BIFURCATIONS AND EXACT SOLUTIONS OF THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS DNLSI-DNLSIII: DYNAMICAL SYSTEM METHOD

Jinsen Zhuang; Yan Zhou; Jibin Li

doi:10.11948/20240390

2025 Volume 15 Issue 5

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Jinsen Zhuang, Yan Zhou, Jibin Li. BIFURCATIONS AND EXACT SOLUTIONS OF THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS DNLSI-DNLSIII: DYNAMICAL SYSTEM METHOD[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2637-2651. doi: 10.11948/20240390

Citation:

Jinsen Zhuang, Yan Zhou, Jibin Li. BIFURCATIONS AND EXACT SOLUTIONS OF THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS DNLSI-DNLSIII: DYNAMICAL SYSTEM METHOD[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2637-2651. doi: 10.11948/20240390

BIFURCATIONS AND EXACT SOLUTIONS OF THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATIONS DNLSI-DNLSIII: DYNAMICAL SYSTEM METHOD

1.
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China
2.
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Author Bio: Email: zzjinsen@hqu.edu.cn(J. Zhuang); Email: zy4233@hqu.edu.cn(Y. Zhou)
Corresponding author: Email: lijb@zjnu.cn(J. Li)
Fund Project: This research is partially supported by the National Natural Science Foundations of China (11871231, 12071162, 11701191), and the Natural Science Foundation of Fujian Province (2021J01303)

Abstract

Abstract

For the derivative nonlinear Schrödinger equations DNLSI-DNLSIII, by using the dynamical system method, we investigate the exact explicit solutions with the form $ q(x, t)=\phi(\xi)\exp{[i(\kappa x-\omega t+\theta(\xi))]}, \xi=x-ct. $ In the given parameter regions, we present exact explicit parametric representations for more than 14 solutions.
- Bifurcation /
- exact solution /
- Hamiltonian system /
- Kaup–Newell equation /
- Chen–Lee–Liu equation /
- Gerdjikov–Ivanov equation
MSC: 34C37, 34C23, 74J30, 58Z05

