Nonstationary contrasting structures in the vicinity of a critical point

详细信息    查看全文

• 作者：A. A. Bykov (1)
  A. S. Sharlo (1)
  
  1. Moscow State University ; Moscow ; Russia
  
• 关键词：internal transitional layer ; contrasting structure ; asymptotic method
• 刊名：Mathematical Models and Computer Simulations
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：7
• 期：2
• 页码：165-178
• 全文大小：883 KB
• 参考文献：1. Pao, CV (1992) Nonlinear Parabolic and Elliptic Equations. Plenum, New York
  2. Zel鈥檇ovich, Ya B, Ruzmaikin, A A, Sokolov, D D (1990) Magnetic Fields in Astrophysics. Taylor & Francis, London
  3. Barenblatt, G I, Zel鈥檇ovich, Ya B (1971) Intermediate asymptotics in mathematical physics. Usp. Mat. Nauk 26: pp. 115-129
  4. Rozhdestvenskii, B P, Yanenko, N N (1978) Systems of Quasi-Linear Equations and their Applications to Gas Dynamics. Nauka, Moscow
  5. Martinson, L K, Malov, Yu I (2002) Differential Equations of Mathematical Physics. MGTU, Moscow
  6. Davydov, A S (1979) Biology and Quantum Mechanics. Naukova Dumla, Kyiv
  7. Ikeda, H, Mimura, M, Tsujikawa, T (1989) Singular perturbation approach to travelling wave solutions of the Hodgkin-Huxley equations and its application to stability problems. Japan J. Appl. Math. 6: pp. 1-66 CrossRef
  8. Volpert, V, Petrovskii, S (2009) Reaction-diffusion waves in biology. Phys. Life Rev. 6: pp. 267-310 CrossRef
  9. Kolmogorov, A, Petrovsky, I, Piskounoff, N (1937) Etude de l鈥檈quations de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Bull Univ.Moskou, Ser. Internat. 1A.
  10. Sveshnikov, A G, Al鈥檚hin, A B, Korpusov, M O, Pletner, Yu D (2007) Linear and Non-Linear Equations of Sobolev Type. Fizmatlit, Moscow
  11. Korpusov, M O, Pletner, Yu D, Sveshnikov, A G (2000) On quasi-steady processes in conducting nondispersive media. Comput. Math. Math. Phys. 40: pp. 1188-1200
  12. Korpusov, M O, Sveshnikov, A G (2005) On the finite-time blowup of solutions to initial-boundary value problems for pseudoparabolic equations with pseudo-Laplacian. Comput. Math. Math. Phys. 45: pp. 261-275
  13. Ma, W X, Fuchssteiner, B (1996) Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation. Int. J. Non-Lin. Mech. 31: pp. 329-338 CrossRef
  14. Wei, J, Yang, J (2009) Solutions with transition layer and spike in an inhomogeneous phase transition model. J. Differ. Equations 246: pp. 3642-3667 CrossRef
  15. Bozhevol鈥檔ov, Yu V, Nefedov, N N (2010) Front motion in the parabolic reaction-diffusion problem. Comput. Math. Math. Phys. 50: pp. 264-274 CrossRef
  16. Nefedov, N N (1992) Nonstationary contrast structures in the reaction-diffusion system. Mat. Model. 4: pp. 58-65
  17. Kozhanov, A I (1999) An initial-boundary value problem for equations of the type of the generalized Boussinesq equation with a nonlinear source. Mat. Zam. 65: pp. 70-75 CrossRef
  18. Vasil鈥檈va, A B, Butuzov, V F, Nefedov, N N (1998) Contrast structures in singularly perturbed problems. Fundam. Appl. Math. 4: pp. 799-851
  19. Bykov, A A, Popov, V Yu (1999) On the lifetime of one-dimensional transient contrasting structures. Comput. Math. Math. Phys. 39: pp. 266-274
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Modeling and IndustrialMathematics
  Simulation and Modeling
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：2070-0490

文摘

Evolution equations of the internal transition layer (ITL) have been derived for the reaction-diffusion equation and for a pseudo-parabolic third-order equation with a small parameter in the highest derivatives, which describes different processes in physics, chemistry, biology, and, in particular, the process of magnetic field generation in the turbulent medium. We consider a case where there is a point with a zero velocity of the ITL drift (critical point), while the drift velocity both on the right and on the left of this point does not change its sign. It is shown that in the case of balanced cubic nonlinearity, which is fairly common in physical applications, the ITL drift velocity in the first-order asymptotic expansion in the power of a small parameter is also zero, but the second-order approximation makes it possible to obtain the drift velocity at the critical point. It is demonstrated that ITL crosses the critical point in finite time.