Generation of higher-dimensional isospectral-nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebra

Haifeng Wang   ; Yufeng Zhang

Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 2, pp. 221-248

Résumé

We construct a new class of higher-dimensional column-vector loop algebras. Based on it, a method for generating higher-dimensional isospectral–nonisospectral integrable hierarchies is proposed. As an application, we derive a generalized nonisospectral integrable Schrödinger hierarchy that can be reduced to the famous derivative nonlinear Schrödinger equation. By using the higher-dimensional column-vector loop algebras, we obtain an extended isospectral–nonisospectral integrable Schrödinger hierarchy that can be reduced to many classical and new equations, such as the extended nonisospectral derivative nonlinear Schrödinger system, the heat equation, and the Fokker–Planck equation, which has a wide range of applications in stochastic dynamical systems. Furthermore, we deduce a $Z_N^\varepsilon$ nonisospectral integrable Schrödinger hierarchy, which means that the coupling results are extended to an arbitrary number of components. Additionally, the Hamiltonian structures of these hierarchies are discussed by using the quadratic form trace identity.

Détail

Keywords: generalized integrable Schrödinger hierarchies, higher-dimensional isospectral–nonisospectral integrable hierarchies, higher-dimensional column-vector loop algebras, Hamiltonian structure.

Bibliographie

[1] D. Levi, “Hierarchies of integrable equations obtained as nonisospectral (in $x$ and $t$) deformations of the Schrödinger spectral problem”, Phys. Lett. A, 119:9 (1987), 453–456 | DOI | MR

[2] D. Levi, O. Ragnisco, “Non-isospectral deformations and Darboux transformations for the third-order spectral problem”, Inverse Problems, 4:3 (1988), 815–828 | DOI | MR

[3] P. R. Gordoa, A. Pickering, “Nonisospectral scattering problems: a key to integrable hierarchies”, J. Math. Phys., 40:11 (1999), 5749–5786 ; “On a new non-isospectral variant of the Boussinesq hierarchy”, J. Phys. A: Math. Gen., 33:3 (2000), 557–567 | DOI | MR | DOI | MR

[4] P. R. Gordoa, A. Pickering, Z. N. Zhu, “New $2+1$ dimensional nonisospectral Toda lattice hierarchy”, J. Math. Phys., 48:2 (2007), 023515, 18 pp. | DOI | MR

[5] F. Calogero, “A method to generate solvable nonlinear evolution equations”, Lett. Nuovo Cimento, 14:12 (1975), 443–447 | DOI | MR

[6] F. Calogero, A. Degasperis, “Solution by the spectral-transform method of a nonlinear evolution equation, including as a special case the cylindrical KdV equation”, Lett. Nuovo Cimento, 23:4 (1978), 150–154 | DOI | MR

[7] C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics, 30, Cambridge Univ. Press, Cambridge, 2002 | DOI | MR

[8] P. Clarkson, P. Gordoa, A. Pickering, “Multicomponent equations associated to non-isospectral scattering problems”, Inverse Problems, 13:6 (1997), 1463–1476 | DOI | MR

[9] Yu. Berezansky, M. Shmoish, “Nonisospectral flows on semi-infinite Jacobi matrices”, Nonlinear Math. Phys., 1:2 (1994), 116–146 | DOI | MR

[10] A. Sakhnovich, “Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions”, J. Phys. A: Math. Theor., 41:15 (2008), 155204, 15 pp. | DOI | MR

[11] P. G. Estévez, 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

[12] B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations. The Inverse Spectral Transform in $2+1$ Dimensions, Springer, New York, 1992 ; Solitons in Multidimensions. Inverse Spectral Transform Method, Singapore, World Sci., 1993 | DOI | MR | DOI | MR

[13] V. E. Zakharov, “The inverse scattering method”, Solitons, Topics in Current Physics, 17, eds. R. K. Bullough, P. J. Caudrey, Springer, Berlin, 1990, 243–285 | DOI | MR

[14] F. Magri, Nonlinear Evolution Equations and Dynamical Systems, Lecture Notes in Physics, 120, eds. M. Boiti, F. Pempinelli, G. Soliani, Springer, Berlin, 1980 | DOI | MR

[15] W.-X. Ma, “An approach for constructing non-isospectral hierarchies of evolution equations”, J. Phys. A: Math. Gen., 25:12 (1992), L719–L726 ; “A simple scheme for generating nonisospectral flows from the zero curvature representation”, Phys. Lett. A, 179:3 (1993), 179–185 | DOI | MR | DOI | MR

[16] 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

[17] 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

[18] W.-X. Ma, “A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction”, Chinese J. Contemp. Math., 13:1 (1992), 79–89 ; “$K$ symmetries and $\tau$ symmetries of evolution equations and their Lie algebras”, J. Phys. A: Math. Gen., 23:13 (1990), 2707–2716 | MR | DOI | MR

