By using the invariants of nonconjugate subgroups of the Poincaré group P(1,4) [conjugation is considered with respect to the group P(1,4)], we propose ansatzes that reduce some linear and nonlinear five-dimensional d’Alembert equations to ordinary differential equations. On the basis of the solutions of the reduced equations, we construct the invariant solutions of these five-dimensional d’Alembert equations.
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References
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies Appl. Math., Vol. 4, SIAM, Philadelphia (1981).
A. F. Barannyk, “On the reduction of the Liouville equation in the Minkowski space R 1,n ,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 3–6 (1990).
B. M. Barbashov and V. V. Nesterenko, Introduction to the Relativistic String Theory, World Scientific, Singapore (1990).
P. L. Bhatnagar, Nonlinear Waves in One-Dimensional Dispersive Systems, Clarendon, Oxford (1979).
V. G. Kadyshevskii, “A new approach to the theory of electromagnetic interactions,” Fiz. Élem. Chastits Atomn. Yadra, 11, No. 1, 5–39 (1980).
R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland, Amsterdam (1982).
Yu. B. Rumer, Investigations in 5-Optics [in Russian], Gostekhteorizdat, Moscow (1956).
V. M. Fedorchuk, “Nonsplit subalgebras of the Lie algebra of the generalized Poincaré group P(1,4),” Ukr. Mat. Zh., 33, No. 5, 696–700 (1981); English translation: Ukr. Math. J., 33, No. 5, 535–538 (1981).
V. M. Fedorchuk, “Splitting subalgebras of the Lie algebra of the generalized Poincaré group P(1,4),” Ukr. Mat. Zh., 31, No. 6, 717–722 (1979); English translation: Ukr. Math. J., 31, No. 6, 554–558 (1979).
R. L. Anderson, A. O. Barut, and R. Rączka, “Bäcklund transformations and new solutions of nonlinear wave equations in fourdimensional space-time,” Lett. Math. Phys., 3, No. 5, 351–358 (1979).
A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, “Theory and applications of the sine-Gordon equation,” Rivista Nuovo Cimento, 1, No. 2, 227–267 (1971).
P. L. Christiansen and P. S. Lomdahl, “Numerical study of 2+1-dimensional sine-Gordon solitons,” Physica, Ser. D, 2, No. 3, 482–494 (1981).
G. P. Dzhordzhadze, A. K. Pogrebkov, and M. K. Polivanov, “Singular solutions of the equation \( \upvarphi +\frac{{\mathrm{m}}^2}{2} \exp \left(\upvarphi \right)=0 \) and the dynamics of singularities,” Theor. Math. Phys., 40, No. 2, 706–715 (1979).
W. I. Fushchich, “Symmetry in problems of mathematical physics,” in: Algebraic-Theoretical Studies in Mathematical Physics [in Russian], Inst. Math., Acad. Sci. Ukr. SSR, Kyiv (1981), pp. 6–28.
W. I. Fushchich, A. F. Barannik, L. F. Barannik, and V. M. Fedorchuk, “Continuous subgroups of the Poincaré group P(1,4),” J. Phys., Ser. A: Math. Gen., 18, No. 15, 2893–2899 (1985).
W. I. Fushchich and Yu. N. Seheda, “Some exact solutions of the many-dimensional sine-Gordon equation,” Lett. Nuovo Cimento, 41, No. 14, 462–464 (1984).
W. I. Fushchich and N. I. Serov, “The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d’Alembert, and eikonal equations,” J. Phys., Ser. A: Math. Gen., 16, No. 15, 3645–3656 (1983).
W. I. Fushchich, W. M. Shtelen, and N. I. Serov, Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht (1993).
A. M. Grundland, J. Harnad, and P. Winternitz, “Solutions of the multidimensional sine-Gordon equation obtained by symmetry reduction,” KINAM Rev. Fís., 4, No. 3, 333–344 (1982).
A. M. Grundland, J. Harnad, and P. Winternitz, “Symmetry reduction for nonlinear relativistically invariant equations,” J. Math. Phys., 25, No. 4, 791–806 (1984).
A. M. Grundland, J. A. Tuszyński, and P. Winternitz, “Applications of the three-dimensional “φ6 ”-model to structural phase transitions,” in: Proc. XV Internat. Colloq. on Group Theoretical Methods in Physics (Philadelphia, PA, 1986), World Sci. Publ., Teaneck, NJ (1987), pp. 589–601.
A. M. Grundland, J. A. Tuszyński, and P. Winternitz, “Group theory and solutions of the classical field theories with polynomial nonlinearities,” Found. Phys., 23, No. 4, 633–665 (1993).
G. P. Jorjadze, A. K. Pogrebkov, M. C. Polivanov, and S. V. Talalov, “Liouville field theory: IST and Poisson bracket structure,” J. Phys., Ser. A: Math. Gen., 19, No. 1, 121–139 (1986).
H. O. Lahno and V. F. Smalij, “Subgroups of Poincaré group and new exact solutions of Maxwell equations,” Proc. Inst. Math., NAS of Ukraine, 43, No. 1, 162–166 (2002).
G. L. Lamb (Jr.), Elements of Soliton Theory, Wiley, New York (1980).
G. Leibbrandt, “New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions,” Phys. Rev. Lett., 41, No. 7, 435–438 (1978).
A. C. Newell, “The inverse scattering transform,” in: R. K. Bullough and P. J. Caudrey (editors), Solitons, Ser. Topics in Current Physics, Vol. 17, Springer, Berlin (1980), pp. 177–242.
A. G. Nikitin and O. Kuriksha, “Invariant solutions for equations of axion electrodynamics,” Comm. Nonlinear Sci. Numer. Simulat., 17, 4585–4601 (2012).
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New York (1984).
P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1986).
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic, New York (1982).
A. K. Pogrebkov and M. K. Polivanov, “Interaction of particles and fields in classical theory,” Soviet J. Particles Nuclei, 14, No. 5, 450–457 (1983).
A. K. Pogrebkov and M. K. Polivanov, “The Liouville and sinh-Gordon equations. Singular solutions, dynamics of singularities and the inverse problem method,” Math. Phys. Rev., 5, 197–271 (1985).
A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, “The soliton: a new concept in applied science,” Proc. IEEE, 61, No. 10, 1443–1483 (1973).
G. B. Whitham, “Comments on some recent multisoliton solutions,” J. Phys., Ser. A: Math. Gen., 12, No. 1, L1–L3 (1979).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 57, No. 4, pp. 27–34, October–December, 2014.
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Fedorchuk, V.I. On the Invariant Solutions of Some Five-Dimensional D’alembert Equations. J Math Sci 220, 27–37 (2017). https://doi.org/10.1007/s10958-016-3165-7
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DOI: https://doi.org/10.1007/s10958-016-3165-7