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References

[1]	K. Abhinav, P. Guha and I. Mukherjee, Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy, J. Math. Phys., 2018, 59, 101507. doi: 10.1063/1.5019268 CrossRef Google Scholar
[2]	T. Aydemir, New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family, Opt. Quant. Electron., 2024, 56(6). DOI: 10.1007/s11082-024-06822-9. CrossRef Google Scholar
[3]	H. M. Baskonus, M. Younis, M. Bilal, U. Younas, Shafqat-ur-Rehman and W. Gao, Modulation instability analysis and perturbed optical soliton and other solutions to the Gerdjikov-Ivanov equation in nonlinear optics, Modern Phys. Lett., 2020, 34(35), 2050404. doi: 10.1142/S0217984920504047 CrossRef Google Scholar
[4]	P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists, Springer, Berlin, 1971. Google Scholar
[5]	H. H. Chen, Y. C. Lee and C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scr., 1979, 20, 490. doi: 10.1088/0031-8949/20/3-4/026 CrossRef Google Scholar
[6]	M. Chen, Bifurcations and exact solutions for the Kundu equation: Dynamical approach, J. Appl. Anal. Comput., 2024, 14(6). DOI: 10.11948/JAAC-2024-0012. CrossRef Google Scholar
[7]	H. H. Dai and E. G. Fan, Variable separation and algebro-geometric solutions of the Gerdjikov–Ivanov equation, Chaos Soliton. Fract., 2004, 22, 93–101. Google Scholar
[8]	M. Dong, L. Tian, J. Wei and Y. Wang, Some localized wave solutions for the coupled Gerdjikov-Ivanov equation, Appl. Math. Lett., 2021, 122, 107483. doi: 10.1016/j.aml.2021.107483 CrossRef Google Scholar
[9]	E. G. Fan, Integrable systems of derivative nonlinear Schrödinger type and their multi-Hamiltonian structure, J. Phys. A Math. Gen., 2001, 34, 513–519. doi: 10.1088/0305-4470/34/3/313 CrossRef Google Scholar
[10]	E. G. Fan, Bi-Hamiltonian structure and Liouville integrablity for a Gerdjikov–Ivanov equation hierarchy, Chin. Phys. Lett., 2001, 18, 1–3. doi: 10.1088/0256-307X/18/1/301 CrossRef Google Scholar
[11]	N. G. Fang, Darboux transformation and soliton-like solutions for the Gerdjikov–Ivanov equation, J. Phys. A: Math. Gen., 2000, 33, 6925–6933. doi: 10.1088/0305-4470/33/39/308 CrossRef Google Scholar
[12]	S. Fromm, Admissible boundary values for the Gerdjikov-Ivanov equation with asymptotically time-periodic boundary data, SIGMA Symmetry Integrability Geom. Methods Appl., 2020, 16, 79. Google Scholar
[13]	V. S. Gerdjikov and M. I. Ivanov, A quadratic pencil of general type and nonlinear evolution equations. Ⅱ. Hierarchies of Hamiltonian structures, Bulg. J. Phys., 1983, 10(1), 13. Google Scholar
[14]	S. C. A. Gómez, A. Jhangeer, H. Rezazadeh, R. A. Talarposhti and A. Bekir, Closed form solutions of the perturbed Gerdjikov-Ivanov equation with variable coefficients, East Asian J. Appl. Math., 2021, 11(1), 207–218. doi: 10.4208/eajam.230620.070920 CrossRef Google Scholar
[15]	H. Indexed, M. Saleh and A. A. Altwaty, Optical solitons of the extended Gerdjikov-Ivanov equation in DWDM system by extended simplest equation method, Appl. Math. Inf. Sci., 2020, 14(5), 901–907. doi: 10.18576/amis/140517 CrossRef Google Scholar
[16]	A. Kalbani, K. Kaltham, K. S. Al-Ghafri, E. V. Krishnan and A. Biswas, Solitons and modulation instability of the perturbed Gerdjikov-Ivanov equation with spatio-temporal dispersion, Chaos Soliton. Fract., 2021, 153, 111523. doi: 10.1016/j.chaos.2021.111523 CrossRef Google Scholar
[17]	D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 1978, 19(4), 798. Google Scholar
[18]	M. M. A. Khater, Abundant wave solutions of the perturbed Gerdjikov-Ivanov equation in telecommunication industry, Modern Phys. Lett., 2021, 35(26), 2150456. doi: 10.1142/S021798492150456X CrossRef Google Scholar
[19]	S. A. Khuri, New optical solitons and traveling wave solutions for the Gerdjikov-Ivanov equation, Optik, 2022, 268(39), 169784. Google Scholar
[20]	J. Li, Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. Google Scholar
[21]	M. Li, Y. Zhang, R. Ye and Y. Lou, Exact solutions of the nonlocal Gerdjikov-Ivanov equation, Commun. Theor. Phys., 2021, 73, 10. Google Scholar
[22]	Y. Li, L. Zhang, B. Hu and R. Wang, The initial-boundary value for the combined Schrödinger and Gerdjikov–Ivanov equation on the half-line via the Riemann–Hilbert approach, Teoret. Mat. Fiz., 2021, 209(2), 258–273. doi: 10.4213/tmf10141 CrossRef Google Scholar
[23]	Y. Lou, Y. Zhang, R. Ye and M. Li, Modulation instability, higher-order rogue waves and dynamics of the Gerdjikov-Ivanov equation, Wave Motion, 2021, 106, 102795. Google Scholar
[24]	P. Lu, Y. Wang and C. Dai, Abundant fractional soliton solutions of a space-time fractional perturbed Gerdjikov-Ivanov equation by a fractional mapping method, Chinese J. Phys., 2021, 74, 96–105. Google Scholar
[25]	J. Luo and E. Fan, Dbar-dressing method for the Gerdjikov-Ivanov equation with nonzero boundary conditions, Appl. Math. Lett., 2021, 120, 107297. Google Scholar
[26]	J. Luo and E. Fan, $\varrho$-dressing method for the coupled Gerdjikov-Ivanov equation, Appl. Math. Lett., 2020, 110, 106589. $\varrho$-dressing method for the coupled Gerdjikov-Ivanov equation" target="_blank">Google Scholar
[27]	S. T. R. Rizvi and S. Shabbir, Optical soliton solution via complete discrimination system approach along with bifurcation and sensitivity analyses for the Gerjikov-Ivanov equation, Optik, 2023, 294(5), 171456. Google Scholar
[28]	T. Tsuchida, Integrable discretizations of derivative nonlinear Schrödinger equations, J. Phys. A Math. Gen. 2002, 35, 7827–7847. Google Scholar
[29]	Vinita and S. Saha Ray, Lie symmetry reductions, power series solutions and conservation laws of the coupled Gerdjikov-Ivanov equation using optimal system of Lie subalgebra, Z. Angew. Math. Phys., 2021, 72(4), 133. Google Scholar
[30]	X. Xiao and Z. Yin, Exact single travelling wave solutions to the fractional perturbed Gerdjikov-Ivanov equation in nonlinear optics, Modern Phys. Lett. 2021, 35, 22. Google Scholar
[31]	S. Zhang, T. Xu, M. Li and X. Zhang, Higher-order algebraic soliton solutions of the Gerdjikov-Ivanov equation: Asymptotic analysis and emergence of rogue waves, Phys. D, 2022, 432, 133128. Google Scholar
[32]	Z. Zhang and E. Fan, Inverse scattering transform and multiple high-order pole solutions for the Gerdjikov-Ivanov equation under the zero/nonzero background, Z. Angew. Math. Phys., 2021, 72(4), 153. Google Scholar
[33]	Z. Zhang and E. Fan, Inverse scattering transform for the Gerdjikov-Ivanov equation with nonzero boundary conditions, Z. Angew. Math. Phys., 2020, 71(5), 149. Google Scholar
[34]	Y. Zhou and J. Li, Bifurcations of traveling wave solutions in the homogeneous Camassa-Holm type equations, J. Appl. Anal. Comput., 2022, 12(1), 392–406. Google Scholar
[35]	J. Zhu and Y. Chen, A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup-Newell equation, J. Math. Phys., 2021, 62, 12. Google Scholar
[36]	J. Zhuang and Y. Zhou, Bifurcations of solitary waves, periodic peakons and compactons of a coupled nonlinear wave equation, J. Appl. Anal. Comput., 2022, 12(5), 1–7. Google Scholar