[19] Y. S. Li, “A kind of evolution equations and the deform of spectral”, Sci. Sinica Ser. A, 25 (1982), 385–387 (in Chinese)

[20] Y.-S. Li, G.-C. Zhu, “New set of symmetries of the integrable equations, Lie algebras and non-isospectral evolution equations. II. AKNS system”, J. Phys. A: Math. Gen., 19:18 (1986), 3713–3725 | DOI | MR

[21] Y. S. Li, D. W. Zhuang, “Nonlinear evolution equations related to characteristic problems dependent on potential energy”, Acta Math. Sinica, 25:4 (1982), 464–474 | MR

[22] 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

[23] Y. Zhang, H. Tam, “A few integrable systems and spatial spectral transformations”, Commun. Nonlinear Sci. Numer. Simul., 14:11 (2009), 3770–3783 | DOI | MR

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

[25] 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

[26] X. K. Chang, X. M. Chen, X. B. Hu, “A generalized nonisospectral Camassa–Holm equation and its multipeakon solutions”, Adv. Math., 263 (2014), 154–177 | DOI | MR

[27] X.-K. Chang, X.-B. Hu, S.-H. Li, “Moment modification, multipeakons, and nonisospectral generalizations”, J. Differ. Equ., 265:9 (2018), 3858–3887 | DOI | MR

[28] H. F. Wang, Y. F. Zhang, “Lump, lumpoff and predictable rogue wave solutions to a dimensionally reduced Hirota bilinear equation”, Chinese Phys. B, 29:4 (2020), 040501, 7 pp. | DOI

[29] Khai-Fen Van, Yui-Fen Chzhan, “Metod $\bar\partial$-odevaniya dlya nekotorykh ($2+1$)-mernykh integriruemykh sparennykh sistem”, TMF, 208:3 (2021), 452–470 | DOI | DOI

[30] V. A. Belinskii, V. E. Zakharov, “Integrirovanie uravnenii Einshteina metodom obratnoi zadachi rasseyaniya i vychislenie tochnykh solitonnykh reshenii”, ZhETF, 75:6 (1978), 1955–1971 | MR

[31] A. V. Mikhailov, A. I. Yaremchuk, “Cylindrically symmetric solutions of the nonlinear chiral field model ($\sigma$ model)”, Nucl. Phys. B, 202:3 (1982), 508–522 | DOI | MR

[32] G. A. Alekseev, “N-solitonnye resheniya uravnenii Einshteina–Maksvella”, Pisma v ZhETF, 32:4 (1980), 301–303

[33] D. Maison, “Are the stationary, axially symmetric Einstein equations completely integrable?”, Phys. Rev. Lett., 41:8 (1978), 521–522 | DOI | MR

[34] F. Calogero, A. Degasperis, “Conservation laws for classes of nonlinear evolution equations solvable by the spectral transform”, Commun. Math. Phys., 63:2 (1978), 155–176 | DOI | MR

[35] G. A. Alekseev, “Metod obratnoi zadachi rasseyaniya i singulyarnye integralnye uravneniya dlya vzaimodeistvuyuschikh bezmassovykh polei”, Dokl. AN SSSR, 283:3 (1985), 577–582 | MR

[36] S. P. Burtsev, V. E. Zakharov, A. V. Mikhailov, “Metod obratnoi zadachi s peremennym spektralnym parametrom”, TMF, 70:3 (1987), 323–342 | DOI | MR | Zbl

[37] A. Yu. Orlov, E. I. Shulman, “O dopolnitelnykh simmetriyakh nelineinogo uravneniya Shredingera”, TMF, 64:2 (1985), 323–328 | DOI | MR | Zbl

[38] A. Yu. Orlov, “Vertex operator, $\bar\partial$-problem, symmetries, variational identities and Hamiltonian formalism for $2+1$ integrable systems”, Plasma Theory and Nonlinear and Turbulent Processes in Physics, v. 1, eds. V. G. Bar'yakhtar, V. M. Chernousenko, N. S. Erokhin, A. G. Sitenko, V. E. Zakharov, World Sci., Singapore, 1988, 116–134 | MR | Zbl

[39] 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. ; “A new multi-component integrable coupling and its application to isospectral and nonisospectral problems”, 105 (2022), 106075, 15 pp. | DOI | MR | DOI | MR

[40] Y. F. Zhang, X. Z. Zhang, “A scheme for generating nonisospectral integrable hierarchies and its related applications”, Acta Math. Sinica. English Ser., 37:5 (2021), 707–730 | DOI | MR

[41] D. J. Kaup, A. C. Newell, “An exact solution for a derivative nonlinear Schrödinger equation”, J. Math. Phys., 19:4 (1978), 798–801 | DOI | MR