The Darboux transformation for WKI system

Yongshuai Zhang   ; Deqin Qiu   ; Yi Cheng   ; Jingsong He

Teoretičeskaâ i matematičeskaâ fizika, Tome 191 (2017) no. 2, pp. 275-290

Résumé

Based on a conservation law, we construct a hodograph transformation for the Wadati–Konno–Ichikawa (WKI) equation, which implies that the WKI equation is equivalent to a modified WKI (mWKI) equation. Applying the Darboux transformation to the mWKI equation, we show that in both the focusing and defocusing cases, the mWKI equation admits an analytic bright soliton solution from the vacuum and the collisions of $n$ solitons are elastic based on the asymptotic analysis. In addition, we find that the mWKI equation still admits the breather and rogue wave solutions, although a modulation instability does not exist for it.

Détail

Mots-clés : Darboux transformation, hodograph transformation, soliton, rogue wave
Keywords: Wadati–Konno–Ichikawa system, modulation instability.

Bibliographie

[1] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, “Method for solving the Korteweg–de Vries equation”, Phys. Rev. Lett., 19:19 (1967), 1095–1097 | DOI | MR | Zbl

[2] V. E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, J. Appl. Mech. Tech. Phys., 9:2 (1968), 190–194 | DOI

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

[4] M. J. Ablowitz, D. J. Kaup, A. Newell, H. Segur, “Method for solving the sine-Gordon equation”, Phys. Rev. Lett., 30:25 (1973), 1262–1264 | DOI | MR

[5] M. Wadati, “The modified Korteweg–de Vries equation”, J. Phys. Soc. Japan, 34:5 (1973), 1289–1296 | DOI | MR | Zbl

[6] A. Fokas, M. Ablowitz, “Inverse scattering transform for the Benjamin–Ono equation – a pivot to multidimensional problems”, Stud. Appl. Math., 68:1 (1983), 1–10 | DOI | MR | Zbl

[7] D. Kaup, Y. Matsuno, “The inverse scattering transform for the Benjamin–Ono equation”, Stud. Appl. Math., 101:1 (1998), 73–98 | DOI | MR | Zbl

[8] Y. Kodama, M. J. Ablowitz, J. Satsuma, “Direct and inverse scattering problems of the nonlinear intermediate long wave equation”, J. Math. Phys., 23:4 (1982), 564–576 | DOI | MR | Zbl

[9] A. S. Fokas, M. J. Ablowitz, “On the inverse scattering on the time-dependent Schrödinger equation and the associated Kadomtsev–Petviashvili equation”, Stud. Appl. Math., 69:3 (1983), 211–228 | DOI | MR | Zbl

[10] S. V. Manakov, “The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev–Petviashvili equation”, Phys. D, 3:1–2 (1981), 420–427 | DOI | Zbl

[11] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “Nonlinear-evolution equations of physical significance”, Phys. Rev. Lett., 31:2 (1973), 125–127 | DOI | MR | Zbl

[12] 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 | Zbl

[13] M. Wadati, K. Konno, Y. H. Ichikawa, “New integrable nonlinear evolution equations”, J. Phys. Soc. Japan, 47:5 (1979), 1698–1700 | DOI | MR | Zbl

[14] K. Konno, A. Jeffrey, “The loop soliton”, Advances in nonlinear waves, v. 1, ed. L. Debnath, Pitman, London, 1984, 162–183 | MR

[15] M. Kruskal, “Nonlinear wave equations”, Dynamical Systems, Theory and Applications, Lecture Notes in Physics, 38, 1975, 310–354 | DOI | MR | Zbl

[16] T. Shimizu, M. Wadati, “A new integrable nonlinear evolution equation”, Progr. Theoret. Phys., 63:3 (1980), 808–820 | DOI | MR | Zbl

[17] M. Wadati, K. Sogo, “Gauge transformations in soliton theory”, J. Phys. Soc. Japan, 52:2 (1983), 394–398 | DOI | MR

[18] Y. Ishimori, “A relationship between the Ablowitz–Kaup–Newell–Segur and Wadati–Konno–Ichikawa schemes of the inverse scattering method”, J. Phys. Soc. Japan, 51:9 (1982), 3036–3041 | DOI | MR

