doi: 10.3934/dcds.2022060
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Long-time asymptotics for the modified complex short pulse equation

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

*Corresponding author: Xianguo Geng

Received  December 2021 Early access April 2022

Fund Project: The second author is supported by the National Natural Science Foundation of China (Grant Nos. 11931017, 11871440)

Based on the spectral analysis and the inverse scattering method, by introducing some spectral function transformations and variable transformations, the initial value problem for the modified complex short pulse (mCSP) equation is transformed into a $ 2\times2 $ matrix Riemann-Hilbert problem. It is proved that the solution of the initial value problem for the mCSP equation has a parametric expression related to the solution of the matrix Riemann-Hilbert problem. Various Deift-Zhou contour deformations and the motivation behind them are given. Through several appropriate transformations and strict error estimates, the original matrix Riemann-Hilbert problem can be reduced to the model Riemann-Hilbert problem, whose solution can be solved explicitly in terms of the parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the initial value problem for the mCSP equation is obtained by using the nonlinear steepest decent method.

Citation: Mingming Chen, Xianguo Geng, Kedong Wang. Long-time asymptotics for the modified complex short pulse equation. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022060
References:
[1] M. J. Ablowitz and P. A. Clakson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.
[2] M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511791246.
[3]

L. K. Arruda and J. Lenells, Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line, Nonlinearity, 30 (2017), 4141-4172.  doi: 10.1088/1361-6544/aa84c6.

[4]

R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math., 37 (1984), 39-90.  doi: 10.1002/cpa.3160370105.

[5]

A. Boutet de MonvelA. Its and 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.  doi: 10.1007/s00220-009-0848-7.

[6]

A. Boutet de MonvelA. KostenkoD. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.  doi: 10.1137/090748500.

[7]

A. Boutet de MonvelJ. Lenells and D. Shepelsky, Long-time asymptotics for the Degasperis-Procesi equation on the half-line, Ann. Inst. Fourier., 69 (2019), 171-230.  doi: 10.5802/aif.3241.

[8]

A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris., 343 (2006), 627-632.  doi: 10.1016/j.crma.2006.10.014.

[9]

A. Boutet de Monvel and D. Shepelsky, The Ostrovsky-Vakhnenko equation: A Riemann-Hilbert approach, C. R. Math. Acad. Sci. Paris., 352 (2014), 189-195.  doi: 10.1016/j.crma.2014.01.001.

[10]

A. Boutet de Monvel and D. Shepelsky, The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach, J. Phys. A. Math. Theor., 48 (2015), 035204, 34 pp. doi: 10.1088/1751-8113/48/3/035204.

[11]

A. Boutet de Monvel, D. Shepelsky and L. Zielinski, The short-wave model for the Camassa-Holm equation: A Riemann-Hilbert approach, Inverse Probl., 27 (2011), 105006, 17 pp. doi: 10.1088/0266-5611/27/10/105006.

[12]

A. Boutet de MonvelD. Shepelsky and L. Zielinski, The short pulse equation by a Riemann-Hilbert approach, Lett. Math. Phys., 107 (2017), 1345-1373.  doi: 10.1007/s11005-017-0945-z.

[13]

J. C. Brunelli, The short pulse hierarchy, J. Math. Phys., 46 (2005), 123507, 9 pp. doi: 10.1063/1.2146189.

[14]

J. C. Brunelli, The bi-Hamiltonian structure of the short pulse equation, Phys. Lett. A., 353 (2006), 475-478.  doi: 10.1016/j.physleta.2006.01.009.

[15]

M. M. Chen, X. G. Geng and K. D. Wang, Spectral analysis and long-time asymptotics for the potential Wadati-Konno-Ichikawa equation, J. Math. Anal. Appl., 501 (2021), Paper No. 125170, 27 pp. doi: 10.1016/j.jmaa.2021.125170.

[16]

P. J. ChengS. Venakides and X. Zhou, Long-time asymptotics for the pure radiation solution of the sine-Gordon equation, Commun. Partial Differ. Equ., 24 (1999), 1195-1262.  doi: 10.1080/03605309908821464.

[17]

Y. ChungC. K. R. T. JonesT. Schäfer and C. E. Wayne, Ultra-short pulses in linear and nonlinear media, Nonlinearity, 18 (2005), 1351-1374.  doi: 10.1088/0951-7715/18/3/021.

[18]

P. Deift, A. R. Its and X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, In Important Developments in Soliton Theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, (1993), 181-204.

[19]

P. A. Deift and J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, Int. Math. Res., 96 (2011), 5505-5624.  doi: 10.1007/s11005-010-0458-5.

