Long-time asymptotics for the generalized Sasa-Satsuma equation

Kedong Wang; Xianguo Geng; Mingming Chen; Ruomeng Li; Kedong Wang; Xianguo Geng; Mingming Chen; Ruomeng Li

doi:10.3934/math.2020475

AIMS Mathematics

2020, Volume 5, Issue 6: 7413-7437. doi: 10.3934/math.2020475

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Research article

Long-time asymptotics for the generalized Sasa-Satsuma equation

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

Received: 30 July 2020 Accepted: 17 September 2020 Published: 21 September 2020
MSC : 35Q53, 35B40

In this paper, we study the long-time asymptotic behavior of the solution of the Cauchy problem for the generalized Sasa-Satsuma equation. Starting with the 3 × 3 Lax pair related to the generalized Sasa-Satsuma equation, we construct a Rieman-Hilbert problem, by which the solution of the generalized Sasa-Satsuma equation is converted into the solution of the corresponding RiemanHilbert problem. Using the nonlinear steepest decent method for the Riemann-Hilbert problem, we obtain the leading-order asymptotics of the solution of the Cauchy problem for the generalized SasaSatsuma equation through several transformations of the Riemann-Hilbert problem and with the aid of the parabolic cylinder function.
- nonlinear steepest descent method,
- generalized Sasa-Satsuma equation,
- long-time asymptotics
Citation: Kedong Wang, Xianguo Geng, Mingming Chen, Ruomeng Li. Long-time asymptotics for the generalized Sasa-Satsuma equation[J]. AIMS Mathematics, 2020, 5(6): 7413-7437. doi: 10.3934/math.2020475

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Abstract

In this paper, we study the long-time asymptotic behavior of the solution of the Cauchy problem for the generalized Sasa-Satsuma equation. Starting with the 3 × 3 Lax pair related to the generalized Sasa-Satsuma equation, we construct a Rieman-Hilbert problem, by which the solution of the generalized Sasa-Satsuma equation is converted into the solution of the corresponding RiemanHilbert problem. Using the nonlinear steepest decent method for the Riemann-Hilbert problem, we obtain the leading-order asymptotics of the solution of the Cauchy problem for the generalized SasaSatsuma equation through several transformations of the Riemann-Hilbert problem and with the aid of the parabolic cylinder function.