[19] D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions”, J. Austral. Math. Soc. Ser. B, 25:1 (1983), 16–43 | DOI | MR | Zbl

[20] P. Dubard, V. B. Matveev, “Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation”, Nat. Hazards Earth Sys. Sci., 11:3 (2011), 667–672 | DOI | MR

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

[22] B. Guo, L. Ling, Q. P. Liu, “High-order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations”, Stud. Appl. Math., 130:4 (2013), 317–344 | DOI | MR | Zbl

[23] L. Guo, Y. Zhang, S. Xu, Z. Wu, J. He, “The higher order rogue wave solutions of the Gerdjikov–Ivanov equation”, Phys. Scripta, 89:3 (2014), 035501 | DOI | MR

[24] S. Xu, J. He, L. Wang, “The Darboux transformation of the derivative nonlinear Schrödinger equation”, J. Phys. A: Math. Theor., 44:30 (2011), 305203, 22 pp. | DOI | MR | Zbl

[25] Y. Zhang, L. Guo, J. He, Z. Zhou, “Darboux transformation of the second–type derivative nonlinear Schrödinger equation”, Lett. Math. Phys., 105:6 (2015), 853–891 | DOI | MR | Zbl

[26] Y. Zhang, L. Guo, S. Xu, Z. Wu, J. He, “The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation”, Commun. Nonlinear Sci. Numer. Simul., 19:6 (2014), 1706–1722 | DOI | MR

[27] Y. Ohta, J. Yang, “Rogue waves in the Davey–Stewartson I equation”, Phys. Rev. E, 86:3 (2012), 036604, 8 pp. | DOI

[28] Y. Ohta, J. Yang, “Dynamics of rogue waves in the Davey–Stewartson II equation”, J. Phys. A: Math. Theor., 46:10 (2013), 105202, 19 pp. | DOI | MR | Zbl

[29] N. Akhmediev, J. M. Soto-Crespo, N. Devine, N. P. Hoffmann, “Rogue wave spectra of the Sasa–Satsuma equation”, Phys. D, 294 (2015), 37–42 | DOI | MR

[30] S. Chen, “Twisted rogue-wave pairs in the Sasa–Satsuma equation”, Phys. Rev. E, 88:2 (2013), 023202, 5 pp. | DOI

[31] F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, M. Onorato, S. Wabnitz, “Vector rogue waves and baseband modulation instability in the defocusing regime”, Phys. Rev. Lett., 113:3 (2014), 034101, 5 pp. | DOI

[32] F. Baronio, A. Degasperis, M. Conforti, S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves”, Phys. Rev. Lett., 109:4 (2012), 044102, 4 pp. | DOI

[33] B.-L. Guo, L.-M. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations”, Chin. Phys. Lett., 28:11 (2011), 110202, 4 pp. | DOI

[34] G. Mu, Z. Qin, R. Grimshaw, “Dynamics of rogue waves on a multisoliton background in a vector nonlinear Schrödinger equation”, SIAM J. Appl. Math., 75:1 (2015), 1–20 | DOI | MR | Zbl

[35] A. Ankiewicz, N. Akhmediev, J. M. Soto-Crespo, “Discrete rogue waves of the Ablowitz–Ladik and Hirota equations”, Phys. Rev. E, 82:2 (2010), 026602, 7 pp. | DOI | MR

[36] A. Ankiewicz, J. M. Soto-Crespo, N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation”, Phys. Rev. E, 81:4 (2010), 046602, 8 pp. | DOI | MR

[37] A. Ankiewicz, Y. Wang, S. Wabnitz, N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions”, Phys. Rev. E, 89:1 (2014), 012907, 9 pp. | DOI | MR

[38] D. Qiu, J. He, Y. Zhang, K. Porsezian, “The Darboux transformation of the Kundu–Eckhaus equation”, Proc. Roy. Soc. A, 471:2180 (2015), 20150236 | DOI | MR

