Integrability of a multicomponent coupled dispersionless integrable

H. Wajahat A. Riaz ; Mahmood ul Hassan

Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 3, pp. 371-384

Résumé

We present a multicomponent coupled dispersionless integrable system and show that it is integrable in the sense of the existence of a Lax pair representation and also the existence of an infinite sequence of conserved quantities, a Darboux transformation, and soliton solutions.

Détail

Keywords: integrable system
Mots-clés : soliton, Darboux transformation.

Bibliographie

[1] M. Pietrzyk, I. Kanattšikov, U. Bandelow, “On the propagation of vector ultra-short pulses”, J. Nonlinear Math. Phys., 15:2 (2008), 162–170 | DOI | MR

[2] 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

[3] Y. Matsuno, “A novel multi-component generalization of the short pulse equation and its multisoliton solutions”, J. Math. Phys., 52:12 (2011), 123702, 22 pp., arXiv: 1111.1792 | DOI | MR

[4] T. Tsuchida, M. Wadati, “Multi-field integrable systems related to WKI-type eigenvalue problems”, J. Phys. Soc. Japan, 68:7 (1999), 2241–2245 | DOI | MR

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

[6] R. Hirota, Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155, Cambridge Univ. Press, Cambridge, 2004 | MR

[7] V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991 | MR

[8] C. Gu, H. Hu, Z. Zhou, Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry, Mathematical Physics Studies, 26, Springer, Berlin, 2005 | DOI | MR

[9] C. X. Li, J. J. C. Nimmo, “Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation”, Proc. R. Soc. London Ser. A, 464:2092 (2008), 951–966, arXiv: 0711.2594 | DOI | MR

[10] M. Hassan, “Darboux transformation of the generalized coupled dispersionless integrable system”, J. Phys. A: Math. Theor., 42:6 (2009), 065203, 11 pp., arXiv: 0912.1671 | DOI | MR

[11] Y. Shi, J. J. C. Nimmo, D. Zhang, “Darboux and binary Darboux transformations for discrete integrable systems I. Discrete potential KdV equation”, J. Phys. A: Math. Theor., 47:2 (2014), 025205, 11 pp. | DOI | MR

[12] H. W. A. Riaz, M. Hassan, “Darboux transformation of a semi-discrete coupled dispersionless integrable system”, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 387–397 | DOI | MR

[13] H. W. A. Riaz, M. Hassan, “Multisoliton solutions of integrable discrete and semi-discrete principal chiral equations”, Commun. Nonlinear Sci. Numer. Simul., 54 (2018), 416–427 | DOI | MR

[14] H. W. A. Riaz, M. Hassan, “A discrete generalized coupled dispersionless integrable system and its multisoliton solutions”, J. Math. Anal. Appl., 458:2 (2018), 1639–1652 | DOI | MR

[15] K. Konno, H. Oono, “New coupled integrable dispersionless equations”, J. Phys. Soc. Japan, 63:2 (1994), 377–378 | DOI

[16] V. E. Adler, A. B. Shabat, “Toward a theory of integrable hyperbolic equations of third order”, J. Phys. A: Math. Theor., 45:39 (2012), 395207, 17 pp. | DOI | MR

[17] H. Kakuhata, K. Konno, “Lagrangian, Hamiltonian and conserved quantities for coupled integrable, dispersionless equations”, J. Phys. Soc. Japan, 65:1 (1995), 1–2 | DOI | MR

[18] H. Kakuhata, K. Konno, “A generalization of coupled integrable, dispersionless system”, J. Phys. Soc. Japan, 65:2 (1996), 340–341 | DOI | MR

[19] T. Alagesan, K. Porsezian, “Painleve analysis and the integrability properties of coupled integrable dispersionless equations”, Chaos, Solitons and Fractals, 7:8 (1996), 1209–1212 | DOI | MR

[20] T. Alagesan, K. Porsezian, “Singularity structure analysis and Hirota's bilinearisation of the coupled integrable dispersionless equations”, Chaos, Solitons and Fractals, 8:10 (1997), 1645–1650 | DOI | MR

[21] S. C. Anco, T. Wolf, “Some symmetry classifications of hyperbolic vector evolution equations”, J. Nonlinear Math. Phys., 12, suppl. 1 (2005), 13–31 | DOI | MR

[22] T. Tsuchida, “New reductions of integrable matrix partial differential equations: $sp(m)$-invariant systems”, J. Math. Phys., 51:5 (2010), 053511, 27 pp., arXiv: 0712.4373 | DOI | MR

[23] V. E. Zakharov, A. V. Mikhailov, “Relyativistski invariantnye dvumernye modeli teorii polya, integriruemye metodom obratnoi zadachi”, ZhETF, 74:6 (1978), 1953–1973 | MR