Geodesic
Motion of dispersive shock edges in nonlinear pulse evolution
Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 3, pp. 415-424 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We formulate a method for calculating the velocities of the edges of dispersive shock waves that are generated after wave breaking of pulses during their propagation in a nonlinear medium. The method is based on the properties of the Whitham modulation system at its degenerate limits obtained for either a vanishing amplitude of oscillations at one edge or a vanishing wave number at the other edge.
Keywords: nonlinear wave, dispersive shock wave, Whitham theory.
Mots-clés : soliton 
@article{TMF_2020_202_3_a7,
     author = {A. M. Kamchatnov},
     title = {Motion of dispersive shock edges in nonlinear pulse evolution},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {415--424},
     year = {2020},
     volume = {202},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a7/}
}
TY  - JOUR
AU  - A. M. Kamchatnov
TI  - Motion of dispersive shock edges in nonlinear pulse evolution
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2020
SP  - 415
EP  - 424
VL  - 202
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a7/
LA  - ru
ID  - TMF_2020_202_3_a7
ER  - 
%0 Journal Article
%A A. M. Kamchatnov
%T Motion of dispersive shock edges in nonlinear pulse evolution
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2020
%P 415-424
%V 202
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a7/
%G ru
%F TMF_2020_202_3_a7
A. M. Kamchatnov. Motion of dispersive shock edges in nonlinear pulse evolution. Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 3, pp. 415-424. http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a7/

[1] R. Z. Sagdeev, “Kollektivnye protsessy i udarnye volny v razrezhennoi plazme”, Voprosy teorii plazmy, 4, ed. M. A. Leontovich, Atomizdat, M., 1964, 20–80

[2] G. B. Whitham, “Non-linear dispersive waves”, Proc. Roy. Soc. London Ser. A, 283:1393 (1965), 238–261 | DOI | MR

[3] Dzh. Uizem, Lineinye i nelineinye volny, Mir, M., 1977

[4] A. V. Gurevich, L. P. Pitaevskii, “Nestatsionarnaya struktura besstolknovitelnoi udarnoi volny”, ZhETF, 65:2 (1973), 590–604

[5] G. A. El, M. A. Hoefer, “Dispersive shock waves and modulation theory”, Phys. D, 333 (2016), 11–65 | DOI | MR

[6] A. V. Gurevich, A. R. Mescherkin, “Rasshiryayuschiesya avtomodelnye razryvy i udarnye volny v dispersionnoi gidrodinamike”, ZhETF, 87:4 (1984), 1277–1292

[7] G. A. El, “Resolution of a shock in hyperbolic systems modified by weak dispersion”, Chaos, 15:3 (2005), 037103, 21 pp., arXiv: nlin/0503010 | DOI | MR

[8] G. A. El, R. H. J. Grimshaw, N. F. Smyth, “Unsteady undular bores in fully nonlinear shallow-water theory”, Phys. Fluids, 18:2 (2006), 027104, 17 pp. | DOI | MR

[9] G. A. El, A. Gammal, E. G. Khamis, R. A. Kraenkel, A. M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media”, Phys. Rev. A, 76:5 (2007), 053813, 18 pp., arXiv: 0706.1112 | DOI

[10] G. A. El, R. H. J. Grimshaw, N. F. Smyth, “Transcritical shallow-water flow past topography: finite-amplitude theory”, J. Fluid Mech., 640 (2009), 187–214 | DOI | MR

[11] J. G. Esler, J. D. Pearce, “Dispersive dam-break and lock-exchange flows in a two-layer fluid”, J. Fluid Mech., 667 (2011), 555–585 | DOI | MR

[12] M. A. Hoefer, “Shock waves in dispersive Eulerian fluids”, J. Nonlinear Sci., 24:3 (2014), 525–577, arXiv: 1303.2541 | DOI | MR

[13] T. Congy, A. M. Kamchatnov, N. Pavloff, “Dispersive hydrodynamics of nonlinear polarization waves in two-component Bose–Einstein condensates”, SciPost Phys., 1:1 (2016), 006, 30 pp., arXiv: 1607.08760 | DOI

[14] M. A. Hoefer, G. A. El, A. M. Kamchatnov, “Oblique spatial dispersive shock waves in nonlinear Schrödinger flows”, SIAM J. Appl. Math., 77:4 (2017), 1352–1374 | DOI | MR

[15] X. An, T. R. Marchant, N. F. Smyth, “Dispersive shock waves governed by the Whitham equation and their stability”, Proc. Roy. Soc. London Ser. A, 474:2216 (2018), 20180278, 18 pp. | DOI | MR

[16] A. M. Kamchatnov, “Dispersive shock wave theory for nonintegrable equations”, Phys. Rev. E, 99:1 (2019), 012203, 18 pp., arXiv: 1809.08553 | DOI

[17] S. K. Ivanov, A. M. Kamchatnov, “Evolution of wave pulses in fully nonlinear shallow-water theory”, Phys. Fluids, 31:5 (2019), 057102, arXiv: 1903.01667 | DOI

[18] L. D. Landau, E. M. Lifshits, Kurs teoreticheskoi fiziki, v. VI, Gidrodinamika, Fizmatlit, M., 2001 | MR | Zbl

[19] O. Akimoto, K. Ikeda, “Steady propagation of a coherent light pulse in a dielectric medium. I”, J. Phys. A: Math. Gen., 10:3 (1977), 425–440 ; K. Ikeda, O. Akimoto, “Steady propagation of a coherent light pulse in a dielectric medium. II. The effect of spatial dispersion”, 12:7 (1979), 1105–1120 | DOI | DOI

[20] S. A. Darmanyan, A. M. Kamchatnov, M. Nevière, “Polariton effect in nonlinear pulse propagation”, ZhETF, 123:5 (2003), 997–1005 | DOI

[21] A. V. Gurevich, A. L. Krylov, N. G. Mazur, “Kvaziprostye volny v gidrodinamike Kortevega–de Vriza”, ZhETF, 95:5 (1989), 1674–1698

[22] V. I. Karpman, “Some asymptotic relations for solutions of the Korteweg–De Vries equation”, Phys. Lett. A, 26:12 (1968), 619–620 | DOI

[23] A. M. Kamchatnov, R. A. Kraenkel, B. A. Umarov, “On asymptotic solutions of integrable wave equations”, Phys. Lett. A, 287:3–4 (2001), 223–232 | DOI | MR

[24] A. M. Kamchatnov, R. A. Kraenkel, B. A. Umarov, “Asymptotic soliton train solutions of the defocusing nonlinear Schrödinger equation”, Phys. Rev. E, 66:3 (2002), 036609, 10 pp. | DOI | MR

[25] G. A. El, R. H. J. Grimshaw, N. F. Smyth, “Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory”, Phys. D, 237:19 (2008), 2423–2435 | DOI | MR