Geodesic
Multidimensional nonlinear wave equations with multivalued solutions
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 2, pp. 272-284 Cet article a éte moissonné depuis la source Math-Net.Ru

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We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d'Alembert equations, nonlinear Klein–Gordon equations, and nonlinear telegraph equations.
Keywords: exact solution of multidimensional hyperbolic equations, breaking wave, multivalued solution, electromagnetic waves in a medium with nonlinear polarization. 
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     author = {V. M. Zhuravlev},
     title = {Multidimensional nonlinear wave equations with multivalued solutions},
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V. M. Zhuravlev. Multidimensional nonlinear wave equations with multivalued solutions. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 2, pp. 272-284. http://geodesic.mathdoc.fr/item/TMF_2013_174_2_a6/

[1] B. L. Rozhdestvenskii, N. N. Yanenko, Cistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR | Zbl

[2] A. G. Kulikovskii, E. I. Sveshnikova, A. P. Chugainova, “Matematicheskie metody izucheniya razryvnykh reshenii nelineinykh giperbolicheskikh sistem uravnenii”, Lekts. kursy NOTs, 16, MIAN, M., 2010, 3–120 | DOI

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

[4] O. I. Bogoyavlenskii, Oprokidyvayuschiesya solitony. Nelineinye integriruemye uravneniya, Nauka, M., 1991 | MR | Zbl

[5] I. G. Kataev, Udarnye elektromagnitnye volny, Sovetskoe radio, M., 1963

[6] A. A. Akopyan, D. L. Oganesyan, Kvantovaya elektronika, 24:7 (1997), 622–624 | DOI

[7] R. Penrose, W. Rindler, Spinors and Space-Time, v. 2, Spinor and Twistor Methods in Space-Time Geometry, Cambridge Univ. Press, Cambridge, 1986 | MR

[8] V. V. Kassandrov, “Algebrodynamics: primordial light, particles-caustics and the flow of time”, Space-Time Structure: Algebra and Geometry, eds. D. G. Pavlov, Gh. Atanasiu, V. Balan, Lilia Print, Moscow, 2007, 422–440; “Quaternionic analysis and the algebrodynamics”, 441–473; arXiv: ; Phys. Atom. Nucl., 72:5 (2009), 813–827, arXiv: 0710.28950907.5425 | DOI

[9] V. M. Zhuravlev, Izvestiya vuzov. Ser. Prikladnaya nelineinaya dinamika, 9:6 (2001), 115–128 | Zbl

[10] V. M. Zhuravlev, “Oprokidyvayuschiesya elektromagnitnye volny v dielektrikakh i provodyaschei srede”, Tezisy dokladov XLVIII Vserossiiskoi konferentsii po problemam fiziki chastits, fiziki plazmy i kondensirovannykh sred, optoelektroniki, posvyaschennoi 100-letiyu professora Ya. P. Terletskogo (Moskva, 15–18 maya 2012 g.), RUDN, M., 2012, 137

[11] S. P. Tsarev, Izv. AN SSSR. Ser. matem., 54:5 (1990), 1048–1068 | DOI | MR | Zbl

[12] E. V. Ferapontov, K. R. Khusnutdinova, Commun. Math. Phys., 248:1 (2004), 187–206, arXiv: nlin/0305044 | DOI | MR | Zbl

[13] E. V. Ferapontov, K. R. Khusnutdinova, M. V. Pavlov, TMF, 144:1 (2005), 35–43 | DOI | DOI | MR | Zbl