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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

Liming Ling 1,

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.
Keywords: high order solution, Darboux transformation, Sasa-Satsuma equation..
Mathematics Subject Classification: Primary: 37K10, 35Q55; Secondary: 35C0.
Citation: Liming Ling. The algebraic representation for high order solution of Sasa-Satsuma equation. Discrete & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Chen, Twisted rogue-wave pairs in the Sasa-Satsuma equation, Phys. Rev. E, 88 (2013), 023202. doi: 10.1103/PhysRevE.88.023202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809. doi: 10.1063/1.1666399.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kodama, Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614. doi: 10.1007/BF01008354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991. doi: 10.1007/978-3-662-00922-2.  Google Scholar

[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.  Google Scholar

[23]

G. Mu and Z. Qin, Dynamic patterns of high-order rogue waves for Sasa-Satsuma equation, Nonlinear Anal. RWA, 31 (2016), 179-209. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y. Ohta, Dark soliton solution of Sasa-Satsuma equation, AIP Conf. Proc., 1212 (2010), 114-121.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Chen, Twisted rogue-wave pairs in the Sasa-Satsuma equation, Phys. Rev. E, 88 (2013), 023202. doi: 10.1103/PhysRevE.88.023202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809. doi: 10.1063/1.1666399.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kodama, Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614. doi: 10.1007/BF01008354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991. doi: 10.1007/978-3-662-00922-2.  Google Scholar

[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.  Google Scholar

[23]

G. Mu and Z. Qin, Dynamic patterns of high-order rogue waves for Sasa-Satsuma equation, Nonlinear Anal. RWA, 31 (2016), 179-209. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y. Ohta, Dark soliton solution of Sasa-Satsuma equation, AIP Conf. Proc., 1212 (2010), 114-121.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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