Classical Darboux transformation and exact soliton solutions of a two-component complex short pulse equation

Qiulan Zhao; Muhammad Arham Amin; Xinyue Li; Qiulan Zhao; Muhammad Arham Amin; Xinyue Li

doi:10.3934/math.2023442

AIMS Mathematics

2023, Volume 8, Issue 4: 8811-8828. doi: 10.3934/math.2023442

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Research article

Classical Darboux transformation and exact soliton solutions of a two-component complex short pulse equation

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China

Received: 18 November 2022 Revised: 25 January 2023 Accepted: 29 January 2023 Published: 08 February 2023
MSC : 35Q35, 37K10, 37K40

This paper investigates soliton solutions to a two-component complex short pulse (c-SP) equation. Based on the known Lax pair representation of this equation, we verify the integrability of a two-component c-SP equation and find an equivalent convenient Lax pair through hodograph transformation. The classical Darboux transformation (DT) is utilized to construct multi-soliton solutions for the two-component c-SP equation as an ordinary determinant. Furthermore, the details of one-soliton and two-soliton solutions are presented and generalized for $ N $-fold soliton solutions. We also derive exact soliton solutions in explicit form using suitable reduction constraints from various "seed" solutions and explore them via graphs.
- integrability,
- hodograph transformation,
- Darboux transformation,
- exact soliton solutions
Citation: Qiulan Zhao, Muhammad Arham Amin, Xinyue Li. Classical Darboux transformation and exact soliton solutions of a two-component complex short pulse equation[J]. AIMS Mathematics, 2023, 8(4): 8811-8828. doi: 10.3934/math.2023442

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Abstract

This paper investigates soliton solutions to a two-component complex short pulse (c-SP) equation. Based on the known Lax pair representation of this equation, we verify the integrability of a two-component c-SP equation and find an equivalent convenient Lax pair through hodograph transformation. The classical Darboux transformation (DT) is utilized to construct multi-soliton solutions for the two-component c-SP equation as an ordinary determinant. Furthermore, the details of one-soliton and two-soliton solutions are presented and generalized for $ N $-fold soliton solutions. We also derive exact soliton solutions in explicit form using suitable reduction constraints from various "seed" solutions and explore them via graphs.

