Application of the $\bar\partial$-dressing method to a $(2+1)$-dimensional equation

Xuedong Chai ; Yufeng Zhang ; Shiyin Zhao

Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 3, pp. 465-474

Résumé

A remarkable method for investigating solutions of nonlinear soliton equation is the $\bar\partial$-dressing method. Although there are other methods that can also be used for that aim, the $\bar\partial$-dressing method is the most transparent and leads directly to the final results. The $(2+1)$-dimensional Sawada–Kotera equation is studied by analyzing the eigenfunction and the Green's function of its Lax representation as well as by the inverse spectral transformation, yielding a new $\bar\partial$ problem. The solution is constructed based on solving the $\bar\partial$-problem by choosing a proper spectral transformation. Furthermore, once the time evolution of the spectral data is determined, we are able to completely obtain a formal solution of the Sawada–Kotera equation.

Détail

Keywords: $\bar\partial$-dressing method, Green's function, eigenfunction
Mots-clés : Sawada–Kotera equation, inverse spectral transformation.

Bibliographie

[1] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149, Cambridge Univ. Press, Cambridge, 1991 | DOI | MR | Zbl

[2] R. Hirota, “A new form of Bäcklund transformations and its relation to the inverse scattering problem”, Progr. Theor. Phys., 52:5 (1974), 1498–1512 | DOI

[3] J. Weiss, M. Tabor, G. Carnevale, “The Painlevé property for partial differential equations”, J. Math. Phys., 24:3 (1983), 522–526 | DOI

[4] R. Hirota, “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27:18 (1971), 1192–1994 | DOI

[5] R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155, eds. A. Nagai, J. Nimmo, C. Gilson, Cambridge Univ. Press, Cambridge, 2004 | MR

[6] R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation”, J. Math. Phys., 14:7 (1973), 805–809 | DOI

[7] R. Hirota, J. Satsuma, “Nonlinear evolution equations generated from the Bäcklund transformation for the Toda lattice”, Progr. Theor. Phys., 57:3 (1977), 797–807 | DOI

[8] C. Rogers, W. F. Shadwick, Bäklund Transformations and Their Applications, Mathematics in Science and Engineering, 161, Academic Press, New York, 1982 | MR | Zbl

[9] A. M. Bruckner, J. B. Bruckner, “Darboux transformations”, Trans. Amer. Math. Soc., 128:1 (1967), 103–111 | DOI | MR

[10] V. E. Zakharov, S. V. Manakov, “Postroenie mnogomernykh nelineinykh integriruemykh sistem i ikh reshenii”, Funkts. anal. i ego pril., 19:2 (1985), 11–25 | MR

[11] M. J. Ablowitz, D. Bar Jaacov, A. S. Fokas, “On the inverse scattering transform for the Kadomtsev–Petviashvili equation”, Stud. Appl. Math., 69:2 (1983), 135–143 | DOI

[12] V. G. Dubrovsky, “The application of the $\bar\partial$-dressing method to some integrable ($2+1$)-dimensional nonlinear equations”, J. Phys. A: Math. Gen., 29:13 (1996), 3617–3630 | DOI

[13] B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations. The Inverse Spectral Transform in $2+1$ Dimensions, Springer Science, Business Media, New York, 2013 | DOI | MR | Zbl

[14] J. Zhu, X. Geng, “A hierarchy of coupled evolution equations with self-consistent sources and the dressing method”, J. Phys. A: Math. Theor., 46:3 (2012), 035204, 18 pp. | DOI

[15] P. A. Deift, X. Zhou, “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation”, Ann. Math., 137:2 (1993), 295–368 | DOI

[16] V. E. Zakharov, “Commutating operators and nonlocal $\bar\partial$-problem”, Plasma Theory and Nonlinear and Turbulent Processes in Physics (Kiev, USSR, 13–25 April, 1987), v. 1, eds. N. S. Erokhin, V. E. Zakharov, A. G. Sitenko, V. M. Chernousenko, V. G. Bar'yakhtar, Naukova Dumka, Kiev, 1988, 152–154 | Zbl

[17] B. G. Konopelchenko, Solitons in MultiDimensions. Inverse Spectral Transform, World Sci., Singapore, 1993 | DOI | MR | Zbl

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

[19] A. S. Fokas, V. E. Zakharov, “The dressing method and nonlocal Riemann–Hilbert problems”, J. Nonlinear Sci., 2:1 (1992), 109–134 | DOI

[20] B. G. Konopelchenko, B. T. Matkarimov, “Inverse spectral transform for nonlinear evolution equation generating the Davey–Stewartson and Ishimory equations”, Stud. Appl. Math., 82:4 (1990), 319–359 | DOI

[21] V. Dubrovsky, A. Topovsky, “Multi-lump solutions of KP equation with integrable boundary via $\bar\partial$-dressing method”, Phys. D, 414 (2020), 132740, 11 pp. | DOI | MR

[22] J. Zhu, Y. Kuang, “CUSP solitons to the long-short waves equation and the $\bar\partial$-dressing method”, Rep. Math. Phys., 75:2 (2015), 199–211 | DOI

[23] Y. Kuang, J. Zhu, “The higher-order soliton solutions for the coupled Sasa–Satsuma system via the $\bar\partial$-dressing method”, Appl. Math. Lett., 66 (2017), 47–53 | DOI

[24] B. G. Konopelchenko, V. G. Dubrovsky, “Some new integrable nonlinear evolution equations in $2+1$ dimensions”, Phys. Lett. A, 102:1–2 (1984), 15–17 | DOI

[25] A. P. Fordy, J. Gibbons, “Factorization of operators I. Miura transformations”, J. Math. Phys., 21:10 (1980), 2508–2510 | DOI

[26] J. M. Dye, A. Parker, “On bidirectional fifth-order nonlinear evolution equations, Lax pairs, and directionally dependent solitary waves”, J. Math. Phys., 42:6 (2001), 2567–2589 | DOI