Geodesic
Classification of solutions of the generalized mixed nonlinear Schrödinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 469-490 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct a generalized Darboux transformation for a generalized mixed nonlinear Schrödinger equation and consider a complete reduction classification of parameters and eigenfunctions of the two-, three-, and four-fold generalized Darboux transformations. According to these different types of reductions, several solutions of the generalized mixed nonlinear Schrödinger equation are found, including a soliton, a quasirational soliton, and a periodic soliton. Finally, a classification corresponding to these reductions is given.
Keywords: generalized mixed nonlinear Schrödinger equation, generalized Darboux transformation
Mots-clés : soliton solution. 
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Deqin Qiu; Yongshuai Zhang. Classification of solutions of the generalized mixed nonlinear Schrödinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 469-490. http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a5/

[1] V. B. Matveev, M. A. Salle, Darboux Transformation and Solitons, Springer, Berlin, 1991 | MR

[2] Soliton Theory and Its Applications, ed. C. Gu, Springer, Berlin, 1995 | DOI | MR

[3] C. Gu, H. Hu, Z. Zhou, Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry, Mathematical Physics Studies, 26, Springer, Dordrecht, 2005 | DOI | MR

[4] V. B. Matveev, “Darboux transformation and the explicit solutions of the Kadomtcev–Petviaschvily equation, depending on functional parameter”, Lett. Math. Phys., 3:3 (1979), 213–216 | DOI | MR

[5] V. B. Matveev, “Darboux transformation and the explicit solutions of differential-difference and difference-difference evolution equations I”, Lett. Math. Phys., 3:3 (1979), 217–222 | DOI | MR

[6] V. B. Matveev, M. A. Salle, “Differential-difference evolution equations. II (Darboux transformation for the Toda lattice)”, Lett. Math. Phys., 3:5 (1979), 425–429 | DOI | MR

[7] V. B. Matveev, “Some comments on the rational solutions of the Zakhrov–Schabat equations”, Lett. Math. Phys., 3:6 (1979), 503–512 | DOI | MR

[8] G. Darboux, “Sur une proposition relative aux équations linéaires”, C. R. Acad. Sci. (Paris), 94 (1882), 1456–1459

[9] M. M. Crum, “Associated Sturm–Liouville systems”, Quart. J. Math., 6:1 (1955), 121–127 | DOI | MR

[10] G. Neugerauer, R. Meinel, “General $N$-soliton solution of the AKNS arbitary background”, Phys. Lett. A, 100:9 (1984), 467–470 | DOI | MR

[11] Y. Li, X. Gu, M. Zou, “Three kinds of Darboux transformation for the evolution equation which connect with AKNS eigenvalue problem”, Acta Math. Sinica (N. S.), 3:2 (1985), 143–151 | DOI | MR

[12] C. Gu, Z. Zhou, “On the Darboux matrix of Bäcklund transformation of the AKNS system”, Lett. Math. Phys., 13:3 (1987), 179–187 | DOI | MR

[13] J. Cieśliński, “An effective method to compute $N$-fold Darboux matrix and $N$-soliton surfaces”, J. Math. Phys., 32:9 (1991), 2395–2399 | DOI | MR

[14] Q. P. Liu, “Darboux transformations for the supersymmetric Korteweg-de Vries equations”, Lett. Math. Phys., 35:2 (1995), 115–122, arXiv: hep-th/9409008 | DOI | MR

[15] H. Steudel, R. Meinel, G. Neugerauer, “Vandermonde-like determinants and $N$-fold Darboux/Bäcklund transformations”, J. Math. Phys., 38:9 (1997), 4692–4695 | DOI | MR

[16] W.-X. Ma, “Darboux transformation for a Lax integrable system in $2n$ dimensions”, Lett. Math. Phys., 39:1 (1997), 33–49, arXiv: solv-int/9605002 | DOI | MR

[17] K. Imai, “Generalization of the Kaup–Newell inverse scattering formulation and Darboux transformation”, J. Phys. Soc. Japan, 68:2 (1999), 355–359 | DOI | MR

[18] E. Fan, “Darboux transformation and soliton-like solutions for the Gerdjikov–Ivanov equation”, J. Phys. A: Math. Gen., 33:39 (2000), 6925–6933 | DOI | MR

[19] J. S. He, L. Zhang, Y. Cheng, Y. S. Li, “Determinant representation of Darboux transformation for the AKNS system”, Sci. China Ser. A, 49:12 (2006), 1867–1878 | MR

[20] V. B. Matveev, “Generalized Wronskian formula for solutions of the KdV equations: first applications”, Phys. Lett. A, 166:3–4 (1992), 205–208 | DOI | MR

[21] V. B. Matveev, “Positons: A new concept in the theory of nonlinear waves”, Nonlinear Coherent Structures in Physics and Biology, Proceedings of a NATO ARW (Bayreuth, Germany, June 1–4, 1993), NATO ASI Series, 329, eds. K. H. Spatschek, F. G. Mertens, Springer, Boston, MA, 1995, 259–262 | DOI

[22] A. A. Stahlhofen, V. B. Matveev, “Positons for the Toda lattice and related spectral problems”, J. Phys. A: Math. Gen., 28:7 (1995), 1957–1965 | DOI | MR

