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Germany; Russia Jena; Yekaterinburg
Section  Mathematics 
Title  Traveling waves in a profile of phase field: exact analytical solutions of a hyperbolic AllenCahn equation 
Author(s)  Nizovtseva I.G.^{a}, Galenko P.K.^{b}, Alexandrov D.V.^{a}, Vikharev S.V.^{a}, Titova E.A.^{a}, Sukhachev I.S.^{a} 
Affiliations  Ural Federal University^{a}, University of Jena^{b} 
Abstract  To obtain solutions of the hyperbolic AllenCahn equation, the first integral method, which follows from wellknown Hilbert Nulltheorem, is used. Exact analytical solutions are obtained in a form of traveling waves, which define complete class of the hyperbolic AllenCahn equation. It is shown that two subclasses of solutions exist within this complete class. The first subclass exhibits continual solutions and the second subclass is represented by solutions with singularity at the origin of coordinate system. Such nonuniqueness of solutions stands a question about stable attractor, i.e., about the traveling wave to which nonstationary solutions may attract. The obtained solutions include earlier solutions for the parabolic AllenCahn equation in a form of finite number of $\tanh$functions. 
Keywords  traveling wave, AllenCahn equation, first integral method, division theorem 
UDC  5172 
MSC  00A79, 35L70 
DOI  10.20537/vm160211 
Received  23 May 2016 
Language  Russian 
Citation  Nizovtseva I.G., Galenko P.K., Alexandrov D.V., Vikharev S.V., Titova E.A., Sukhachev I.S. Traveling waves in a profile of phase field: exact analytical solutions of a hyperbolic AllenCahn equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 245257. 
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