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The algebraic representation for high order solution of Sasa-Satsuma equation
1. | School of Mathematics, South China University of Technology, Guangzhou 510640, China |
References:
[1] |
U. Bandelow and N. Akhmediev, Sasa-Satsuma equation: Soliton on a background and its limiting cases, Phys. Rev. E, 86 (2012), 026606. |
[2] |
D. Bian, B. Guo and L. Ling, High-order soliton solution of Landau-Lifshitz equation, Stud. Appl. Math., 134 (2015), 181-214.
doi: 10.1111/sapm.12051. |
[3] |
H. H. Chen, Y. C. Lee and C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scr., 20 (1979), 490-492.
doi: 10.1088/0031-8949/20/3-4/026. |
[4] |
S. Chen, Twisted rogue-wave pairs in the Sasa-Satsuma equation, Phys. Rev. E, 88 (2013), 023202.
doi: 10.1103/PhysRevE.88.023202. |
[5] |
S. Ghosh, A. Kundu and S. Nandy, Soliton solutions, liouville integrability and gauge equivalence of Sasa Satsuma equation, J. Math. Phys., 40 (1999), 1993-2000.
doi: 10.1063/1.532845. |
[6] |
C. Gilson, J. Hietarinta, J. Nimmo and Y. Ohta, Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions, Phys. Rev. E, 68 (2003), 016614, 10pp.
doi: 10.1103/PhysRevE.68.016614. |
[7] |
C. H. Gu, H. S. Hu and Z. X. Zhou, Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry, Springer, Dordrecht, 2005.
doi: 10.1007/1-4020-3088-6. |
[8] |
B. Guo and L. Ling, Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations, Chin. Phys. Lett., 28 (2011), 110202.
doi: 10.1088/0256-307X/28/11/110202. |
[9] |
B. Guo, L. Ling and Q. P. Liu, Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions, Phys. Rev. E, 85 (2012), 026607.
doi: 10.1103/PhysRevE.85.026607. |
[10] |
B. Guo, L. Ling and Q. P. Liu, High-Order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations, Stud. Appl. Math., 130 (2013), 317-344.
doi: 10.1111/j.1467-9590.2012.00568.x. |
[11] |
R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809.
doi: 10.1063/1.1666399. |
[12] |
Y. Jiang and B. Tian, Dark and dark-like-bright solitons for a higher-order nonlinear Schröinger equation in optical fibers, EPL, 102 (2013), 10010. |
[13] |
D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 19 (1978), 798-801.
doi: 10.1063/1.523737. |
[14] |
D. Kaup and J. Yang, The inverse scattering transform and squared eigenfunctions for a degenerate $3\times 3$ operator, Inverse Problems, 25 (2009), 105010, 21pp.
doi: 10.1088/0266-5611/25/10/105010. |
[15] |
Y. Kodama, Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614.
doi: 10.1007/BF01008354. |
[16] |
Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron, 23 (1987), 510-524.
doi: 10.1109/JQE.1987.1073392. |
[17] |
J. Lenells, Initial-boundary value problems for integrable evolution equations with 3$\times$3 Lax pairs, Physica D, 241 (2012), 857-875.
doi: 10.1016/j.physd.2012.01.010. |
[18] |
L. Ling, L. Zhao and B. Guo, Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations, Commun Nonlinear Sci Numer Simulat, 32 (2016), 285-304.
doi: 10.1016/j.cnsns.2015.08.023. |
[19] |
L. Ling, B. Guo and L. Zhao, High-order rogue waves in vector nonlinear Schrödinger equations, Phys. Rev. E, 89 (2014), 041201(R).
doi: 10.1103/PhysRevE.89.041201. |
[20] |
L. Ling, L.-C. Zhao and B. Guo, Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 3243-3261.
doi: 10.1088/0951-7715/28/9/3243. |
[21] |
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991.
doi: 10.1007/978-3-662-00922-2. |
[22] |
D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu and N. Truta, Soliton solutions for a perturbed nonlinear Schrödinger equation, J. Phys. A: Math. Gen., 26 (1993), L757-L765.
doi: 10.1088/0305-4470/26/17/001. |
[23] |
G. Mu and Z. Qin, Dynamic patterns of high-order rogue waves for Sasa-Satsuma equation, Nonlinear Anal. RWA, 31 (2016), 179-209. |
[24] |
S. Nandy, Inverse scattering approach to coupled higher-order nonlinear Schröinger equation and N-soliton solutions, Nucl. Phys. B, 679 (2004), 647-659.
