Kulish–Sklyanin-type models: Integrability and reductions

V. S. Gerdjikov

Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 2, pp. 187-206

Résumé

We start with a Riemann–Hilbert problem (RHP) related to BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s+1)\otimes O(2s))$, $s\ge1$. We consider two RHPs: the first is formulated on the real axis $\mathbb R$ in the complex-$\lambda$ plane; the second, on $\mathbb R\oplus i\mathbb R$. The first RHP for $s=1$ allows solving the Kulish–Sklyanin (KS) model; the second RHP is related to a new type of KS model. We consider an important example of nontrivial deep reductions of the KS model and show its effect on the scattering matrix. In particular, we obtain new two-component nonlinear Schrödinger equations. Finally, using the Wronski relations, we show that the inverse scattering method for KS models can be understood as generalized Fourier transforms. We thus find a way to characterize all the fundamental properties of KS models including the hierarchy of equations and the hierarchy of their Hamiltonian structures.

Détail

Keywords: symmetric space, multicomponent nonlinear Schrödinger equation, Lax representation, reduction group.

Bibliographie

[1] P. P. Kulish, E. K. Sklyanin, “$\rm{O}(N)$-invariant nonlinear Schrödinger equation – a new completely integrable system”, Phys. Lett. A, 84:7 (1981), 349–352 | MR

[2] S. V. Manakov, “K teorii dvumernoi statsionarnoi samofokusirovki elektromagnitnykh voln”, ZhETF, 65:2 (1974), 505–516

[3] J. Ieda, T. Miyakawa, M. Wadati, “Exact analysis of soliton dynamics in spinor Bose–Einstein condensates”, Phys. Rev. Lett., 93:19 (2004), 194102, 4 pp. | DOI

[4] L. Li, Z. Li, B. A. Malomed, D. Mihalache, W. M. Liu, “Exact soliton solutions and nonlinear modulation instability in Spinor Bose–Einstein condensates”, Phys. Rev. A, 72:3 (2005), 033611, 11 pp. | DOI

[5] C. V. Ciobanu, S.-K. Yip, T.-L. Ho, “Phase diagrams of $F=2$ spinor Bose–Einstein condensates”, Phys. Rev. A, 61:3 (2000), 033607, 5 pp. | DOI

[6] T. Ohmi, K. Machida, “Bose–Einstein condensation with internal degrees of freedom in alkali atom gases”, J. Phys. Soc. Japan, 67:6 (1998), 1822–1825 | DOI

[7] T.-L. Ho, “Spinor Bose condensates in optical traps”, Phys. Rev. Lett., 81:4 (1998), 742–745 | DOI

[8] A. P. Fordy, P. P. Kulish, “Nonlinear Schrödinger equations and simple Lie algebras”, Commun. Math. Phys., 89:3 (1983), 427–443 | DOI | MR | Zbl

[9] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR | MR | Zbl

[10] L. A. Takhtadzhyan, L. D. Faddeev, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | MR | Zbl

[11] M. J. Ablowitz, B. Prinari, A. D. Trubatch, Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Note Series, 302, Cambridge Univ. Press, Cambridge, 2004 | MR

[12] V. S. Gerdjikov, “Basic aspects of soliton theory”, Geometry, Integrability and Quantization, Proceedings of the 6th International Conference (Sts. Constantine and Elena, Bulgaria, June 3–10, 2004), eds. I. M. Mladenov, A. C. Hirshfeld, Softex, Sofia, 2005, 78–125, arXiv: nlin/0604004 | MR | Zbl

[13] V. S. Gerdjikov, “Algebraic and analytic aspects of soliton type equations”, The Legacy of the Inverse Scattering Transform in Applied Mathematics (Mount Holyoke College, South Hadley, MA, USA, June 17–21, 2001), Contemporary Mathematics, 301, eds. J. Bona, R. Choudhury, D. Kaup, AMS, Providence, RI, 2002, 35–68, arXiv: nlin/0206014v1 | DOI | MR | Zbl

[14] V. S. Gerdjikov, G. Vilasi, A. B. Yanovski, Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods, Lecture Notes in Physics, 748, Springer, Berlin, 2008 | DOI | MR

[15] V. E. Zakharov, A. B. Shabat, “Skhema integrirovaniya nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya. I”, Funkts. analiz i ego pril., 8:3 (1974), 43–53 | DOI | MR | Zbl

[16] V. E. Zakharov, A. V. Mikhailov, “On the integrability of classical spinor models in two-dimensional space-time”, Commun. Math. Phys., 74:1 (1980), 21–40 | DOI | MR