[23] V. B. Matveev, “Pozitony: medlenno ubyvayuschie analogi solitonov”, TMF, 131:1 (2002), 44–61 | DOI | DOI | MR | Zbl

[24] B. Guo, L. Ling, Q. P. Liu, “Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions”, Phys. Rev. E, 85:2 (2012), 026607, 9 pp., arXiv: 1108.2867 | DOI

[25] B. Guo, L. Ling, Q. P. Liu, “High-order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations”, Stud. Appl. Math., 130:4 (2013), 317–344 | DOI | MR

[26] L. Ling, B. Guo, L.-C. Zhao, “High-order rogue waves in vector nonlinear Schrödinger equations”, Phys. Rev. E, 89:4 (2014), 041201, 5 pp., arXiv: 1311.2720 | DOI

[27] L. Ling, L.-C. Zhao, B. Guo, “Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations”, Commun. Nonlinear Sci. Numer. Simul., 32 (2016), 285–304 | DOI | MR

[28] X.-Y. Wen, Y. Yang, Z. Yan, “Generalized perturbation $(n,M)$-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation”, Phys. Rev. E, 92:1 (2015), 012917, 19 pp., arXiv: 1704.02557 | DOI | MR

[29] A. Kundu, “Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations”, J. Math. Phys., 25:12 (1984), 3433–3438 | DOI | MR

[30] D. J. Kaup, A. C. Newell, “An exact solution for a derivative nonlinear Schrödinger equation”, J. Math. Phys., 19:4 (1978), 798–801 | DOI | MR

[31] H. H. Chen, Y. C. Lee, C. S. Liu, “Integrability of nonlinear Hamiltonian systems by inverse scattering method”, Phys. Scr., 20:3–4 (1979), 490–492 | DOI | MR

[32] V. S. Gerdjikov, M. I. Ivanov, “A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures”, Bulg. J. Phys., 10:2 (1983), 130–143 | MR

[33] A. S. Smirnov, “Spectral curves for the derivative nonlinear Schrödinger equations”, Symmetry, 13:7 (2021), 1203, 18 pp. | DOI

[34] S.-Z. Liu, Y.-S. Zhang, J.-S. He, “Smooth positons of the second-type derivative nonlinear Schrödinger equation”, Commun. Theor. Phys., 71:4 (2019), 357–361 | DOI | MR

[35] F. Calogero, W. Eckhaus, “Nonlinear evolution equations, rescalings, model PDEs and their integrability. I”, Inverse Problems, 3:2 (1987), 229–262 | DOI | MR

[36] Y. Xiang, X. Dai, S. Wen, J. Guo, D. Fan, “Controllable Raman soliton self-frequency shift in nonlinear metamaterials”, Phys. Rev. A, 84:3 (2011), 033815, 7 pp. | DOI

[37] A. Choudhuri, K. Porsezian, “Dark-in-the-Bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms”, Opt. Commun., 285:3 (2012), 364–367 | DOI

[38] R. S. Johnson, “On the modulation of water waves in the neighbourhood of $kh\approx 1.363$”, Proc. Roy. Soc. London Ser. A, 357:1689 (1977), 131–141 | DOI | MR

[39] P. A. Clarkson, C. M. Cosgrove, “Painlevé analysis of the non-linear Schrödinger family of equations”, J. Phys. A: Math. Gen., 20:8 (1987), 2003–2024 | DOI | MR

[40] X. Lü, M. Peng, “Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics”, Commun. Nonlinear Sci. Numer. Simul., 18:9 (2013), 2304–2312 | DOI | MR

[41] X. Lü, “Soliton behavior for a generalized mixed nonlinear Schrödinger model with $N$-fold Darboux transformation”, Chaos, 23:3 (2013), 033137, 8 pp. | DOI | MR

[42] D. Qiu, Q. P. Liu, “Darboux transformation of the generalized mixed nonlinear Schrödinger equation revisited”, Chaos, 30:12 (2020), 123111, 17 pp. | DOI | MR

[43] S. Kakei, N. Sasa, J. Satsuma, “Bilinearization of a generalized derivative nonlinear Schrödinger equation”, J. Phys. Soc. Japan, 64:5 (1995), 1519–1523 | DOI | MR

[44] B. Yang, J. Chen, J. Yang, “Rogue waves in the generalized derivative nonlinear Schrödinger equations”, J. Nonlinear Sci., 30:6 (2020), 3027–3056, arXiv: 1912.05589 | DOI | MR

[45] J. Chen, B.-F. Feng, “A note on the bilinearization of the generalized derivative nonlinear Schrödinger equation”, J. Phys. Soc. Japan, 90:2 (2021), 023001, 3 pp. | DOI

[46] L. Wang, D.-Y. Jiang, F.-H. Qi, Y.-Y. Shi, Y.-C. Zhao, “Dynamics of the higher-order rogue waves for a generalized mixed nonlinear Schrödinger model”, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 502–519 | DOI | MR

[47] S. Xu, J. He, L. Wang, “The Darboux transformation of the derivative nonlinear Schrödinger equation”, J. Phys. A: Math. Theor., 44:30 (2011), 305203, 22 pp. | DOI | MR

[48] Y. Zhang, L. Guo, J. He, Z. Zhou, “Darboux transformation of the second-type derivative nonlinear Schrödinger equation”, Lett. Math. Phys., 105:6 (2015), 853–891 | DOI | MR