[20]

P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math., 137 (1993), 295-368.  doi: 10.2307/2946540.

[21]

X. G. GengM. M. Chen and K. D. Wang, Long-time asymptotics of the coupled modified Korteweg-de Vries equation, J. Geom. Phys., 142 (2019), 151-167.  doi: 10.1016/j.geomphys.2019.04.009.

[22]

X. G. GengR. M. Li and B. Xue, A vector general nonlinear Schrödinger equation with $(m+n)$ components, J. Nonlinear Sci., 30 (2020), 991-1013.  doi: 10.1007/s00332-019-09599-4.

[23]

X. G. Geng and H. Liu, The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation, J. Nonlinear Sci., 28 (2018), 739-763.  doi: 10.1007/s00332-017-9426-x.

[24]

X. G. GengK. D. Wang and M. M. Chen, Long-time asymptotics for the spin-1 Gross-Pitaevskii equation, Commun. Math. Phys., 382 (2021), 585-611.  doi: 10.1007/s00220-021-03945-y.

[25]

X. G. Geng and J. P. Wu, Riemann-Hilbert approach and $N$-soliton solutions for a generalized Sasa-Satsuma equation, Wave Motion, 60 (2016), 62-72.  doi: 10.1016/j.wavemoti.2015.09.003.

[26]

X. G. GengX. Zeng and J. Wei, The application of the theory of trigonal curves to the discrete coupled nonlinear Schrödinger hierarchy, Ann. Henri Poincaré., 20 (2019), 2585-2621.  doi: 10.1007/s00023-019-00798-z.

[27]

X. G. GengY. Y. Zhai and H. H. Dai, Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math., 263 (2014), 123-153.  doi: 10.1016/j.aim.2014.06.013.

[28]

K. Grunert and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom., 12 (2009), 287-324.  doi: 10.1007/s11040-009-9062-2.

[29]

B. L. Guo and L. M. Ling, Riemann-Hilbert approach and $N$-soliton formula for coupled derivative Schrödinger equation, J. Math. Phys., 53 (2012), 073506, 20 pp. doi: 10.1063/1.4732464.

[30]

A. V. Kitaev and A. H. Vartanian, Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation: Solitonless sector, Inverse Probl., 13 (1997), 1311-1339.  doi: 10.1088/0266-5611/13/5/014.

[31]

A. V. Kitaev and 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.  doi: 10.1137/S0036141098332019.

[32]

R. M. Li and X. G. Geng, On a vector long wave-short wave-type model, Stud. Appl. Math., 144 (2020), 164-184.  doi: 10.1111/sapm.12293.

[33]

R. M. Li and X. G. Geng, Rogue periodic waves of the sine-Gordon equation, Appl. Math. Lett., 102 (2020), 106147, 8 pp. doi: 10.1016/j.aml.2019.106147.

[34]

H. LiuX. G. Geng and B. Xue, The Deift-Zhou steepest descent method to long-time asymptotics for the Sasa-Satsuma equation, J. Differ. Equ., 265 (2018), 5984-6008.  doi: 10.1016/j.jde.2018.07.026.

[35]

Y. Matsuno, Integrable multi-component generalization of a modified short pulse equation, J. Math. Phys., 57 (2016), 111507, 23 pp. doi: 10.1063/1.4967952.

[36]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991.

[37]

M. L. Robelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math., 81 (1989), 221-248.  doi: 10.1002/sapm1989813221.

[38]

A. Sakovich and S. Sakovich, The short pulse equation is integrable, J. Phys. Soc. Jpn., 74 (2005), 239-241.  doi: 10.1088/0305-4470/39/22/L03.

[39]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.  doi: 10.1016/j.physd.2004.04.007.

[40]

V. S. Shchesnovich and J. K. Yang, General soliton matrices in the Riemann-Hilbert problem for integrable nonlinear equations, J. Math. Phys., 44 (2003), 4604-4639.  doi: 10.1063/1.1605821.

[41]

S. F. ShenB. F. Feng and Y. Ohta, A modified complex short pulse equation of defocusing type, J. Nonlinear Math. Phys., 24 (2017), 195-209.  doi: 10.1080/14029251.2017.1306946.

[42]

A. H. Vartanian, Higher order asymptotics of the modified non-linear Schrödinger equation, Commun. Partial Differ. Equ., 25 (2000), 1043-1098.  doi: 10.1080/03605300008821541.

[43]

J. WeiX. G. Geng and X. Zeng, The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices, Trans. Amer. Math. Soc., 371 (2019), 1483-1507.  doi: 10.1090/tran/7349.

