Geodesic
A type of multicomponent nonisospectral generalized nonlinear Schrödinger hierarchies
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 3, pp. 437-464 Cet article a éte moissonné depuis la source Math-Net.Ru

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We introduce a Lie algebra $A_1$ with an arbitrary constant $\alpha$ that can be used to solve nonisospectral problems. For a given higher-dimensional Lie algebra, we introduce two new classes of higher-dimensional Lie algebras extended by $A_1$. By solving the extended nonisospectral zero-curvature equations that correspond to nonisospectral problems, we derive several multicomponent nonisospectral hierarchies. For one of them, with the aid of the $Z^\varepsilon_N$-trace identity and given the Lax pairs, we obtain the bi-Hamilton structures.
Keywords: multicomponent nonisospectral hierarchy, $Z^\varepsilon_N$-trace identity, bi-Hamiltonian structure, nonisospectral problem. 
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Jianduo Yu; HaiFeng Wang; Chuanzhong Li. A type of multicomponent nonisospectral generalized nonlinear Schrödinger hierarchies. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 3, pp. 437-464. http://geodesic.mathdoc.fr/item/TMF_2023_215_3_a6/

[1] G.-Z. Tu, “The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems”, J. Math. Phys., 30:2 (1989), 330–338 | DOI | MR

[2] W.-X. Ma, “A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction”, Chinese J. Contemp. Math., 13:1 (1992), 79–89 | MR

[3] W.-X. Ma, M. Chen, “Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras”, J. Phys. A: Math. Gen., 39:34 (2006), 10787–10801 | DOI | MR

[4] X. G. Geng, W.-X. Ma, “A multipotential generalization of the nonlinear diffusion equation”, J. Phys. Soc. Japan, 69:4 (2000), 985–986 | DOI | MR

[5] X.-B. Hu, “A powerful approach to generate new integrable systems”, J. Phys. A, 27:7 (1994), 2497–2514 | DOI | MR

[6] Y. F. Zhang, J. Q. Mei, H. Y. Guan, “A method for generating isospectral and nonisospectral hierarchies of equations as well as symmetries”, J. Geom. Phys., 147 (2020), 103538, 15 pp. | DOI | MR

[7] Y.-F. Zhang, H.-W. Tam, “Generation of nonlinear evolution equations by reductions of the self-dual Yang–Mills equations”, Commun. Theor. Phys. (Beijing), 61:2 (2014), 203–206 | DOI | MR

[8] Y. F. Zhang, H. W. Tam, F. K. Guo, “Invertible linear transformations and the Lie algebras”, Commun. Nonlinear Sci. Numer. Simul., 13:4 (2008), 682–702 | DOI | MR

[9] Y. F. Zhang, H. Q. Zhang, “A direct method for integrable couplings of TD hierarchy”, J. Math. Phys., 43:1 (2002), 466–472 | DOI | MR

[10] W.-X. Ma, “A simple scheme for generating nonisospectral flows from zero curvature representation”, Phys. Lett. A, 179:3 (1993), 179–185 | DOI | MR

[11] W.-X. Ma, “The algebraic structures of isospectral Lax operators and applications to integrable equations”, J. Phys. A: Math. Gen., 25:20 (1992), 5329–5343 | DOI | MR

[12] W.-X. Ma, “Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations”, J. Math. Phys., 33:7 (1992), 2464–2476 | DOI | MR

[13] Z. J. Qiao, “Algebraic structure of the operator related to stationary systems”, Phys. Lett. A, 206:5–6 (1995), 347–358 | DOI | MR

[14] Z. J. Qiao, “New hierarchies of isospectral and non-isospectral integrable NLEEs derived from the Harry–Dym spectral problem”, Phys. A, 252:3–4 (1998), 377–387 | DOI | MR

[15] W.-X. Ma, “An approach for constructing nonisospectral hierarchies of evolution equations”, J. Phys. A: Math. Gen., 25:12 (1992), L719–L726 | DOI | MR

[16] Y. F. Zhang, W. J. Rui, “A few continuous and discrete dynamical systems”, Rep. Math. Phys., 78:1 (2016), 19–32 | DOI | MR

[17] X.-X. Xu, “An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy”, Appl. Math. Comput., 216:1 (2010), 344–353 | MR

[18] X.-H. Zhao, B. Tiao, H.-M. Li, Y.-J. Guo, “Solitons, periodic waves, breathers and integrability for a non-isospectral and variable-coefficient fifth-order Korteweg–de Vries equation in fluids”, Appl. Math. Lett., 65 (2017), 48–55 | DOI | MR

