$ \overline{\partial} $-dressing method for the three-coupled fourth-order nonlinear Schrödinger system

Jin-Jin Mao; Xue-Wei Yan; Jin-Jin Mao; Xue-Wei Yan

doi:10.3934/math.20251250

AIMS Mathematics

2025, Volume 10, Issue 12: 28407-28435. doi: 10.3934/math.20251250

Previous Article Next Article

Research article

$ \overline{\partial} $-dressing method for the three-coupled fourth-order nonlinear Schrödinger system

Jin-Jin Mao ^,,
Xue-Wei Yan

College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

Received: 09 September 2025 Revised: 11 November 2025 Accepted: 21 November 2025 Published: 03 December 2025
MSC : 30E25, 35Q15

This article mainly investigated the three-coupled fourth-order nonlinear Schrödinger (NLS) systems, which characterized the alpha-helical proteins with interspine coupling at the fourth-order interaction. By the $ \overline{\partial} $-dressing method, we studied the three-coupled fourth-order NLS systems. We derived a new spectral problem starting from the $ \overline{\partial} $-problem. Then, we obtained the three-coupled fourth-order NLS hierarchy associated with the spectral problem using a recursive operator. Finally, we analyzed the $ N $-soliton solutions of the three-coupled fourth-order NLS systems by determining the spectral transform matrix in the $ \overline{\partial} $-problem. We also provided detailed descriptions of the $ 1 $-, $ 2 $-, and $ 3 $-soliton solutions, along with dynamic features and graphical illustration.
- the three-coupled fourth-order nonlinear Schrödinger systems,
- $ \overline{\partial} $-dressing method,
- Lax pair,
- recursive operator,
- soliton solutions
Citation: Jin-Jin Mao, Xue-Wei Yan. $ \overline{\partial} $-dressing method for the three-coupled fourth-order nonlinear Schrödinger system[J]. AIMS Mathematics, 2025, 10(12): 28407-28435. doi: 10.3934/math.20251250

Related Papers:

Abstract

This article mainly investigated the three-coupled fourth-order nonlinear Schrödinger (NLS) systems, which characterized the alpha-helical proteins with interspine coupling at the fourth-order interaction. By the $ \overline{\partial} $-dressing method, we studied the three-coupled fourth-order NLS systems. We derived a new spectral problem starting from the $ \overline{\partial} $-problem. Then, we obtained the three-coupled fourth-order NLS hierarchy associated with the spectral problem using a recursive operator. Finally, we analyzed the $ N $-soliton solutions of the three-coupled fourth-order NLS systems by determining the spectral transform matrix in the $ \overline{\partial} $-problem. We also provided detailed descriptions of the $ 1 $-, $ 2 $-, and $ 3 $-soliton solutions, along with dynamic features and graphical illustration.