[44] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511608759.
[45]

J. Xu, Long-time asymptotics for the short pulse equation, J. Differ. Equ., 265 (2018), 3494-3532.  doi: 10.1016/j.jde.2018.05.009.

[46]

J. Xu and E. G. Fan, Long-time asymptotic behavior for the complex short pulse equation, J. Differ. Equ., 269 (2020), 10322-10349.  doi: 10.1016/j.jde.2020.07.009.

show all references

References:
[1] M. J. Ablowitz and P. A. Clakson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.
[2] M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511791246.
[3]

L. K. Arruda and J. Lenells, Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line, Nonlinearity, 30 (2017), 4141-4172.  doi: 10.1088/1361-6544/aa84c6.

[4]

R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math., 37 (1984), 39-90.  doi: 10.1002/cpa.3160370105.

[5]

A. Boutet de MonvelA. Its and 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.  doi: 10.1007/s00220-009-0848-7.

[6]

A. Boutet de MonvelA. KostenkoD. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.  doi: 10.1137/090748500.

[7]

A. Boutet de MonvelJ. Lenells and D. Shepelsky, Long-time asymptotics for the Degasperis-Procesi equation on the half-line, Ann. Inst. Fourier., 69 (2019), 171-230.  doi: 10.5802/aif.3241.

[8]

A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris., 343 (2006), 627-632.  doi: 10.1016/j.crma.2006.10.014.

[9]

A. Boutet de Monvel and D. Shepelsky, The Ostrovsky-Vakhnenko equation: A Riemann-Hilbert approach, C. R. Math. Acad. Sci. Paris., 352 (2014), 189-195.  doi: 10.1016/j.crma.2014.01.001.

[10]

A. Boutet de Monvel and D. Shepelsky, The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach, J. Phys. A. Math. Theor., 48 (2015), 035204, 34 pp. doi: 10.1088/1751-8113/48/3/035204.

[11]

A. Boutet de Monvel, D. Shepelsky and L. Zielinski, The short-wave model for the Camassa-Holm equation: A Riemann-Hilbert approach, Inverse Probl., 27 (2011), 105006, 17 pp. doi: 10.1088/0266-5611/27/10/105006.

[12]

A. Boutet de MonvelD. Shepelsky and L. Zielinski, The short pulse equation by a Riemann-Hilbert approach, Lett. Math. Phys., 107 (2017), 1345-1373.  doi: 10.1007/s11005-017-0945-z.

[13]

J. C. Brunelli, The short pulse hierarchy, J. Math. Phys., 46 (2005), 123507, 9 pp. doi: 10.1063/1.2146189.

[14]

J. C. Brunelli, The bi-Hamiltonian structure of the short pulse equation, Phys. Lett. A., 353 (2006), 475-478.  doi: 10.1016/j.physleta.2006.01.009.

[15]

M. M. Chen, X. G. Geng and K. D. Wang, Spectral analysis and long-time asymptotics for the potential Wadati-Konno-Ichikawa equation, J. Math. Anal. Appl., 501 (2021), Paper No. 125170, 27 pp. doi: 10.1016/j.jmaa.2021.125170.

[16]

P. J. ChengS. Venakides and X. Zhou, Long-time asymptotics for the pure radiation solution of the sine-Gordon equation, Commun. Partial Differ. Equ., 24 (1999), 1195-1262.  doi: 10.1080/03605309908821464.

[17]

Y. ChungC. K. R. T. JonesT. Schäfer and C. E. Wayne, Ultra-short pulses in linear and nonlinear media, Nonlinearity, 18 (2005), 1351-1374.  doi: 10.1088/0951-7715/18/3/021.

[18]

P. Deift, A. R. Its and X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, In Important Developments in Soliton Theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, (1993), 181-204.

[19]

P. A. Deift and J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, Int. Math. Res., 96 (2011), 5505-5624.  doi: 10.1007/s11005-010-0458-5.

[20]

P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math., 137 (1993), 295-368.  doi: 10.2307/2946540.

[21]

X. G. GengM. M. Chen and K. D. Wang, Long-time asymptotics of the coupled modified Korteweg-de Vries equation, J. Geom. Phys., 142 (2019), 151-167.  doi: 10.1016/j.geomphys.2019.04.009.

[22]

X. G. GengR. M. Li and B. Xue, A vector general nonlinear Schrödinger equation with $(m+n)$ components, J. Nonlinear Sci., 30 (2020), 991-1013.  doi: 10.1007/s00332-019-09599-4.

[23]

X. G. Geng and H. Liu, The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation, J. Nonlinear Sci., 28 (2018), 739-763.  doi: 10.1007/s00332-017-9426-x.