[19] P. G. Estévz, C. Savdón, “Miura-reciprocal transformations for non-isospectral Camassa–Holm hierarchies in $2+1$ dimensions”, J. Nonlinear Math. Phys., 20:4 (2013), 552–564 | DOI | MR

[20] P. G. Estévz, J. D. Lejarreta, C. Sardón, “Non-isospectral $1+1$ hierarchies arising from a Camassa–Holm hierarchy in $2+1$ dimensions”, J. Nonlinear Math. Phys., 18:1 (2011), 9–28 | DOI | MR

[21] H. F. Wang, Y. F. Zhang, “A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system”, Internat. J. Geom. Methods Modern Phys., 18:10 (2021), 2150156, 17 pp. | DOI | MR

[22] K. M. Tamizhmani, M. Lakshmanan, “Complete integrability of the Korteweg–de Vries equation under perturbation around its solution: Lie–Backlund symmetry approach”, J. Phys. A: Math. Gen., 16:16 (1983), 3773–3782 | DOI | MR

[23] B. Fuchssteiner, “Coupling of completely integrable systems: the perturbation bundle”, Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (Exeter, UK, July 14–19, 1992), Nato Science Series C, 413, ed. P. A. Clarkson, Kluwer, Dordrecht, 1993, 125–138 | DOI | MR

[24] W.-X. Ma, X.-X. Xu, Y. F. Zhang, “Semi-direct sums of Lie algebras and continuous integrable couplings”, Phys. Lett. A, 351:3 (2006), 125–130 | DOI | MR

[25] H. F. Wang, Y. F. Zhang, “Two nonisospectral integrable hierarchies and its integrable coupling”, Internat. J. Theoret. Phys., 59:8 (2020), 2529–2539 | DOI | MR

[26] W.-X. Ma, J. H. Meng, H. Q. Zhang, “Integrable couplings, variational identities and Hamiltonian formulations”, Global J. Math. Sci., 1:1 (2012), 1–17

[27] M. Mcanally, W.-X. Ma, “Two integrable couplings of a generalized D-Kaup–Newell hierarchy and their Hamiltonian and bi-Hamiltonian structures”, Nonlinear Anal., 191 (2020), 111629, 13 pp. | DOI | MR

[28] F. K. Guo, Y. F. Zhang, “The quadratic-form identity for constructing the Hamiltonian structure of integrable systems”, J. Phys. A: Math. Gen., 38:40 (2005), 8537–8548 | DOI | MR

[29] X.-G. Geng, W.-X. Ma, “A generalized Kaup–Newell spectral problem, soliton equations and finite-dimensional integrable systems”, Nuovo Cimento A, 108:4 (2010), 477–486 | DOI | MR

[30] F. K. Guo, Y. F. Zhang, “A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling”, J. Math. Phys., 44:12 (2003), 5793–5803 | DOI | MR

[31] Y. F. Zhang, H. Tam, B. L. Feng, “A generalized Zakharov–Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter”, Chaos Solitons Fractals, 44:11 (2011), 968–976 | DOI | MR

[32] Y. F. Zhang, E. G. Fan, H. Tam, “A few expanding Lie algebras of the Lie algebra $A_1$ and applications”, Phys. Lett. A, 359:5 (2006), 471–480 | DOI | MR

[33] W.-X. Ma, “Riemann–Hilbert problems and $N$-soliton solutions for a coupled mKdV system”, J. Geom. Phys., 132 (2018), 45–54 | DOI | MR

[34] H. F. Wang, Y. F. Zhang, “A kind of generalized integrable couplings and their bi-Hamiltonian structure”, Internat. J. Theoret. Phys., 60:5 (2021), 1797–1812 | DOI | MR

[35] H. F. Wang, Y. F. Zhang, “A kind of nonisospectral and isospectral integrable couplings and their Hamiltonian systems”, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105822, 15 pp. | DOI | MR

[36] H. F. Wang, Y. F. Zhang, “A new multi-component integrable coupling and its application to isospectral and nonisospectral problems”, Commun. Nonlinear Sci. Numer. Simul., 105 (2022), 106075, 15 pp. | DOI | MR

[37] H. F. Wang, The multi-component nonisospectral KdV hierarchies associated with a new class of $N$-dimensional Lie algebra, arXiv: 2201.03205