References

[1]	N. Sasa, J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schrödinger equation, J. Phys. Soc. Jpn, 60 (1991), 409-417.
[2]	Y. Kivshar, G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Waltham: Academic Press, 2003.
[3]	K. Porsezian, Soliton models in resonant and nonresonant optical fibers, Pramana, 57 (2001), 1003-1039.
[4]	A. V. Slunyaev, A high-order nonlinear envelope equation for gravity waves in finite-depth water, J. Exp. Theor. Phys., 101 (2005), 926-941.
[5]	M. Trippenbach, Y. B. Band, Effects of self-steepening and self-frequency shifting on short-pulse splitting in disper-sive nonlinear media, Phys. Rev. A, 57 (1991), 4791-4803.
[6]	J. K. Yang, D. J. Kaup, Squared eigenfunctions for the Sasa-Satsuma equation, J. Math. Phys., 50 (2009), 023504.
[7]	C. Gilson, J. Hietarinta, J. Nimmo, et al. Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions, Phys. Rev. E, 68 (2003), 016614.
[8]	J. J. C. Nimmo, H. Yilmaz, Binary Darboux transformation for the Sasa-Satsuma equation, J. Phys. A, 48 (2015), 425202.
[9]	X. G. Geng, R. M. Li, B. Xue, A vector general nonlinear Schrödinger equation with (m + n) components, J. Nonlinear Sci., 30 (2020), 991-1013.
[10]	R. M. Li, X. G. Geng, On a vector long wave-short wave-type model, Stud, Appl. Math., 144 (2020), 164-184.
[11]	R. M. Li, X. G. Geng, Rogue periodic waves of the sine-Gordon equation, Appl. Math. Lett., 102 (2020), 106147.
[12]	J. Xu, Q. Z. Zhu, E. G. Fan, The initial-boundary value problem for the Sasa-Satsuma equation on a finite interval via the Fokas method, J. Math. Phys., 59 (2018), 073508.
[13]	Y. Y. Zhai, X. G. Geng, The coupled Sasa-Satsuma hierarchy: trigonal curve and finite genus solutions, Anal. Appl., 15 (2017), 667-697.
[14]	X. G. Geng, L. H. Wu, G. L. He, Quasi-periodic solutions of the Kaup-Kupershmidt hierarchy, J. Nonlinear Sci., 23 (2013), 527-555.
[15]	X. G. Geng, Y. Y. Zhai, H. H. Dai, Algebro-geometric solutions of the coupled modified Kortewegde Vries hierarchy, Adv. Math., 263 (2014), 123-153.
[16]	J. Wei, X. G. Geng, X. Zeng, The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices, Trans. Amer. Math. Soc., 371 (2019), 1483-1507.
[17]	J. Wei, X. G. Geng, A super Sasa-Satsuma hierarchy and bi-Hamiltonian structures, Appl. Math. Lett., 83 (2018), 46-52.
[18]	P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math., 137 (1993), 295-368.
[19]	K. Grunert, G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom., 12 (2009), 287-324.
[20]	P. J. Cheng, S. Venakides, X. Zhou, Long-time asymptotics for the pure radiation solution of the sine-Gordon equation, Comm. Partial Differential Equations, 24 (1999), 1195-1262.
[21]	A. V. Kitaev, A. H. Vartanian, Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation: Solitonless sector, Inverse Problem, 13 (1997), 1311-1339.
[22]	A. V. Kitaev, A. H. Vartanian, Asymptotics of solutions to the modified nonlinear Schrödinger equation: Solution on a nonvanishing continuous background, SIAM J. Math. Anal., 30 (1999), 787-832.
[23]	A. H. Vartanian, Higher order asymptotics of the modified nonlinear Schrödinger equation, Comm. Partial Differential Equations, 25 (2000), 1043-1098.
[24]	A. Boutet de Monvel, A. Kostenko, D. Shepelsky, G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.
[25]	L. K. Arruda, J. Lenells, Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line, Nonlinearity, 30 (2017), 4141-4172.
[26]	A. Boutet de Monvel, A. Its, V. Kotlyarov, Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition on the half-line, Comm. Math. Phys., 290 (2009), 479-522.
[27]	P. Deift, J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, Int. Math. Res. Not. IMRN, 24 (2011), 5505-5624.
[28]	I. Egorova, J. Michor, G. Teschl, Rarefaction waves for the Toda equation via nonlinear steepest descent, Discrete Contin. Dyn. Syst., 38 (2018), 2007-2028.
[29]	H. Yamane, Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, J. Math. Soc. Japan, 66 (2014), 765-803.
[30]	J. Lenells, The nonlinear steepest descent method: asymptotics for initial-boundary value problems, SIAM J. Math. Anal., 48 (2016), 2076-2118.
[31]	A. Boutet de Monvel, D. Shepelsky, A Riemann-Hilbert approach for the Degasperis-Procesi equation, Nonlinearity, 26 (2013), 2081-2107.
[32]	A. Boutet de Monvel, J. Lenells, D. Shepelsky, Long-time asymptotics for the Degasperis-Procesi equation on the half-line, Ann. Inst. Fourier, 69 (2019), 171-230.
[33]	X. G. Geng, H. Liu, The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation, J. Nonlinear Sci., 28 (2018), 739-763.
[34]	H. Liu, X. G. Geng, B. Xue, The Deift-Zhou steepest descent method to long-time asymptotics for the Sasa-Satsuma equation, J. Differential Equations, 265 (2018), 5984-6008.
[35]	B. B. Hu, T. C. Xia, W. X. Ma, Riemann-Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line, Appl. Math. Comput., 332 (2018), 148-159.
[36]	B. B. Hu, T. C. Xia, N. Zhang, et al. Initial-boundary value problems for the coupled higher-order nonlinear Schrödinger equations on the half-line, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 83-92.
[37]	N. Zhang, T. C. Xia, E. G. Fan, A Riemann-Hilbert approach to the Chen-Lee-Liu equation on the half line, Acta Math. Appl. Sin. Engl. Ser., 34 (2018), 493-515.
[38]	X. G. Geng, J. P. Wu, Riemann-Hilbert approach and N-soliton solutions for a generalized SasaSatsuma equation, Wave Motion, 60 (2016), 62-72.
[39]	M. J. Ablowitz, A. S. Fokas, Complex variables: Introduction and applications, Cambridge: Cambridge University Press, 2003.
[40]	R. Beals, R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math., 37 (1984), 39-90.
[41]	R. Beals, R. Wong, Special functions and orthogonal polynomials, Cambridge: Cambridge University Press, 2016.
[42]	E. T. Whittaker, G. N. Watson, A Course of Modern Analysis 4^th Ed, Cambridge: Cambridge University Press, 1927.

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