[24]

X. G. GengK. D. Wang and M. M. Chen, Long-time asymptotics for the spin-1 Gross-Pitaevskii equation, Commun. Math. Phys., 382 (2021), 585-611.  doi: 10.1007/s00220-021-03945-y.

[25]

X. G. Geng and J. P. Wu, Riemann-Hilbert approach and $N$-soliton solutions for a generalized Sasa-Satsuma equation, Wave Motion, 60 (2016), 62-72.  doi: 10.1016/j.wavemoti.2015.09.003.

[26]

X. G. GengX. Zeng and J. Wei, The application of the theory of trigonal curves to the discrete coupled nonlinear Schrödinger hierarchy, Ann. Henri Poincaré., 20 (2019), 2585-2621.  doi: 10.1007/s00023-019-00798-z.

[27]

X. G. GengY. Y. Zhai and H. H. Dai, Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math., 263 (2014), 123-153.  doi: 10.1016/j.aim.2014.06.013.

[28]

K. Grunert and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom., 12 (2009), 287-324.  doi: 10.1007/s11040-009-9062-2.

[29]

B. L. Guo and L. M. Ling, Riemann-Hilbert approach and $N$-soliton formula for coupled derivative Schrödinger equation, J. Math. Phys., 53 (2012), 073506, 20 pp. doi: 10.1063/1.4732464.

[30]

A. V. Kitaev and A. H. Vartanian, Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation: Solitonless sector, Inverse Probl., 13 (1997), 1311-1339.  doi: 10.1088/0266-5611/13/5/014.

[31]

A. V. Kitaev and 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.  doi: 10.1137/S0036141098332019.

[32]

R. M. Li and X. G. Geng, On a vector long wave-short wave-type model, Stud. Appl. Math., 144 (2020), 164-184.  doi: 10.1111/sapm.12293.

[33]

R. M. Li and X. G. Geng, Rogue periodic waves of the sine-Gordon equation, Appl. Math. Lett., 102 (2020), 106147, 8 pp. doi: 10.1016/j.aml.2019.106147.

[34]

H. LiuX. G. Geng and B. Xue, The Deift-Zhou steepest descent method to long-time asymptotics for the Sasa-Satsuma equation, J. Differ. Equ., 265 (2018), 5984-6008.  doi: 10.1016/j.jde.2018.07.026.

[35]

Y. Matsuno, Integrable multi-component generalization of a modified short pulse equation, J. Math. Phys., 57 (2016), 111507, 23 pp. doi: 10.1063/1.4967952.

[36]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991.

[37]

M. L. Robelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math., 81 (1989), 221-248.  doi: 10.1002/sapm1989813221.

[38]

A. Sakovich and S. Sakovich, The short pulse equation is integrable, J. Phys. Soc. Jpn., 74 (2005), 239-241.  doi: 10.1088/0305-4470/39/22/L03.

[39]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.  doi: 10.1016/j.physd.2004.04.007.

[40]

V. S. Shchesnovich and J. K. Yang, General soliton matrices in the Riemann-Hilbert problem for integrable nonlinear equations, J. Math. Phys., 44 (2003), 4604-4639.  doi: 10.1063/1.1605821.

[41]

S. F. ShenB. F. Feng and Y. Ohta, A modified complex short pulse equation of defocusing type, J. Nonlinear Math. Phys., 24 (2017), 195-209.  doi: 10.1080/14029251.2017.1306946.

[42]

A. H. Vartanian, Higher order asymptotics of the modified non-linear Schrödinger equation, Commun. Partial Differ. Equ., 25 (2000), 1043-1098.  doi: 10.1080/03605300008821541.

[43]

J. WeiX. G. Geng and X. Zeng, The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices, Trans. Amer. Math. Soc., 371 (2019), 1483-1507.  doi: 10.1090/tran/7349.

[44] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511608759.
[45]

J. Xu, Long-time asymptotics for the short pulse equation, J. Differ. Equ., 265 (2018), 3494-3532.  doi: 10.1016/j.jde.2018.05.009.

[46]

J. Xu and E. G. Fan, Long-time asymptotic behavior for the complex short pulse equation, J. Differ. Equ., 269 (2020), 10322-10349.  doi: 10.1016/j.jde.2020.07.009.

Figure 1.  The oriented contour on $ \mathbb{R} $
Figure 2.  The signature table for Re $ (i\theta) $ in the complex $ k $-plane
Figure 3.  The oriented contour on $ \mathbb{R} $
Figure 4.  The oriented contour $ \Sigma $
Figure 5.  The oriented contour $ \Sigma^\prime $
Figure 6.  The oriented contour $ \Sigma_A $ or $ \Sigma_B $ (reoriented)
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