Geodesic
On soliton solutions and soliton interactions of Kulish–Sklyanin and Hirota–Ohta systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 213 (2022) no. 1, pp. 20-40 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a simplest two-dimensional reduction of the remarkable three-dimensional Hirota–Ohta system. The Lax pair of the Hirota–Ohta system was extended to a Lax triad by adding extra third linear equation, whose compatibility conditions with the Lax pair of the Hirota–Ohta imply another remarkable systems: the Kulish–Sklyanin system (KSS) together with its first higher commuting flow, which we can call the vector complex mKdV. This means that any common particular solution of both these two-dimensional integrable systems yields a corresponding particular solution of the three-dimensional Hirota–Ohta system. Using the Zakharov–Shabat dressing method, we derive the $N$-soliton solutions of these systems and analyze their interactions, i.e., explicitly derive the shifts of the relative center-of-mass coordinates and the phases as functions of the discrete eigenvalues of the Lax operator. Next, we relate Hirota–Ohta-type system to these nonlinear evolution equations and obtain its $N$-soliton solutions.
Keywords: two-dimensional Kulish–Sklyanin system, three-dimensional Hirota–Ohta system, Lax representation, dressing method, two-dimensional reductions.
Mots-clés : multisoliton solutions 
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V. S. Gerdjikov; Nianhua Li; V. B. Matveev; A. O. Smirnov. On soliton solutions and soliton interactions of Kulish–Sklyanin and Hirota–Ohta systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 213 (2022) no. 1, pp. 20-40. http://geodesic.mathdoc.fr/item/TMF_2022_213_1_a1/

[1] M. J. Ablowitz, S. Chakravarty, A. D. Trubatch, J. Villarroel, “A novel class of solutions of the non-stationary Schrödinger equation and the Kadomtsev–Petviashvili I equation”, Phys. Lett. A, 267:2–3 (2000), 132–146 | DOI | MR

[2] V. B. Matveev, Abelian functions and solitons, Preprint No. 373, University of Wroclaw, Wroclaw, 1976

[3] V. B. Matveev, “Darboux transformation and explicit solutions of the Kadomtsev–Petviashvili equation, depending on functional parameters”, Lett. Math. Phys., 3:3 (1979), 213–216 | DOI | MR

[4] V. B. Matveev, M. A. Salle, “New families of the explicit solutions of the Kadomtsev–Petviashvili equation and their application to Johnson equation”, Some Topics on Inverse Problems, Proceedings of the 16th Workshop on Interdisciplinary Study of Inverse Problems (Montpellier, France, November 30 – December 4, 1987), ed. P. C. Sabatier, World Sci., Singapore, 1988, 304–315 | MR | Zbl

[5] V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer, Berlin, 1991 | MR

[6] M. Boiti, F. Pempinelli, A. K. Pogrebkov, “Properties of solutions of the Kadomtsev–Petviaschvili-I equation”, J. Math. Phys., 35:9 (1994), 4683–4718 | DOI | MR

[7] M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Ob ekvivalentnosti razlichnykh podkhodov k postroeniyu mnogosolitonnykh reshenii uravneniya Kadomtseva–Petviashvili-II”, TMF, 165:1 (2010), 3–24 | DOI | DOI

[8] M. Boiti, F. Pempinelli, A. K. Pogrebkov, “KPII: Cauchy–Jost function, Darboux transformations and totally nonnegative matrices”, J. Phys. A, 50:30 (2017), 304001, 22 pp., arXiv: 1611.04198 | DOI | MR

[9] A. K. Pogrebkov, “Kadomtsev–Petviashvili hierarchy: negative times”, Mathematics, 9:16 (2021), 1988, 10 pp. | DOI

[10] 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

[11] V. E. Zakharov, A. B. Shabat, “Integrirovanie nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya. II”, Funkts. analiz i ego pril., 13:3 (1979), 13–22 | DOI | MR | Zbl

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

[13] J. Satsuma, M. J. Ablowitz, “Two-dimensional lumps in nonlinear dispersive systems”, J. Math. Phys., 20 (1979), 1496–1503 | DOI | MR | Zbl

[14] S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, “Two-dimensional solitons of the Kadomtsev–Petviaschvili equation and their interaction”, Phys. Lett. A, 63:3 (1977), 205–206 | DOI

