Abstract
Themethod of differential constraints is applied to the study of some inverse problems for nonlinear one-dimensional differential equations of general type that include the classical equations of soliton theory. Under consideration is the problem of finding a potential for an equation of continuum mechanics in the one-dimensional case in the presence of some differential constraint.
Similar content being viewed by others
References
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981; Mir,Moscow, 1987).
Yu. E. Anikonov, N. B. Ayupova, V. G. Bardakov, V. P. Golubyatnikov, and M. V. Neshchadim, “Inversion of Mapping and Inverse Problems,” Siberian Electron. Math. Rep. 9, 382–432 (2012) [URL: http://semr.math.nsc.ru].
Yu. E. Anikonov and M. V. Neshchadim, “On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations. I,” Sibirsk. Zh. Industr. Mat. 14 (1), 27–39 (2011) [J. Appl. Indust. Math. 5 (4), 506–518 (2011)].
Yu. E. Anikonov and M. V. Neshchadim, “On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations. II,” Sibirsk. Zh. Industr. Mat. 14 (2), 28–33 (2011) [J. Appl. Indust. Math. 6 (1), 6–11 (2012)].
Yu. E. Anikonov and M. V. Neshchadim, “Representations for the Solutions and Coefficients of Second-Order Differential Equations,” Sibirsk.Zh. Industr. Mat. 15 (4), 17–23 (2012) [J. Appl. Indust. Math. 7 (1), 15–21 (2013)].
Yu. E. Anikonov and M. V. Neshchadim, “Representations for the Solutions and Coefficients of Evolution Equations,” Sibirsk.Zh. Industr. Mat. 16 (2), 40–49 (2013) [J. Appl. Indust.Math. 7 (3), 326–334 (2013)].
D. R. Blend, Nonlinear Dynamic Elasticity (Blaisdell, Waltham, 1969; Mir,Moscow, 1972).
E. K. Vdovina, “TheHarry DymEquation Is Reducible to a LinearOrdinaryDifferential Equation,” Obozrenie Prikl. i Promyshl. Mat. 18 (4), 626–627 (2011).
S. V. Golovin, “Exact Solutions for Evolution Submodels of Gas Dynamics,” Prikl. Mekh. i Teoret. Fiz. 43 (4), 3–14 (2002).
A. Goriely, Integrability and Nonintegrability of Dynamical Systems (World Scientific, Singapore, 2001; Inst. Comput. Issled., Izhevsk, 2006).
E. Infeld and G. Rowlands, Nonlinear Waves, Solitons, and Chaos (Cambridge Univ. Press, Cambridge, 1990; Fizmatlit,Moscow, 2006).
F. Calogero and A. Degasperis, Spectral Transforms and Solitons (North-Holland, Amsterdam, 1982; Mir, Moscow, 1985).
A. F. Sidorov, V. P. Shapeev, and N. N. Yanenko, Method of Differential Constraints and Its Application to Gas Dynamics (Nauka, Novosibirsk, 1984) [in Russian].
N. N. Yanenko, “About Invariant Differential Constraints for Hyperbolic Systems of Quasilinear Equations,” Izv. Vyssh. Uchebn. Zaved. Mat. No. 3, 186–194 (1961).
N. N. Yanenko, Selected Works.Mathematics. Mechanics (Nauka, Moscow, 1991) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.E. Anikonov, M.V. Neshchadim, 2015, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2015, Vol. XVIII, No. 2, pp. 36–47.
Rights and permissions
About this article
Cite this article
Anikonov, Y.E., Neshchadim, M.V. The method of differential constraints and nonlinear inverse problems. J. Appl. Ind. Math. 9, 317–327 (2015). https://doi.org/10.1134/S1990478915030035
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478915030035