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December  2016, 9(6): 1975-2010. doi: 10.3934/dcdss.2016081

The algebraic representation for high order solution of Sasa-Satsuma equation

1. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China

Received  July 2015 Revised  September 2016 Published  November 2016

In this paper, we reestablish the elementary Darboux transformation for Sasa-Satsuma equation with the aid of loop group method. Furthermore, the generalized Darboux transformation is given with the limit technique. As direct applications, we give the single solitonic solutions for the focusing and defocusing case. The general high order solution formulas with the determinant form are obtained through generalized DT and the formal series method.
Citation: Liming Ling. The algebraic representation for high order solution of Sasa-Satsuma equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1975-2010. doi: 10.3934/dcdss.2016081
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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