References

[1]	G. P. Agrawal, Nonlinear Fiber Optics, Berlin, Heidelberg: Springer Berlin Heidelberg, 542 (2000), 195–211. https://doi.org/10.1007/3-540-46629-0_9
[2]	D. R. Solli, C. Ropers, P. Koonath, B. Jalali, Optical rogue waves, Nature, 450 (2007), 1054–1057. https://doi.org/10.1038/nature06402
[3]	D. R. Solli, C. Ropers, B. Jalali, Active control of rogue waves for stimulated supercontinuum generation, Phys. Rev. Lett., 101 (2008), 233902. https://doi.org/10.1103/PhysRevLett.101.233902 doi: 10.1103/PhysRevLett.101.233902
[4]	W. M. Moslem, P. Shukal, B. Eliasson, EPL, 96 (2011), 25002.
[5]	Z. Du, B. Tian, Q. X. Qu, H. P. Chai, X. Y. Wu, Semirational rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein, Superlattice. Microst., 112 (2017), 362–373. https://doi.org/10.1016/j.spmi.2017.09.046 doi: 10.1016/j.spmi.2017.09.046
[6]	A. Chabchoub, N. Hoffmann, H. Branger, C. Kharuf, N. Akhmediev, Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model, Phys. Fluids., 25 (2013), 101704. https://doi.org/10.1063/1.4824706 doi: 10.1063/1.4824706
[7]	Z. Y. Yan, Financial rogue waves, Commun. Theor. Phys., 54 (2010), 947–949. https://doi.org/10.1088/0253-6102/54/5/31 doi: 10.1088/0253-6102/54/5/31
[8]	L. Stenflo, P. Shukla, J. Plasma, Nonlinear acoustic-gravity waves, J. Plasma Phys., 75 (2009), 841–847. https://doi.org/10.1017/S0022377809007892 doi: 10.1017/S0022377809007892
[9]	L. Ling, L. C. Zhao, B. Guo, Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 3243–3261. https://doi.org/10.1088/0951-7715/28/9/3243 doi: 10.1088/0951-7715/28/9/3243
[10]	G. Zhang, Z. Yan, Three-component nonlinear Schrödinger equations: modulational instability, Nth-order vector rational and semi-rational rogue waves, and dynamics, Commun. Nonlinear. Sci., 62 (2018), 117–133. https://doi.org/10.1016/j.cnsns.2018.02.008 doi: 10.1016/j.cnsns.2018.02.008
[11]	G. Mu, Z. Qin, R. Grimshaw, N. Akhmediev, Intricate dynamics of rogue waves governed by the Sasa-Satsuma equation, Physica D, 402 (2020), 132252. https://doi.org/10.1016/j.physd.2019.132252 doi: 10.1016/j.physd.2019.132252
[12]	C. R. Zhang, B. Tian, Q. X. Qu, L. Liu, H. Y. Tian, Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber, Z. Angew. Math. Phys., 71 (2020), 1–19. https://doi.org/10.1007/s00033-019-1225-9 doi: 10.1007/s00033-019-1225-9
[13]	R. Hirota, J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A., 85 (1981), 407–408. https://doi.org/10.1016/0375-9601(81)90423-0 doi: 10.1016/0375-9601(81)90423-0
[14]	L. Liu, B. Tian, W. R. Sun, H. L. Zhen, W. R. Shen, Bright-dark vector soliton solutions for a generalized coupled Hirota system in the optical glass fiber, Commun. Nonlinear. Sci., 39 (2016), 545–555. https://doi.org/10.1016/j.cnsns.2016.04.001 doi: 10.1016/j.cnsns.2016.04.001
[15]	A. C. Newell, Solitons in Mathematics and Physics, Society for Industrial and applied Mathematics, 1985. https://epubs.siam.org/doi/pdf/10.1137/1.9781611970227.fm
[16]	M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, Society for Industrial and applied Mathematics, 1981. https://epubs.siam.org/doi/pdf/10.1137/1.9781611970883.bm
[17]	S. Novikov, S. V. Manakov, L. P. Pitaevskii, V. E. Zakharov, Theory of solitons: The inverse scattering method, Springer Science Business Media, 1984.
[18]	M. J. Ablowitz, Z. H. Musslimani, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29 (2016), 915–946. https://doi.org/10.1088/0951-7715/29/3/915 doi: 10.1088/0951-7715/29/3/915
[19]	D. Wang, Z. Liu, H. Zhao, H. Qin, G. Bai. C. Chen, et al., Launching by cavitation, Science, 389 (2025), 935–939. https://doi.org/10.1126/science.adu8943 doi: 10.1126/science.adu8943
[20]	D. S. Mou, Z. Z. Si, W. X. Qiu, C. Q. Dai, Optical soliton formation and dynamic characteristics in photonic Moiré lattices, Opt. Laser Technol., 181 (2025), 111774. https://doi.org/10.1016/j.optlastec.2024.111774 doi: 10.1016/j.optlastec.2024.111774
[21]	Z. Z. Si, D. L. Wang, B. W. Zhu, Z. T. Ju, X. P. Wang, W. Liu, et al., Deep learning for dynamic modeling and coded information storage of vector-soliton pulsations in mode-locked fiber lasers, Laser Photonics Rev., 18 (2024), 2400097. https://doi.org/10.1002/lpor.202400097 doi: 10.1002/lpor.202400097
[22]	Z. Z. Si, Y. Y. Wang, C. Q. Dai. Switching, explosion, and chaos of multi-wavelength soliton states in ultrafast fiber lasers, Sci. China Phys. Mech., 67 (2024), 274211. https://doi.org/10.1007/s11433-023-2365-7 doi: 10.1007/s11433-023-2365-7
[23]	A. C. Scott, Launching a Davydov soliton: Ⅰ. Soliton analysis, Physica Scripta, 29 (1984), 279. https://doi.org/10.1088/0031-8949/29/3/016 doi: 10.1088/0031-8949/29/3/016
[24]	S. S. Veni, M. M. Latha, A generalized Davydov model with interspine coupling and its integrable discretization, Phys. Scripta, 86 (2012), 25003. https://doi.org/10.1088/0031-8949/86/02/025003 doi: 10.1088/0031-8949/86/02/025003
