The hyperbolic Allen-Cahn equation: Exact solutions / Nizovtseva I.G., Galenko P.K., Alexandrov D.V. // Journal of Physics A: Mathematical and Theoretical. - 2016. - V. 49, l. 43.

ISSN:
17518113
Type:
Article
Abstract:
Using the first integral method, a general set of analytical solutions is obtained for the hyperbolic Allen-Cahn equation. The solutions are presented by (i) the class of continual solutions described by tanh-profiles for traveling waves of the order parameter, and (ii) the class of singular solutions which exhibit unbounded discontinuity in the profile of the order parameter at the origin of the coordinate system. It is shown that the solutions include the previous analytical results for the parabolic Allen-Cahn equation as a limited class of tanh-functions, in which the inertial effects are omitted.
Author keywords:
Cahn-Allen; division theorem; exact solutions; first integral method; traveling wave
Index keywords:
нет данных
DOI:
10.1088/1751-8113/49/43/435201
Смотреть в Scopus:
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84991574367&doi=10.1088%2f1751-8113%2f49%2f43%2f435201&partnerID=40&md5=925bfae69f89d50daa627e5fcb3ec2ce
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Art. No. 435201
Link https://www.scopus.com/inward/record.uri?eid=2-s2.0-84991574367&doi=10.1088%2f1751-8113%2f49%2f43%2f435201&partnerID=40&md5=925bfae69f89d50daa627e5fcb3ec2ce
Affiliations Fridrich-Schiller-Universität-Jena, Physikalisch-Astronomische Fakultät, Löbdergraben Strasse 32, Jena, Germany; Department of Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Lenin ave., 51, Ekaterinburg, Russian Federation
Author Keywords Cahn-Allen; division theorem; exact solutions; first integral method; traveling wave
Funding Details ID 1160779, Alexander von Humboldt-Stiftung
Funding Text The authors acknowledge the support from the Russian Science Foundation (project no. 16-11-10095). I G N especially acknowledges the support of Alexander von Humboldt Foundation (ID 1160779). P K G especially acknowledges the support from German Research Foundation (DFG Project RE 1261/8-2).
References Cahn, J.W., Allen, S.M., A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics (1977) J. Physique, 38, pp. C751-C754; Allen, S.M., Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening (1979) Acta Metall., 27, pp. 1085-1095. , 1085-95; Wheeler, A., Boettinger, W.J., McFadden, G.B., Phase-field model for isothermal phase transitions in binary alloys (1992) Phys. Rev., 45, pp. 7424-7439. , 7424-39; Gouyet, J.F., Generalized Allen-Cahn equations to describe far-from-equilibrium order-disorder dynamics (1995) Phys. Rev., 51, pp. 1695-1710. , 1695-710; Fife, P.C., Lacey, A.A., Motion by curvature in generalized Cahn-Allen models (1994) J. Stat. Phys., 77, pp. 173-181. , 173-81; Benes, M., Chalupecký, V., Mikula, K., Geometrical image segmentation by the Allen-Cahn equation (2004) Appl. Numer. Math., 51, pp. 187-205. , 187-205; Alfaro, M., Hilhorst, D., Generation of interface for an Allen-Cahn equation with nonlinear diffusion (2010) Math. Model. Nat. Phenom., 5, pp. 1-2. , 1-2; Caginalp, G., Chen, X., Phase field equations in the singular limit of sharp interface problems (1992) IMA Vol. Math. Appl., 43, pp. 1-28. , 1-28; Wheeler, A.A., Phase-field theory of edges in an anisotropic crystal (2006) Proc. R. Soc., 462, pp. 3363-3384. , 3363-84; Bates, P.W., Chen, F., Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation (2002) J. Math. Anal. Appl., 273, pp. 45-57. , 45-57; Galenko, P., Jou, D., Diffuse-interface model for rapid phase transformations in nonequilibrium systems (2005) Phys. Rev. E, 71; Yang, Y., Humadi, H., Buta, D., Laird, B.B., Sun, D., Hoyt, J.J., Asta, M., Atomistic simulations of nonequilibrium crystal-growth kinetics from alloy melts (2011) Phys. Rev. Lett., 107; Jou, D., Galenko, P., Coarse graining for the phase-field model of fast phase transitions (2013) Phys. Rev. E, 88; Field, R.J., Burger, M., (1985) Oscillations and Traveling Waves in Chemical Systems, , (New York: Wiley); Vladimirov, V.A., Kutafina, E.V., Exact travelling wave solutions of some nonlinear evolutionary equations (2004) Rep. Math. Phys., 54, pp. 261-271. , 261-71; Wazwaz, A.