[17] R. I. Ivanov, “On the dressing method for the generalized Zakharov–Shabat system”, Nucl. Phys. B, 694:3 (2004), 509–524 | DOI | MR | Zbl

[18] V. S. Gerdjikov, G. G. Grahovski, R. I. Ivanov, N. A. Kostov, “$N$-wave interactions related to simple Lie algebras. ${\mathbb Z}_2$-reductions and soliton solutions”, Inverse Problems, 17:4 (2001), 999–1015 | DOI | MR | Zbl

[19] V. S. Gerdzhikov, G. G. Grakhovski, N. A. Kostov, “O mnogokomponentnykh uravneniyakh tipa nelineinogo uravneniya Shredingera na simmetrichnykh prostranstvakh i ikh reduktsiyakh”, TMF, 144:2 (2005), 313–323 | DOI | DOI | MR | Zbl

[20] A. V. Mikhailov, “The reduction problem and the inverse scattering method”, Physica D, 3:1–2 (1981), 73–117 | DOI | Zbl

[21] V. S. Gerdjikov, R. I. Ivanov, G. G. Grahovski, “On Integrable Wave Interactions and Lax pairs on symmetric spaces”, Wave Motion, 71 (2017), 53–70, arXiv: 1607.06940 | DOI | MR

[22] G. G. Grahovski, V. S. Gerdjikov, N. A. Kostov, V. A. Atanasov, “New integrable multi-component NLS type equations on symmetric spaces: $Z_4$- and $Z_6$-reductions”, Geometry, Integrability and Quantization, Proceedings of the 7th International Conference (Sts. Constantine and Elena, Bulgaria, June 2–10, 2005), eds. I. Mladenov, M. De León, Softex, Sofia, 2006, 154–175, arXiv: nlin/0603066 | MR | Zbl

[23] V. S. Gerdjikov, “Derivative nonlinear Schrödinger equations with ${\mathbb Z}_N$ and $\mathbb D_N $-reductions”, Romanian J. Phys., 58:5–6 (2013), 573–582 | MR

[24] V. S. Gerdjikov, A. A. Stefanov, “New types of two component NLS-type equations”, Pliska Stud. Math. Bulgar., 26 (2016), 53–66, arXiv: 1703.01314

[25] V. S. Gerdzhikov, M. I. Ivanov, P. P. Kulish, “Kvadratichnyi puchok i nelineinye uravneniya”, TMF, 44:3 (1980), 342–357 | DOI | MR | Zbl

[26] V. S. Gerdzhikov, M. I. Ivanov, “Kvadratichnyi puchok obschego vida i nelineinye evolyutsionnye uravneniya. Razlozheniya po “kvadratam” reshenii – obobschennye preobrazovaniya Fure”, Bolg. fiz. zh., 10:1 (1983), 13–26 | MR | MR | Zbl

[27] V. S. Gerdzhikov, M. I. Ivanov, Bolg. fiz. zh., 10:2 (1983), 130–143 | MR

[28] V. S. Gerdjikov, “Riemann–Hilbert Problems with canonical normalization and families of commuting operators”, Pliska Stud. Math. Bulgar., 21 (2012), 201–216, arXiv: 1204.2928v1 | MR

[29] S. Helgason, Differential Geometry, Lie groups and Symmetric Spaces, Graduate Studies in Mathematics, 34, AMS, Providence, RI, 2001 | DOI | MR

[30] V. G. Drinfeld, V. V. Sokolov, “Algebry Li i uravneniya tipa Kortevega–de Friza”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Nov. dostizh., 24 (1984), 81–180 | DOI | MR | Zbl | Zbl

[31] D. J. Kaup, “Closure of the squared Zakharov–Shabat eigenstates”, J. Math. Anal. Appl., 54:3 (1976), 849–864 | DOI | MR | Zbl

[32] F. Calogero, F. Degasperis, Spectral Transform and Solitons, v. 1, Lecture Notes in Computer Science, 144, Tools to Solve and Investigate Nonlinear Evolution Equations, North Holland, Amsterdam—New York, 1982 | MR

[33] V. S. Gerdjikov, “Generalized Fourier transforms for the soliton equations. Gauge-covariant formulation”, Inverse Problems, 2:1 (1986), 51–74 | DOI | MR | Zbl

[34] V. E. Zakharov, E. I. Schulman, “To the integrability of the system of two coupled nonlinear Schrödinger equations”, Physica D, 4 (1982), 270–274 | DOI | MR | Zbl