References

[1]	Y. Song, X. Shi, C. Wu, D. Tang, H. Zhang, Recent progress of study on optical solitons in fiber lasers, Appl. Phys. Rev., 6 (2019), 021313. http://dx.doi.org/10.1063/1.5091811 doi: 10.1063/1.5091811
[2]	K. Rajitha, C. Mishra, T. Dey, P. Panigrahi, Phase-controlled stable solitons in nonlinear fibers, J. Opt. Soc. Am. B, 36 (2019), 1–6. http://dx.doi.org/10.1364/JOSAB.36.000001 doi: 10.1364/JOSAB.36.000001
[3]	G. Agrawal, Nonlinear fiber optics, In: Nonlinear science at the dawn of the 21st century, Berlin: Springer, 2000. http://dx.doi.org/10.1007/3-540-46629-0_9
[4]	Y. Chung, C. Jones, T. Schäfer, C. Wayne, Ultra-short pulses in linear and nonlinear media, Nonlinearity, 18 (2005), 1351. http://dx.doi.org/10.1088/0951-7715/18/3/021 doi: 10.1088/0951-7715/18/3/021
[5]	S. Shen, Z. Yang, Z. Pang, Y. Ge, The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics, Appl. Math. Lett., 125 (2022), 107755. http://dx.doi.org/10.1016/j.aml.2021.107755 doi: 10.1016/j.aml.2021.107755
[6]	J. Guo, Z. Yang, L. Song, Z. Pang, Propagation dynamics of tripole breathers in nonlocal nonlinear media, Nonlinear Dyn., 101 (2020), 1147–1157. http://dx.doi.org/10.1007/s11071-020-05829-7 doi: 10.1007/s11071-020-05829-7
[7]	S. Shen, Z. Yang, X. Li, S. Zhang, Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media, Commun. Nonlinear Sci., 103 (2021), 106005. http://dx.doi.org/10.1016/j.cnsns.2021.106005 doi: 10.1016/j.cnsns.2021.106005
[8]	V. Zakharov, A. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Journal of Experimental and Theoretical Physics, 34 (1972), 118–134.
[9]	B. Feng, Complex short pulse and coupled complex short pulse equations, Physica D, 297 (2015), 62–75. http://dx.doi.org/10.1016/j.physd.2014.12.002 doi: 10.1016/j.physd.2014.12.002
[10]	T. Schäfer, C. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D, 196 (2004), 90–105. http://dx.doi.org/10.1016/j.physd.2004.04.007 doi: 10.1016/j.physd.2004.04.007
[11]	J. Brunelli, The bi-hamiltonian structure of the short pulse equation, Phys. Lett. A, 353 (2006), 475–478. http://dx.doi.org/10.1016/j.physleta.2006.01.009 doi: 10.1016/j.physleta.2006.01.009
[12]	A. Sakovich, S. Sakovich, Solitary wave solutions of the short pulse equation, J. Phys. A: Math. Gen., 39 (2006), 361. http://dx.doi.org/10.1088/0305-4470/39/22/L03 doi: 10.1088/0305-4470/39/22/L03
[13]	M. Rabelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math., 81 (1989), 221–248. http://dx.doi.org/10.1002/sapm1989813221 doi: 10.1002/sapm1989813221
[14]	A. Sakovich, S. Sakovich, The short pulse equation is integrable, J. Phys. Soc. Jpn., 74 (2005), 239–241. http://dx.doi.org/10.1143/jpsj.74.239 doi: 10.1143/jpsj.74.239
[15]	Y. Matsuno, Multiloop soliton and multibreather solutions of the short pulse model equation, J. Phys. Soc. Jpn., 76 (2007), 084003. http://dx.doi.org/10.1143/JPSJ.76.084003 doi: 10.1143/JPSJ.76.084003
[16]	Q. Zhang, Y. Xia, Discontinuous Galerkin methods for short pulse type equations via hodograph transformations, J. Comput. Phys., 399 (2019), 108928. http://dx.doi.org/10.1016/j.jcp.2019.108928 doi: 10.1016/j.jcp.2019.108928
[17]	R. Hirota, The direct method in soliton theory, Cambridge: Cambridge University Press, 2004. http://dx.doi.org/10.1017/cbo9780511543043
[18]	Z. Li, S. Tian, J. Yang, On the soliton resolution and the asymptotic stability of N-soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions, Adv. Math., 409 (2022), 108639. http://dx.doi.org/10.1016/j.aim.2022.108639 doi: 10.1016/j.aim.2022.108639
[19]	Z. Li, S. Tian, J. Yang, Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data, Ann. Henri Poincaré, 23 (2022), 2611–2655. http://dx.doi.org/10.1007/s00023-021-01143-z doi: 10.1007/s00023-021-01143-z
[20]	B. Feng, L. Ling, Darboux transformation and solitonic solution to the coupled complex short pulse equation, Physica D, 437 (2022), 133332. http://dx.doi.org/10.1016/j.physd.2022.133332 doi: 10.1016/j.physd.2022.133332
[21]	H. Sun, Z. Zhu, Darboux transformation and soliton solutions of the spatial discrete coupled complex short pulse equation, Physica D, 436 (2022), 133312. http://dx.doi.org/10.1016/j.physd.2022.133312 doi: 10.1016/j.physd.2022.133312
[22]	Z. Li, S. Tian, J. Yang, E. Fan, Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions, J. Differ. Equations, 329 (2022), 31–88. http://dx.doi.org/10.1016/j.jde.2022.05.003 doi: 10.1016/j.jde.2022.05.003
[23]	H. Wang, X. Wen, Soliton elastic interactions and dynamical analysis of a reduced integrable nonlinear schrödinger system on a triangular-lattice ribbon, Nonlinear Dyn., 100 (2020), 1571–1587. http://dx.doi.org/10.1007/s11071-020-05587-6 doi: 10.1007/s11071-020-05587-6
[24]	L. Ling, L. Zhao, B. Guo, Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 3243. http://dx.doi.org/10.1088/0951-7715/28/9/3243 doi: 10.1088/0951-7715/28/9/3243
[25]	V. Matveev, M. Salle, Darboux transformations and solitons, Berlin: Springer-Verlag, 1991.
[26]	Z. Lin, X. Wen, Dynamical analysis of position-controllable loop rogue wave and mixed interaction phenomena for the complex short pulse equation in optical fiber, Nonlinear Dyn., 108 (2022), 2573–2593. http://dx.doi.org/10.1007/s11071-022-07315-8 doi: 10.1007/s11071-022-07315-8
[27]	M. Manas, L. Alonso, A hodograph transformation which applies to the heavenly equation, arXiv: nlin/0209050.
[28]	Q. Zhao, M. Amin, Explicit solutions of rational integrable differential-difference equations, Partial Differential Equations in Applied Mathematics, 5 (2022), 100338. http://dx.doi.org/10.1016/j.padiff.2022.100338 doi: 10.1016/j.padiff.2022.100338
[29]	Y. Yang, E. Fan, Soliton resolution for the short-pulse equation, J. Differ. Equations, 280 (2021), 644–689. http://dx.doi.org/10.1016/j.jde.2021.01.036 doi: 10.1016/j.jde.2021.01.036
[30]	U. Saleem, M. ul Hassan, Darboux transformation and multisoliton solutions of the short pulse equation, J. Phys. Soc. Jpn., 81 (2012), 094008. http://dx.doi.org/10.1143/JPSJ.81.094008 doi: 10.1143/JPSJ.81.094008
[31]	J. He, L. Zhang, Y. Cheng, Y. Li, Determinant representation of darboux transformation for the AKNS system, Sci. China Ser. A, 49 (2006), 1867–1878. http://dx.doi.org/10.1007/s11425-006-2025-1 doi: 10.1007/s11425-006-2025-1
[32]	H. Sarfraz, U. Saleem, Y. Hanif, Loop dynamics of a fully discrete short pulse equation, arXiv: 2209.00738.

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