doi: 10.1016/j.nuclphysb.2003.12.018. |
[25] |
J. Nimmo and H. Yilmaz, Binary Darboux transformation for the Sasa-Satsuma equation, J. Phys. A, 48 (2015), 425202, 16pp.
doi: 10.1088/1751-8113/48/42/425202. |
[26] |
Y. Ohta, Dark soliton solution of Sasa-Satsuma equation, AIP Conf. Proc., 1212 (2010), 114-121. |
[27] |
Q.-H. Park, H. J. Shin and J. Kim, Integrable coupling of optical waves in higher-order nonlinear Schrödinger equations, Phys. Lett. A, 263 (1999), 91-97.
doi: 10.1016/S0375-9601(99)00713-6. |
[28] |
N. Sasa and J. Satsuma, New type of soliton solutions for a higher-order nonlinear Schrödinger equation, J. Phys. Soc. Japan, 60 (1991), 409-417.
doi: 10.1143/JPSJ.60.409. |
[29] |
C. L. Terng and K. Uhlenbeck, Bäcklund transformations and loop group actions, Commun. Pure Appl. Math., 53 (2000), 1-75.
doi: 10.1002/(SICI)1097-0312(200001)53:1<1::AID-CPA1>3.0.CO;2-U. |
[30] |
O. C. Wright, Sasa-Satsuma equation, unstable plane waves and heteroclinic connections, Chaos Solitons and Fractals, 33 (2007), 374-387.
doi: 10.1016/j.chaos.2006.09.034. |
[31] |
J. Xu and E. Fan, The unified transform method for the Sasa-Satsuma equation on the half-line, Proc R Soc A, 469 (2013), 20130068, 25pp.
doi: 10.1098/rspa.2013.0068. |
[32] |
T. Xu, M. Li and L. Li, Anti-dark and mexican-hat solitons in the Sasa-Satsuma equation on the continuous wave background, EPL, 109 (2015), 30006.
doi: 10.1209/0295-5075/109/30006. |
[33] |
J. Yang and D. Kaup, Squared eigenfunctions for the Sasa-Satsuma equation, J. Math. Phys., 50 (2009), 023504, 21pp.
doi: 10.1063/1.3075567. |
[34] |
L.-C. Zhao, S.-C. Li and L. Ling, Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation, Phys. Rev. E, 89 (2014), 023210. |
show all references
References:
[1] |
U. Bandelow and N. Akhmediev, Sasa-Satsuma equation: Soliton on a background and its limiting cases, Phys. Rev. E, 86 (2012), 026606. |
[2] |
D. Bian, B. Guo and L. Ling, High-order soliton solution of Landau-Lifshitz equation, Stud. Appl. Math., 134 (2015), 181-214.
doi: 10.1111/sapm.12051. |
[3] |
H. H. Chen, Y. C. Lee and C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scr., 20 (1979), 490-492.
doi: 10.1088/0031-8949/20/3-4/026. |
[4] |
S. Chen, Twisted rogue-wave pairs in the Sasa-Satsuma equation, Phys. Rev. E, 88 (2013), 023202.
doi: 10.1103/PhysRevE.88.023202. |
[5] |
S. Ghosh, A. Kundu and S. Nandy, Soliton solutions, liouville integrability and gauge equivalence of Sasa Satsuma equation, J. Math. Phys., 40 (1999), 1993-2000.
doi: 10.1063/1.532845. |
[6] |
C. Gilson, J. Hietarinta, J. Nimmo and Y. Ohta, Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions, Phys. Rev. E, 68 (2003), 016614, 10pp.
doi: 10.1103/PhysRevE.68.016614. |
[7] |
C. H. Gu, H. S. Hu and Z. X. Zhou, Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry, Springer, Dordrecht, 2005.
doi: 10.1007/1-4020-3088-6. |
[8] |
B. Guo and L. Ling, Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations, Chin. Phys. Lett., 28 (2011), 110202.
doi: 10.1088/0256-307X/28/11/110202. |
[9] |
B. Guo, L. Ling and Q. P. Liu, Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions, Phys. Rev. E, 85 (2012), 026607.
doi: 10.1103/PhysRevE.85.026607. |
[10] |
B. Guo, L. Ling and Q. P. Liu, High-Order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations, Stud. Appl. Math., 130 (2013), 317-344.
doi: 10.1111/j.1467-9590.2012.00568.x. |
[11] |
R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809.
