S07 doi: 10.1088/0305-4470/39/39/S07
[32]	R. Buckingham, R. Jenkins, P. Miller, Semiclassical soliton ensembles for the three-wave resonant interaction equations, Commun. Math. Phys., 354 (2017), 1015–1100. https://doi.org/10.1007/s00220-017-2897-7 doi: 10.1007/s00220-017-2897-7
[33]	G. Huang, Exact solitary wave solutions of three-wave interaction equations with dispersion, J. Phys. A: Math. Gen., 33 (2000), 8477. https://doi.org/10.1088/0305-4470/33/47/310 doi: 10.1088/0305-4470/33/47/310
[34]	G. Valiulis, K. Stali$\rm\bar{u}$nas, On the subject of the integrability and soliton solutions of three-wave interaction equations, Lith. J. Phys., 31 (1991), 61.
[35]	F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, Rogue waves emerging from the resonant interaction of three waves, Phys. Rev. Lett., 111 (2013), 114101. https://doi.org/10.1103/PhysRevLett.111.114101 doi: 10.1103/PhysRevLett.111.114101
[36]	A. Degasperis, S. Lombardo, Rational solitons of wave resonant-interaction models, Phys. Rev. E, 88 (2013), 052914. https://doi.org/10.1103/PhysRevE.88.052914 doi: 10.1103/PhysRevE.88.052914
[37]	X. Wang, J. Cao, Y. Chen, Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation, Phys. Scr., 90 (2015), 105201. https://doi.org/10.1088/0031-8949/90/10/105201 doi: 10.1088/0031-8949/90/10/105201
[38]	B. Yang, J. Yang, General rogue waves in the three-wave resonant interaction systems, IMA J. Appl. Math., 86 (2021), 378–425. https://doi.org/10.1093/imamat/hxab005 doi: 10.1093/imamat/hxab005
[39]	X. Geng, Y. Li, B. Xue, A second-order three-wave interaction system and its rogue wave solutions, Nonlinear. Dyn., 105 (2021), 2575–2593. https://doi.org/10.1007/s11071-021-06727-2 doi: 10.1007/s11071-021-06727-2
[40]	G. He, X. Geng, L. Wu, Algebro-geometric quasiperiodic solutions to the three-wave resonant interaction hierarchy, SIAM J. Math. Anal., 46 (2014), 1348–1384. https://doi.org/10.1137/130918794 doi: 10.1137/130918794
[41]	X. Geng, Y. Zhai, H. Dai, Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math., 263 (2014), 123–153. https://doi.org/10.1016/j.aim.2014.06.013 doi: 10.1016/j.aim.2014.06.013
[42]	J. Wei, X. Geng, X. Zeng, The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices, T. Am. Math. Soc., 371 (2019), 1483–1507. https://doi.org/10.1090/tran/7349 doi: 10.1090/tran/7349
[43]	G. Arutyunov, Elements of classical and quantum integrable systems, Switzerland: Springer, 2019. https://doi.org/10.1007/978-3-030-24198-8

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