[39] D. Qiu, Y. Zhang, J. He, “The rogue wave solutions of a new $(2+1)$-dimensional equation”, Commun. Nonlinear Sci. Numer. Simul., 30:1–3 (2016), 307–315 | DOI | MR

[40] A. Chabchoub, N. Akhmediev, “Observation of rogue wave triplets in water waves”, Phys. Lett. A, 377:38 (2013), 2590–2593 | DOI | Zbl

[41] A. Chabchoub, N. P. Hoffmann, N. Akhmediev, “Rogue wave observation in a water wave tank”, Phys. Rev. Lett., 106:20 (2011), 204502, 4 pp. | DOI | MR

[42] A. Chabchoub, N. P. Hoffmann, N. Akhmediev, “Observation of rogue wave holes in a water wave tank”, J. Geophys. Res. Oceans, 117:C11 (2012), C00J02 | DOI

[43] A. Chabchoub, N. Hoffmann, M. Onorato, N. Akhmediev, “Super rogue waves: observation of a higher-order breather in water waves”, Phys. Rev. X, 2:1 (2012), 011015, 6 pp. | DOI | MR

[44] A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank”, Phys. Rev. E, 86:5 (2012), 056601, 6 pp. | DOI

[45] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics”, Nature Phys., 6:10 (2010), 790–795 | DOI

[46] D. Solli, C. Ropers, P. Koonath, B. Jalali, “Optical rogue waves”, Nature, 450:7172 (2007), 1054–1057 | DOI

[47] H. Bailung, S. Sharma, Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions”, Phys. Rev. Lett., 107:25 (2011), 255005, 4 pp. | DOI

[48] M. Shats, H. Punzmann, H. Xia, “Capillary rogue waves”, Phys. Rev. Lett., 104:10 (2010), 104503, 4 pp. | DOI

[49] C. Kharif, J. Touboul, “Under which conditions the Benjamin–Feir instability may spawn an extreme wave event: a fully nonlinear approach”, Eur. Phys. J. Spec. Top., 185 (2010), 159–168 | DOI

[50] A. Slunyaev, “Freak wave events and the wave phase coherence”, Eur. Phys. J. Spec. Top., 185:1 (2010), 67–80 | DOI | MR

[51] V. Zakharov, A. Gelash, “Nonlinear stage of modulation instability”, Phys. Rev. Lett., 111:5 (2013), 054101, 5 pp. | DOI

[52] P. A. Clarkson, A. S. Fokas, M. J. Ablowitz, “Hodograph transformations of linearizable partial differential equations”, SIAM J. Appl. Math., 49:4 (1989), 1188–1209 | DOI | MR | Zbl

[53] S. Kawamoto, “An exact transformation from the Harry Dym equation to the modified KdV equation”, J. Phys. Soc. Japan, 54:5 (1985), 2055–2056 | DOI

[54] D. Levi, O. Ragnisco, A. Sym, “The Bäcklund transformations for nonlinear evolution equations which exhibit exotic solitons”, Phys. Lett. A, 100:1 (1984), 7–10 | DOI | MR

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

[56] J. He, L. Zhang, Y. Cheng, Y. Li, “Determinant representation of Darboux transformation for the AKNS system”, Sci. China Ser. A, 49:12 (2006), 1867–1878 | DOI | MR | Zbl

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

[58] M. Wadati, Y. H. Ichikawa, T. Shimizu, “Cusp soliton of a new integrable nonlinear evolution equation”, Prog. Theor. Phys., 64:6 (1980), 1959–1967 | DOI | MR | Zbl

[59] J. S. He, H. R. Zhang, L. H. Wang, K. Porsezian, A. S. Fokas, “Generating mechanism for higher-order rogue waves”, Phys. Rev. E, 87:5 (2013), 052914, 10 pp. | DOI

[60] T. B. Benjamin, J. E. Feir, “The disintegration of wave trains on deep water. Part 1. Theory”, J. Fluid Mech., 27:3 (1967), 417–430 | DOI | MR | Zbl

[61] G. Whitham, “Non-linear dispersion of water waves”, J. Fluid Mech., 27:2 (1967), 399–412 | DOI | MR | Zbl