Nonlinear waves in a parabolic equation with a spatial argument rescaling operator and with time delay

E. P. Kubyshkin ; V. A. Kulikov

Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 2, pp. 298-326

Résumé

Bifurcations of nonlinear waves (spatially inhomogeneous solutions) from homogeneous equilibrium states of an initial boundary value problem in a circle for a nonlinear parabolic equation with a spatial argument stretching operator and a time delay arising in nonlinear optics are studied. In the plane of the basic parameters of the equation, the regions of stability (instability) of homogeneous equilibrium states are constructed, the dynamics of stability regions depending on the magnitude of the delay is studied. The mechanisms of loss of stability by homogeneous equilibrium states, possible bifurcations of spatially inhomogeneous self-oscillatory solutions and their stability are investigated. The possibility of bifurcation of stable rotational and spiral waves is shown.

Détail

Keywords: parabolic equation with spatial argument transformation and delay, nonlinear waves, spatially inhomogeneous solutions, rotational and spiral wave
Mots-clés : bifurcation

Bibliographie

[1] S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, “Krupnomasshtabnye poperechnye nelineinye vzaimodeistviya v lazernykh puchkakh; novye tipy nelineinykh voln, vozniknovenie ‘opticheskoi turbulentnosti’ ”, Pisma v ZhETF, 47:12 (1988), 611–614

[2] S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, “Generatsiya struktur v opticheskikh sistemakh s dvumernoi obratnoi svyazyu: na puti k sozdaniyu nelineino-opticheskikh analogov neironnykh setei”, Novye fizicheskie printsipy opticheskoi obrabotki informatsii, eds. S. A. Akhmanov, M. A. Vorontsov, Nauka, M., 1990, 263–325

[3] S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, A. V. Larichev, N. I. Zheleznykh, “Controlling transverse-wave interactions in nonlinear optics: generation and interaction of spatiotemporal structures”, J. Optical Soc. Amer. Ser. B, 9:1 (1992), 78–90 | DOI

[4] A. V. Razgulin, Nelineinye modeli opticheskoi sinergetiki, MAKS Press, M., 2008

[5] A. V. Razgulin, “Ob avtokolebaniyakh v nelineinoi parabolicheskoi zadache s preobrazovannym argumentom”, Zh. vychisl. matem. i matem. fiz., 33:1 (1993), 69–80 | MR | Zbl

[6] A. V. Razgulin, “Ustoichivost bifurkatsionnykh avtokolebanii v nelineinoi parabolicheskoi zadache s preobrazovannym argumentom”, Zh. vychisl. matem. i matem. fiz., 33:10 (1993), 1499–1508 | MR | Zbl

[7] E. P. Belan, “O dinamike beguschikh voln v parabolicheskom uravnenii s preobrazovaniem sdviga prostranstvennoi peremennoi”, Zhurn. matem. fiz., anal., geom., 1:1 (2005), 3–34 | MR | Zbl

[8] S. A. Kaschenko, “Asimptotika prostranstvenno-neodnorodnykh struktur v kogerentnykh nelineino-opticheskikh sistemakh”, Zh. vychisl. matem. i matem. fiz., 31:3 (1991), 467–473 | MR | Zbl

[9] A. Yu. Kolesov, N. Kh. Rozov, “Opticheskaya bufernost i mekhanizmy ee vozniknoveniya”, TMF, 140:1 (2004), 14–28 | DOI | DOI | MR | Zbl

[10] A. V. Razgulin, “Rotatsionnye volny v opticheskoi sisteme s dvumernoi obratnoi svyazyu”, Matem. modelirovanie, 5:4 (1993), 105–119 | Zbl

[11] A. L. Skubachevskii, “O bifurkatsii Khopfa dlya kvazilineinogo parabolicheskogo funktsionalno-differentsialnogo uravneniya”, Differents. uravneniya, 34:10 (1998), 1394–1401 | MR

[12] A. L. Skubachevskii, “Bifurcation of periodic solutions for nonlinear parabolic functional-differential equations arising in optoelectronics”, Nonlinear Anal., 32:2 (1998), 261–278 | DOI | MR

[13] A. V. Razgulin, T. E. Romanenko, “Vraschayuschiesya volny v parabolicheskom funktsionalno-differentsialnom uravnenii s povorotom prostranstvennogo argumenta i zapazdyvaniem”, Zh. vychisl. matem. i matem. fiz., 53:11 (2013), 1804–1821 | DOI | DOI | MR

[14] S. S. Budzinskiy, A. V. Razgulin, “Rotating and standing waves in a diffractive nonlinear optical system with delayed feedback under $O(2)$ Hopf bifurcation”, Commun. Nonlinear Sci. Numer. Simul., 49 (2017), 17–29 | DOI | MR

[15] E. P. Kubyshkin, V. A. Kulikov, “Analysis of occurrence conditions for spatially inhomogeneous structures of light waves in optical information transmission systems”, Aut. Control Comp. Sci., 54:7 (2020), 752–757 | DOI

[16] E. P. Kubyshkin, V. A. Kulikov, “Bifurkatsii avtokolebatelnykh reshenii nelineinogo parabolicheskogo uravneniya s povorotom prostranstvennogo argumenta i zapazdyvaniem”, Zh. vychisl. matem. i matem. fiz., 61:3 (2021), 428–449 | DOI | DOI

[17] Yu. I. Neimark, “$D$-razbienie prostranstva kvazipolinomov (k ustoichivosti linearizovannykh raspredelennykh sistem)”, PMM, 13:4 (1949), 349–380 | MR

[18] A. N. Kulikov, “O gladkikh invariantnykh mnogoobraziyakh polugruppy nelineinykh operatorov v banakhovom prostranstve”, Issledovaniya po ustoichivosti i teorii kolebanii, Sbornik nauchnykh trudov, Izd-vo YarGU, Yaroslavl, 1976, 114–129 | MR

[19] Dzh. Marsden, M. Mak-Kraken, Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, Mir, M., 1980 | DOI | MR | MR | Zbl

[20] A. D. Bryuno, Lokalnyi metod nelineinogo analiza differentsialnykh uravnenii, Nauka, M., 1979 | MR

[21] Dzh. Kheil, Teoriya funktsionalno-differentsialnykh uravnenii, Mir, M., 1984 | DOI

[22] Yu. A. Kuznetsov, Bifurkatsiya Andronova–Khopfa v chetyrekhmernoi sisteme s krugovoi simmetriei, Preprint NIVTs AN SSSR, Puschino, 1984

[23] Dzh. Kheil, Kolebaniya v nelineinykh sistemakh, Mir, M., 1966 | MR | Zbl

[24] E. P. Kubyshkin, A. R. Moriakova, “Features of bifurcations of periodic solutions of the Ikeda equation”, Russ. J. Nonlinear Dyn., 14:3 (2018), 301–324 | DOI | MR

[25] E. P. Kubyshkin, A. R. Moryakova, “Osobennosti bifurkatsii periodicheskikh reshenii uravneniya Mekki–Glassa”, Zh. vychisl. matem. i matem. fiz., 59:8 (2019), 1340–1357 | DOI | DOI

[26] M. A. Krasnoselskii, G. M. Vainikko, P. P. Zabreiko, Ya.B. Rutitskii, V. Ya. Stetsenko, Priblizhennoe reshenie operatornykh uravnenii, Nauka, M., 1969 | MR