[15] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems”, Stud. Appl. Math., 53:4 (1974), 249–315 | DOI | MR

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

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

[18] E. V. Doktorov, S. B. Leble, A Dressing Method in Mathematical Physics, Mathematical Physics Study, 28, Springer, Dordrecht, 2007 | MR

[19] V. E. Zakharov, A. B. Shabat, “Tochnaya teoriya dvumernoi samofokusirovki i odnomernoi avtomodulyatsii voln v nelineinykh sredakh”, ZhETF, 61:1 (1971), 118–134

[20] V. E. Zakharov, A. B. Shabat, “O vzaimodeistvii solitonov v ustoichivoi srede”, ZhETF, 64:5 (1973), 1627–1639

[21] V. B. Matveev, A. O. Smirnov, “Resheniya tipa ‘voln-ubiits’ uravnenii ierarkhii Ablovitsa–Kaupa–Nyuella–Sigura: edinyi podkhod”, TMF, 186:2 (2016), 191–220 | DOI | DOI | MR

[22] P. Dubard, V. B. Matveev, “Multi-rogue waves solutions: from the NLS to the KP-I equation”, Nonlinearity, 26:12 (2013), R93–R125 | DOI | MR | Zbl

[23] 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 | DOI | MR

[24] S. Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, 34, AMS, Providence, RI, 2012 | MR

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

[26] 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

[27] V. S. Gerdjikov, N. A. Kostov, T. I. Valchev, “Solutions of multi-component NLS models and Spinor Bose–Einstein condensates”, Phys. D, 238:15 (2009), 1306–1310, arXiv: 0802.4398 | DOI | MR

[28] V. S. Gerdjikov, “Bose–Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations”, Discrete and Continuous Dynamical Systems, 4:5 (2011), 1181–1197, arXiv: 1001.0164 | DOI | MR

[29] S. Li, B. Prinari, G. Biondini, “Solitons and rogue waves in spinor Bose–Einstein condensates”, Phys. Rev. E, 97:2, 022221, 18 pp., arXiv: 1802.06471 | DOI | MR

[30] H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, R. Carretero-Gonzales, “Bright-dark soliton complexes in spinor Bose–Einstein condensates”, Phys. Rev. A, 77:3 (2008), 033612, 13 pp., arXiv: 0705.1324 | DOI

[31] V. S. Gerdjikov, G. G. Grahovski, “On $N$-wave and NLS type systems: generating operators and the gauge group action: the $so(5) $ case”, Symmetry in Nonlinear Mathematical Physics, Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. Mathematics and its Applications, 50, Part 1, No 1, eds. A. G. Nikitin, V. M. Bojko, R. O. Popovych, I. A. Jegorchenko, Institute of Mathematics of NAS of Ukraine, Kyïv, 2004, 388–395 | MR | Zbl

[32] S. V. Manakov, “K teorii dvumernoi statsionarnoi samofokusirovki elektromagnitnykh voln”, ZhETF, 65:2 (1973), 505–516

[33] V. E. Zakharov, S. V. Manakov, “K teorii rezonansnogo vzaimodeistviya volnovykh paketov v nelineinykh sredakh”, ZhETF, 69:5 (1975), 1654–1673 | MR

[34] V. S. Gerdjikov, “Basic aspects of soliton theory”, Geometry, Integrability and Quantization (Varna, Bulgaria, June 3–10, 2004), eds. I. M. Mladenov, A. C. Hirshfeld, Softex, Sofia, 2005, 78–125, arXiv: nlin.SI/0604004 | Zbl

[35] V. S. Gerdzhikov, “Modeli tipa Kulisha–Sklyanina: integriruemost i reduktsii”, TMF, 192:2 (2017), 187–206 | DOI | DOI | MR

[36] 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

[37] T. Kanna, M. Lakshmanan, “Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape-changing collisions, logic gates, and partially coherent solitons”, Phys. Rev. E, 67:4 (2003), 046617, 25 pp., arXiv: nlin/0303025 | DOI

[38] V. S. Gerdjikov, D. J. Kaup, N. A. Kostov, T. I. Valchev, “On classification of soliton solutions of multicomponent nonlinear evolution equations”, J. Phys. A: Math. Theor., 41:31 (2008), 315213, 36 pp. | DOI | MR

