Dynamics of three nonisospectral nonlinear Schrödinger equations

doi:10.1088/1674-1056/28/2/020202

Abstract

Dynamics of three nonisospectral nonlinear Schrödinger equations (NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrödinger equation (NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover, we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes, velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.

Keywords: nonisospectral nonlinear Schrö dinger equations, gauge transformations, bilinear forms, solitons, rogue waves

Received: 26 November 2018
Revised: 18 December 2018
Accepted manuscript online:

PACS:	02.30.Ik	(Integrable systems)
	02.30.Ks	(Delay and functional equations)
	05.45.Yv	(Solitons)

Fund:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11601312, 11631007, and 11875040).

Corresponding Authors: Da-Jun Zhang
E-mail: djzhang@staff.shu.edu.cn

Cite this article:

Abdselam Silem, Cheng Zhang(张成), Da-Jun Zhang(张大军) Dynamics of three nonisospectral nonlinear Schrödinger equations 2019 Chin. Phys. B 28 020202

[1]	Zakharov V E and Shabat A B 1972 Sov. Phys. JETP 34 62
[2]	Ablowitz M J, Kaup D J, Newell A C and Segur H 1973 Phys. Rev. Lett. 31 125
[3]	Hirota R 1973 J. Math. Phys. 14 805
[4]	Matveev V B and Salle M A 1991 Darboux Transformations and Solitons (Berlin: Springer)
[5]	Satsuma J 1979 J. Phys. Soc. Jpn. 46 359
[6]	Nimmo J J C 1983 Phys. Lett. A 99 279
[7]	Zhang D J and Hietarinta J 2005 “Generalized double-Wronskian solutions to the nonlinear Schrödinger equation”, preprint
[8]	Benney D J and Newell A C 1967 J. Math. Phys. 46 133
[9]	Zakharov V E 1968 J. Appl. Mech. Tech. Phys. 9 190
[10]	Hasimoto H 1972 J. Fluid Mech. 51 477
[11]	Hasegawa A and Tappert F 1973 Appl. Phys. Lett. 23 142
[12]	Hasegawa A and Tappert F 1973 Appl. Phys. Lett. 23 171
[13]	Zakharov V E 1972 Sov. Phys. JETP 35 908
[14]	Manakov S V 1973 Zh. Eksp. Tecr. Fiz. 38 505
[15]	Peregrine D H 1983 J. Austral. Math. Soc. Ser. B 25 16
[16]	Eleonskii V M, Krichever I M and Kulagin N E 1986 Sov. Phys. Dokl. 31 226
[17]	Akhmediev N, Ankiewicz A and Soto-Crespo J M 2009 Phys. Rev. E 80 026601
[18]	Akhmediev N and Pelinovsky E 2010 Eur. Phys. J. Special Topics 185 1
[19]	Onorato M, Residori S, Bortolozzo U, Montina A and Arecchi F T 2013 Phys. Rep. 528 47
[20]	Chen H H and Liu C S 1976 Phys. Rev. Lett. 37 693
[21]	Gupta M R 1979 Phys. Lett. A 72 420
[22]	Calogero F and Degasperis A 1978 Commun. Math. Phys. 63 155
[23]	Hirota R and Satsuna J 1976 J. Phys. Soc. Jpn. 41 2141
[24]	Li Y S 1982 Sci. Sin. Ser. A 25 911
[25]	Ning T K, Chen D Y and Zhang D J 2004 Physica A 339 248
[26]	Ning T K, Zhang W G and Jia G 2009 Chaos, Solitons and Fractals 42 1100
[27]	Calogero F and Degasperis A 1982 The spectral transform and solitons (North Holland)
[28]	Chen D Y and Zhang D J 1996 J. Math. Phys. 37 5524
[29]	Burtsev S P, Zakharov V E and Mikhailov A V 1987 Theor. Math. Phys. 70 227
[30]	Ma W X 1992 J. Math. Phys. 33 2464
[31]	Wadati M and Sogo K 1983 J. Phys. Soc. Jpn. 52 394
[32]	Wadati M, Sanuki H and Konno K 1975 Prog. Theor. Phys. 53 419
[33]	Wadati M, Sanuki H and Konno K 1975 Prog. Theor. Phys. 53 1653
[34]	Zhang D J and Chen D Y 2004 J. Phys. A: Math. Gen. 37 851
[35]	Zhang Y, Deng S F, Zhang D J and Chen D Y 2004 Physica A 339 228
[36]	Sun Y P, Bi J B and Chen D Y 2005 Chaos, Solitons and Fractals 26 905
[37]	Hirota R 2004 The Direct Method in Soliton Theory (Cambridge: Cambridge University Press)
[38]	Liu Q M 1990 J. Phys. Soc. Jpn. 59 3520
[39]	Zhang D J 2006 “Notes on solutions in Wronskian form to soliton equations: Korteweg-de Vries-type”, arXiv: nlin.SI/0603008
[40]	Zhang D J, Zhao S L, Sun Y Y and Zhou J 2014 Rev. Math. Phys. 26 14300064
[41]	Guo B L, Ling L M and Liu Q P 2012 Phys. Rev. E 85 026607
[42]	Ohta Y and Yang J 2012 Proc. R. Soc. A 468 1716
[43]	Tao Y S, He J S and Porsezian K 2013 Chin. Phys. B 22 074210
[44]	Solli D R, Ropers C, Koonath P and Jalali B 2007 Nature 450 1054
[45]	Qian C, Rao J G, Liu Y B and He J S 2016 Chin. Phys. Lett. 33 110201
[46]	Liu Y K and Li B 2017 Chin. Phys. Lett. 34 010202
[47]	Akhmediev N, Ankiewicz A and Taki M 2009 Phys. Lett. A 373 675
[48]	Kou X 2011 “Exact solutions of two soliton equations”, Master Thesis, Shanghai University
[49]	Zhu X M, Zhang D J and Chen D Y 2011 Commun. Theor. Phys. 55 13

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