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Pelinovsky, Dmitry E.; White, Robert E.

Localized structures on librational and rotational travelling waves in the sine-Gordon equation. (English) Zbl 1472.35010

Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2242, Article ID 20200490, 18 p. (2020).
Summary: We derive exact solutions to the sine-Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) that decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution represents a kink propagating on the periodic background and decaying algebraically in the transverse direction to its propagation. These solutions model the universal patterns in the dynamics of fluxon condensates in the semi-classical limit. The different dynamics are related to modulational instability of the librational waves and modulational stability of the rotational waves.

Cited in 13 Documents

MSC:

35A18 Wave front sets in context of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)

Keywords:

sine-Gordon equation; librational and rotational waves; rogue waves; kinks and antikinks; Darboux transformation; differential equations; mathematical physics; wave motion
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References:

[1] Buckingham RJ, Miller PD. 2012 The sine-Gordon equation in the semi-classical limit: critical behavior near a separatrix. J. d’Analyse Math. 118, 397-492. (doi:10.1007/s11854-012-0041-3) · Zbl 1307.35255 · doi:10.1007/s11854-012-0041-3
[2] Buckingham RJ, Miller PD. 2013 The sine-Gordon equation in the semi-classical limit: dynamics of fluxon condensates. Mem. AMS 225, 1-136. (doi:10.1090/S0065-9266-2012-00672-1) · Zbl 1312.35147 · doi:10.1090/S0065-9266-2012-00672-1
[3] Lu BY, Miller PD. 2019 Universality near the gradient catastrophe point in the semi-classical sine-Gordon equation. (http://arxiv.org/abs/1912.09037)
[4] Scott AC, Chu FYF, Reible SA. 1976 Magnetic-flux propagation on a Josephson transmission line. J. Appl. Phys. 47, 3272-3286. (doi:10.1063/1.323126) · doi:10.1063/1.323126
[5] Braun OM, Kivshar Yu.. 2004 The Frenkel-Kontorova model: concepts, methods, and applications. New York, NY: Springer. · Zbl 1140.82001
[6] Cuevas-Maraver J, Kevrekidis PG, Williams F (eds). 2014 The Sine-Gordon model and its applications. In Nonlinear systems and complexity, vol. 10. New York, NY: Springer. · Zbl 1341.37046
[7] Jones CKRT, Marangell R, Miller PD, Plaza RG. 2013 On the stability analysis of periodic Sine-Gordon traveling waves. Physica D 251, 63-74. (doi:10.1016/j.physd.2013.02.003) · Zbl 1278.35046 · doi:10.1016/j.physd.2013.02.003
[8] Jones CKRT, Marangell R, Miller PD, Plaza RG. 2014 Spectral and modulational stability of periodic wavetrains for the nonlinear Klein-Gordon equation. J. Differ. Equ. 257, 4632-4703. (doi:10.1016/j.jde.2014.09.004) · Zbl 1304.35079 · doi:10.1016/j.jde.2014.09.004
[9] Marangell R, Miller PD. 2015 Dynamical Hamiltonian-Hopf instabilities of periodic traveling waves in Klein-Gordon equations. Physica D 308, 87-93. (doi:10.1016/j.physd.2015.06.006) · Zbl 1364.35299 · doi:10.1016/j.physd.2015.06.006
[10] Deconinck B, McGill P, Segal and BL. 2017 The stability spectrum for elliptic solutions to the sine-Gordon equation. Physica D 360, 17-35. (doi:10.1016/j.physd.2017.08.010) · Zbl 1378.35028 · doi:10.1016/j.physd.2017.08.010
[11] Upsal J, Deconinck B. 2020 Real Lax spectrum implies spectral stability. Stud. Appl. Math. 145, 765-790. (doi:10.1111/sapm.12335) · Zbl 1457.37094 · doi:10.1111/sapm.12335
[12] Clarke WA, Marangell R. 2020 A new Evans function for quasi-periodic solutions of the linearized sine-Gordon equation. (http://arxiv.org/abs/2005.08511) · Zbl 1462.35025
[13] Chen J, Pelinovsky DE. 2018 Rogue periodic waves in the modified Korteweg-de Vries equation. Nonlinearity 31, 1955-1980. (doi:10.1088/1361-6544/aaa2da) · Zbl 1393.35201 · doi:10.1088/1361-6544/aaa2da
[14] Chen J, Pelinovsky DE. 2019 Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background. J. Nonlin. Sci. 29, 2797-2843. (doi:10.1007/s00332-019-09559-y) · Zbl 1427.35226 · doi:10.1007/s00332-019-09559-y
[15] Chen J, Pelinovsky DE. 2018 Rogue periodic waves in the focusing nonlinear Schrödinger equation. Proc. R. Soc. A 474, 20170814. (doi:10.1098/rspa.2017.0814) · Zbl 1402.35256 · doi:10.1098/rspa.2017.0814
[16] Chen J, Pelinovsky DE, White RE. 2019 Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation. Phys. Rev. E 100, 052219. (doi:10.1103/PhysRevE.100.052219) · doi:10.1103/PhysRevE.100.052219
[17] Chen J, Pelinovsky DE, White RE. 2020 Periodic standing waves in the focusing nonlinear Schrödinger equation: rogue waves and modulation instability. Physica D 405, 132378. (doi:10.1016/j.physd.2020.132378) · Zbl 1490.35399 · doi:10.1016/j.physd.2020.132378
[18] Cao CW, Geng XG. 1990 Classical integrable systems generated through nonlinearization of eigenvalue problems. In Nonlinear physics (Shanghai, 1989), pp. 68-78. Research Reports in Physics. Berlin, Germany: Springer. · Zbl 0728.70012
[19] Feng BF, Ling L, Takahashi DA. 2020 Multi-breathers and high order rogue waves for the nonlinear Schrödinger equation on the elliptic function background. Stud. Appl. Math. 144, 46-101. (doi:10.1111/sapm.12287) · Zbl 1454.35338 · doi:10.1111/sapm.12287
[20] Li R, Geng X. 2020 Rogue periodic waves of the sine-Gordon equation. Appl. Math. Lett. 102, 106147. (doi:10.1016/j.aml.2019.106147) · Zbl 1440.35030 · doi:10.1016/j.aml.2019.106147
[21] Ablowitz MJ, Kaup DJ, Newell AC, Segur H. 1974 The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249-315. (doi:10.1002/sapm1974534249) · Zbl 0408.35068 · doi:10.1002/sapm1974534249
[22] Hou C, Bu L, Baronio F, Mihalache D, Chen S. 2020 Sine-Gordon breathers and formation of extreme waves in self-induced transparency media. Rom. Rep. Phys. 72, 405.
[23] White RE. 2020 Rogue wave potentials occurring in the sine-Gordon equation. MSc thesis, McMaster University, Hamilton, Ontario.
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