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

[40] V. S. Gerdjikov, “Algebraic and analytic aspects of $N$-wave type equations”, The Legacy of the Inverse Scattering Transform in Applied Mathematics (South Hadley, June 17–21, 2001), Contemporary Mathematics, 301, eds. J. Bona, R. Choudhury, D. Kaup, AMS, Providence, RI, 2002, 35–68 | DOI | MR | Zbl

[41] V. S. Gerdjikov, A. O. Smirnov, V. B. Matveev, “From generalized Fourier transforms to spectral curves for the Manakov hierarchy. I. Generalized Fourier transforms”, Eur. Phys. J. Plus, 135:8 (2020), 659, 45 pp. | DOI

[42] A. O. Smirnov, V. S. Gerdjikov, V. B. Matveev, “From generalized fourier transforms to spectral curves for the Manakov hierarchy. II. Spectral curves for the Manakov hierarchy”, Eur. Phys. J. Plus, 135:7 (2020), 561, 20 pp. | DOI

[43] V. S. Gerdjikov, P. P. Kulish, “The generating operator for the $n \times n$ linear system”, Phys. D, 3:3 (1981), 549–564 | DOI | MR

[44] A. O. Smirnov, “Spectral curves for the derivative nonlinear Schrödinger equations”, Symmetry, 13:7 (2021), 1203, 18 pp. | DOI

[45] A. O. Smirnov, V. S. Gerdjikov, E. E. Aman, “The Kulish–Sklyanin type hierarchy and spectral curves”, IOP Conf. Ser.: Mater. Sci. Eng., 1047 (2021), 012114, 7 pp. | DOI

[46] V. B. Matveev, A. O. Smirnov, “AKNS and NLS hierarchies, MRW solutions, $P_n$ breathers, and beyond”, J. Math. Phys., 59:9 (2018), 091419, 42 pp. | DOI | MR

[47] V. S. Gerdjikov, “The Zakharov–Shabat dressing method and the representation theory of the semisimple Lie algebras”, Phys. Lett. A, 126:3 (1987), 184–188 | DOI | MR

[48] V. S. Gerdjikov, G. G. Grahovski, “Multi-component NLS models on symmetric spaces: spectral properties versus representations theory.”, SIGMA, 6 (2010), 044, 29 pp., arXiv: 1006.0301 | DOI | MR

[49] V. S. Gerdjikov, G. G. Grahovski, “Two soliton interactions of BD. I multicomponent NLS equations and their gauge equivalent”, AIP Conf. Proc., 1301 (2010), 561–572

[50] 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

[51] V. S. Gerdjikov, “On soliton interactions of vector nonlinear Schrödinger equations”, AIP Conf. Proc., 1404:1 (2011), 57–67 | DOI

[52] 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

[53] Y. Ohta, R. Hirota, “New type of soliton equations”, J. Phys. Soc. Japan, 76:2 (2007), 024005, 14 pp. | DOI

[54] Y. Ohta, R. Hirota, “Hierarchies of coupled soliton equations. I”, J. Phys. Soc. Japan, 60:3 (1991), 798–809 | DOI | MR

[55] V. E. Adler, V. V. Postnikov, “Linear problems and Bäcklund transformations for the Hirota–Ohta system”, Phys. Lett. A, 375:3, 468–473, arXiv: 1007.4698 | DOI | MR

[56] 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

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

[58] A. B. Shabat, “Obratnaya zadacha rasseyaniya dlya sistemy differentsialnykh uravnenii”, Funkts. analiz i ego pril., 9:3 (1975), 75–78 | DOI | MR | Zbl

[59] A. B. Shabat, “Obratnaya zadacha rasseyaniya”, Differents. uravneniya, 15:10 (1979), 1824–1834 | MR | Zbl

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

[61] V. S. Gerdjikov, “The generalized Zakharov–Shabat system and the soliton perturbations”, TMF, 99:2 (1994), 292–299 | DOI | MR | Zbl

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

[63] R. Beals, R. Coifman, “Inverse scattering and evolution equations”, Commun. Pure Appl. Math., 38:1 (1985), 29–42 | DOI | MR