-M., The tanh method for traveling wave solutions of nonlinear equations (2004) Appl. Math. Comput., 154, pp. 713-723. , 713-23; Kim, H., Sakthivel, R., Travelling wave solutions for time-delayed nonlinear evolution equations (2010) Appl. Math. Lett., 23, pp. 527-532. , 527-32; Galenko, P.K., Abramova, E.V., Jou, D., Danilov, D.A., Lebedev, V.G., Herlach, D.M., Solute trapping in rapid solidification of a binary dilute system: A phase-field study (2011) Phys. Rev., 84; Salhoumi, A., Galenko, P.K., Gibbs-Thomson condition for the rapidly moving interface in a binary system (2016) Physica, 447, pp. 161-171. , 161-71; Zh, F., The first-integral method to study the Burgers-Korteweg-de Vries equation (2002) J. Phys. A: Math. Gen., 35 (2), pp. 343-349. , 343-9; Lu, B., Zhang, H.-Q., Xie, F.-D., Traveling wave solutions of nonlinear partial differential equations by using the first integral method (2010) Appl. Math. Comput., 216, pp. 1329-1336. , 1329-36; Feng, Z., Wang, X., The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation (2003) Phys. Lett., 308, pp. 173-178. , 173-8; Fan, E., Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method (2002) J Phys. A: Math Gen., 35 (32), pp. 6853-6872. , 6853-72; Kudryashov, N.A., Seven common errors in finding exact solutions of nonlinear differential equations (2009) Commun. Nonlinear Sci. Numer. Simulat., 14, pp. 3507-3529. , 3507-29; Feng, X., Exploratory approach to explicit solution of nonlinear evolution equations (2000) Int. J. Theor. Phys., 39, pp. 207-222. , 207-22; Hu, J., Zhang, H., A new method for finding exact traveling wave solutions to nonlinear partial differential equations (2001) Phys. Lett., 286, pp. 175-179. , 175-9; Rehman, T., Gambino, G., Choudhury, S.R., Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations (2013) Commun. Nonlinear Sci. Numer. Simul., 19, pp. 1746-1769. , 1746-69; Ahmed Ali, A.H., Raslan, K.R., New solutions for some important partial differential equations (2007) Int. J. Nonlinear Sci., 4, pp. 109-117. , 109-17; Taşcan, F., Bekir, A., Travelling wave solutions of Cahn-Allen equation by using first integral method (2009) Appl. Math. Comput., 207, pp. 279-282. , 279-82; Bourbaki, N., (1972) Commutative Algebra, , (Paris: Addison-Wesley); Hilhorst, D., Nara, M., Singular limit of a damped wave equation with a bistable nonlinearity (2014) SIAM J. Math. Anal., 46, pp. 1701-1730. , 1701-30; Provatas, N., Elder, K., (2010) Phase-Field Methods in Materials Science and Engineering, , (Weinheim: Wiley-VCH); Choi, J.-W., Lee, H.G., Jeong, D., Kim, J., An unconditionally gradient stable numerical method for solving the Allen-Cahn equuation (2009) Physica, 388, pp. 1791-1803. , 1791-803; Vladimorov, V.A., Kutafina, E.V., Pudelko, A., Constructing soliton and kink solutions of PDE models in transport and biology (2006) Symmetry, Integrability Geom.: Methods Appl., 2, pp. 1-5. , 1-5; Galenko, P.K., Sanches, F.I., Elder, K.R., Traveling wave profiles for a crystaline front invading liquid states (2015) Physica D, 308, pp. 1-2. , 1-0; Emmerich, H., Löwen, H., Wittkowski, R., Gruhn, T., Toth, G.I., Tegze, G., Gránásy, L., Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: An overview (2012) Adv. Phys., 61, pp. 665-743. , 665-743
Publisher Institute of Physics Publishing
Language of Original Document English
Abbreviated Source Title J. Phys. Math. Theor.
Source Scopus