doi: 10.1063/1.1666399. |
[12] |
Y. Jiang and B. Tian, Dark and dark-like-bright solitons for a higher-order nonlinear Schröinger equation in optical fibers, EPL, 102 (2013), 10010. |
[13] |
D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 19 (1978), 798-801.
doi: 10.1063/1.523737. |
[14] |
D. Kaup and J. Yang, The inverse scattering transform and squared eigenfunctions for a degenerate $3\times 3$ operator, Inverse Problems, 25 (2009), 105010, 21pp.
doi: 10.1088/0266-5611/25/10/105010. |
[15] |
Y. Kodama, Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614.
doi: 10.1007/BF01008354. |
[16] |
Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron, 23 (1987), 510-524.
doi: 10.1109/JQE.1987.1073392. |
[17] |
J. Lenells, Initial-boundary value problems for integrable evolution equations with 3$\times$3 Lax pairs, Physica D, 241 (2012), 857-875.
doi: 10.1016/j.physd.2012.01.010. |
[18] |
L. Ling, L. Zhao and B. Guo, Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations, Commun Nonlinear Sci Numer Simulat, 32 (2016), 285-304.
doi: 10.1016/j.cnsns.2015.08.023. |
[19] |
L. Ling, B. Guo and L. Zhao, High-order rogue waves in vector nonlinear Schrödinger equations, Phys. Rev. E, 89 (2014), 041201(R).
doi: 10.1103/PhysRevE.89.041201. |
[20] |
L. Ling, L.-C. Zhao and B. Guo, Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 3243-3261.
doi: 10.1088/0951-7715/28/9/3243. |
[21] |
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991.
doi: 10.1007/978-3-662-00922-2. |
[22] |
D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu and N. Truta, Soliton solutions for a perturbed nonlinear Schrödinger equation, J. Phys. A: Math. Gen., 26 (1993), L757-L765.
doi: 10.1088/0305-4470/26/17/001. |
[23] |
G. Mu and Z. Qin, Dynamic patterns of high-order rogue waves for Sasa-Satsuma equation, Nonlinear Anal. RWA, 31 (2016), 179-209. |
[24] |
S. Nandy, Inverse scattering approach to coupled higher-order nonlinear Schröinger equation and N-soliton solutions, Nucl. Phys. B, 679 (2004), 647-659.
doi: 10.1016/j.nuclphysb.2003.12.018. |
[25] |
J. Nimmo and H. Yilmaz, Binary Darboux transformation for the Sasa-Satsuma equation, J. Phys. A, 48 (2015), 425202, 16pp.
doi: 10.1088/1751-8113/48/42/425202. |
[26] |
Y. Ohta, Dark soliton solution of Sasa-Satsuma equation, AIP Conf. Proc., 1212 (2010), 114-121. |
[27] |
Q.-H. Park, H. J. Shin and J. Kim, Integrable coupling of optical waves in higher-order nonlinear Schrödinger equations, Phys. Lett. A, 263 (1999), 91-97.
doi: 10.1016/S0375-9601(99)00713-6. |
[28] |
N. Sasa and J. Satsuma, New type of soliton solutions for a higher-order nonlinear Schrödinger equation, J. Phys. Soc. Japan, 60 (1991), 409-417.
doi: 10.1143/JPSJ.60.409. |
[29] |
C. L. Terng and K. Uhlenbeck, Bäcklund transformations and loop group actions, Commun. Pure Appl. Math., 53 (2000), 1-75.
doi: 10.1002/(SICI)1097-0312(200001)53:1<1::AID-CPA1>3.0.CO;2-U. |
[30] |
O. C. Wright, Sasa-Satsuma equation, unstable plane waves and heteroclinic connections, Chaos Solitons and Fractals, 33 (2007), 374-387.
doi: 10.1016/j.chaos.2006.09.034. |
[31] |
J. Xu and E. Fan, The unified transform method for the Sasa-Satsuma equation on the half-line, Proc R Soc A, 469 (2013), 20130068, 25pp.
doi: 10.1098/rspa.2013.0068. |
[32] |
T. Xu, M. Li and L. Li, Anti-dark and mexican-hat solitons in the Sasa-Satsuma equation on the continuous wave background, EPL, 109 (2015), 30006.
doi: 10.1209/0295-5075/109/30006. |
[33] |
J. Yang and D. Kaup, Squared eigenfunctions for the Sasa-Satsuma equation, J. Math. Phys., 50 (2009), 023504, 21pp.
doi: 10.1063/1.3075567. |
[34] |
L.-C. Zhao, S.-C. Li and L. Ling, Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation, Phys. Rev. E, 89 (2014), 023210. |
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