[25]	W. R. Sun, B. Tian, Y. F. Wang, H. L. Zhen, Soliton excitations and interactions for the three-coupled fourth-order nonlinear Schrödinger equations in the alpha helical proteins, Eur. Phys, J. D, 69 (2015), 1–9. https://doi.org/10.1140/epjd/e2015-60027-6 doi: 10.1140/epjd/e2015-60027-6
[26]	Z. Du, B. Tian, H. P. Chai, Y. Q. Yuan, Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein, Commun. Nonlinear. Sci., 67 (2019), 49–59. https://doi.org/10.1016/j.cnsns.2018.06.014 doi: 10.1016/j.cnsns.2018.06.014
[27]	Z. Du, B. Tian, H. P. Chai, X. H. Zhao, Lax pair, Darboux transformation and rogue waves for the three-coupled fourth-order nonlinear Schrödinger system in an alpha helical protein, Wave. Random Complex, 31 (2021), 1051–1071. https://doi.org/10.1080/17455030.2019.1644466 doi: 10.1080/17455030.2019.1644466
[28]	M. J. Dong, L. X. Tian, J. D. Wei, Infinitely many conservation laws and Darboux-dressing transformation for the three-coupled fourth-order nonlinear Schrödinger equations, Eur. Phys. J. Plus., 137 (2022), 1–15. https://doi.org/10.1140/epjp/s13360-021-02200-6 doi: 10.1140/epjp/s13360-021-02200-6
[29]	L. C. Zhao, J. Liu, Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation, Phys. Rev. E., 87 (2013), 013201. https://doi.org/10.1103/PhysRevE.87.013201 doi: 10.1103/PhysRevE.87.013201
[30]	X. B. Wang, B. Han, The three-component coupled nonlinear Schrödinger equation: Rogue waves on a multi-soliton background and dynamics, EPL, 126 (2019), 15001. https://doi.org/10.1209/0295-5075/126/15001 doi: 10.1209/0295-5075/126/15001
[31]	F. Baronio, A. Degasperis, M. Conforti, S. Wabnitz, Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic rogue waves, Phys. Rev. Lett., 109 (2012), 044102. https://doi.org/10.1103/PhysRevLett.109.044102 doi: 10.1103/PhysRevLett.109.044102
[32]	V. E. Zakharov, A. B. Shabat, A scheme for integrating the nonlinear equations of mathermatical physics by the method of the inverse scattering problem, I. Funct. Anal. Appl., 8 (1974), 43–53. https://doi.org/10.1007/BF01075696 doi: 10.1007/BF01075696
[33]	Q. Y. Cheng, Y. L. Yang, E. G. Fan, Long-time asymptotic behavior of a mixed Schrödinger equation with weighted Sobolev initial data, J. Math. Phys., 62 (2021), 093507. https://doi.org/10.1063/5.0045970 doi: 10.1063/5.0045970
[34]	Y. L. Yang, E. G. Fan, On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions, 2021. arXiv preprint arXiv: 2101.02489. https://doi.org/10.48550/arXiv.2101.02489
[35]	R. Beals, R. R. Coifman, The D-bar approach to inverse scattering and nonlinear evolutions, Physica D, 18 (1986), 242–249. https://doi.org/10.1016/0167-2789(86)90184-3 doi: 10.1016/0167-2789(86)90184-3
[36]	R. Beals, R. R. Coifman, Linear spectral problems, non-linear equations and the $\overline{\partial}$-method, Inverse Probl., 5 (1989), 87–130. https://doi.org/10.1088/0266-5611/5/2/002 doi: 10.1088/0266-5611/5/2/002
[37]	A. S. Fokas, V. E. Zakharov, The dressing method and nonlocal Riemann-Hilbert peoblem, J. Nonlinear Sci., 2 (1992), 109–134. https://doi.org/10.1007/BF02429853 doi: 10.1007/BF02429853
[38]	J. Zhu, X. Geng, The AB equations and the $\overline{\partial}$-dressing method in semi-characteristic coordinates, Math. Phys. Anal. Geom., 17 (2014), 49–65. https://doi.org/10.1007/s11040-014-9140-y doi: 10.1007/s11040-014-9140-y
[39]	V. G. Dubrovsky, The $\overline{\partial}$-dressing method and the solutions with constant asymptotic values at infinity of DS-Ⅱ equation, J. Math. Phys., 38 (1997), 6382–6400. https://doi.org/10.1063/1.532218 doi: 10.1063/1.532218
[40]	J. Zhu, S. Zhou, Z. Qiao, Forced $(2+1)$-dimensional discrete three-wave equation, Commun. Theor. Phys., 72 (2020), 015004. https://doi.org/10.1088/1572-9494/ab5fb4 doi: 10.1088/1572-9494/ab5fb4
[41]	P. V. Nabeleka, V. E. Zakharov, Solutions to the Kaup-Broer system and its $(2+1)$ dimensional integrable generalization via the dressing method, Physica D, 409 (2020), 132478. https://doi.org/10.1016/j.physd.2020.132478 doi: 10.1016/j.physd.2020.132478
[42]	J. Luo, E. Fan, $\overline{\partial}$-dressing method for the coupled Gerdjikov-Ivanov equation, Appl. Math. Lett., 110 (2020), 106589. https://doi.org/10.1016/j.aml.2020.106589 doi: 10.1016/j.aml.2020.106589
[43]	Z. Q. Li, S. F. Tian, A hierarchy of nonlocal nonlinear evolution equations and $\overline{\partial}$-dressing method, Appl. Math. Lett., 120 (2021), 107254. https://doi.org/10.1016/j.aml.2021.107254 doi: 10.1016/j.aml.2021.107254
[44]	X. Wang, J. Zhu, Z. Qiao, New solutions to the differential-difference KP equation, Appl. Math. Lett., 113 (2021), 106836. https://doi.org/10.1016/j.aml.2020.106836 doi: 10.1016/j.aml.2020.106836
[45]	M. J. Ablowitz, D. B. Yaacov, A. S. Fokas, On the inverse scattering transform for the Kadomtesev-Petviashvili equation, Stud. Appl. Math., 69 (1983), 135–143. https://doi.org/10.1002/sapm1983692135 doi: 10.1002/sapm1983692135

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)