ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Kashchenko S. A. Dynamics of two-component parabolic systems of schrodinger type. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 5, pp. 81-100. DOI: 10.18500/0869-6632-2018-26-5-81-100

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 292)
Language: 
Russian
Article type: 
Article
UDC: 
517.9

Dynamics of two-component parabolic systems of schrodinger type

Autors: 
Kashchenko Sergej Aleksandrovich, Yaroslavl State University
Abstract: 

Issue. The paper considers the local dynamics of important for applications class of two-component nonlinear systems of parabolic equations. These systems contain a small parameter appearing in the diffusion coefficients and characterizing «closeness» of the initial system of a parabolic type to a hyperbolic one. On quite natural conditions critical cases in the problem about balance state stability are realized to linearized equation coefficients. Innovation. An important thing here is the fact that these critical cases have an infinite dimension: infinitely many roots of a standard equation go to the imaginary axis when a small parameter vanishes. The specificity of all considered critical cases is typical of Schrodinger type systems and of a classical Schrodinger equation, in particular. These peculiarities are connected with the arrangement of roots of a standard equation. Three most important cases are stood here. Note that they fundamentally differ from each other. This difference is basically determined by the presence of specific resonance relations in the considered cases. It is these relations that define the structure of nonlinear functions included in normal forms. Investigation methods. A normalization algorithm is offered, that is the reduction of the initial system to the infinite system of ordinary differential equations for slowly changing amplitudes. Results. The situations when the corresponding systems can be compactly written as boundary-value problems with special nonlinearities are picked out. These boundary-value problems play the role of normal forms for initial parabolic systems. Their nonlocal dynamics determines the behavior of the solutions of the initial system with the initial conditions from some sufficiently small and not depending on a small parameter balance state neighborhood. Scalar complex parabolic Schrodinger equations are considered as important applications. Conclusions. The problem about the local dynamics of two-component parabolic systems of Schrodinger type is reduced to the investigation of nonlocal behavior of the solutions of ? special nonlinear evolutionary equations.    

Reference: 
  1. Ablowitz M.J., Segur H. Solitons and the inverse scattering transform. Philadelphia: SIAM, 1981. 435 p. (SIAM Studies in Applied Mathematics; 4).
  2. Novikov S.P., Manakov S.V., Pitaevskii L.P., et al. Theory of Solitons: The Inverse Scattering Method. New York: Springer US, 1984. 287 p. (Contemporary Soviet Mathematics). 
  3. Naumkin P.I. Solution asymptotics at large times for the non-linear Schrodinger equation // Izvestiya. Mathematics. 1997. Vol. 61, no. 4. P. 757–794. DOI: 10.1070/im1997v061n04ABEH000137.
  4. Hayashi N., Naumkin P.I. Asymptotics of odd solutions for cubic nonlinear Schro-dinger equations // Journal of Differential Equations. 2009. Vol. 246, no. 4. P. 1703– 1722. DOI: 10.1016/j.jde.2008.10.020.
  5. Naumkin P.I. The dissipative property of a cubic non-linear Schrodinger equation // Izvestiya. Mathematics. 2015. Vol. 79, no. 2. P. 346–374. DOI: 10.1070/IM2015v079n02ABEH002745.
  6. Shatah J. Normal forms and quadratic nonlinear Klein–Gordon equations // Communications on Pure and Applied Mathematics. 1985. Vol. 38, no. 5. P. 685–696. DOI: 10.1002/cpa.3160380516.
  7. Gourley S.A., Sou J.W.-H., Wu J.H. Nonlocality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics // Journal of Mathematical Sciences. 2004. Vol. 124, no. 4. P. 5119–5153. DOI: 10.1023/B:JOTH.0000047249.39572.6d.
  8. Haken H. Brain Dynamics: Synchronization and Activity Patterns in Pulse-coupled Neural Nets with Delays and Noise. Berlin: Springer Verlag, 2007. 257 p. (Springer eries in Synergetics).
  9. Kuang Y. Delay Differential Equations : With Applications in Population Dynamics. Boston : Academic Press, 1993. 410 p. (Mathematics in science and engineering; 191).
  10. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence. Berlin: Springer Verlag, 1984. 164 p. (Springer Series in Synergetics; 19). DOI: 10.1007/978-3-642-69689-3.
  11. Marsden J.E., McCracken M.F. The Hopf Bifurcation and Its Applications. New York: Springer, 1976. 421 p. (Applied Mathematical Sciences; 19). DOI: 10.1007/978-1-4612-6374-6. 
  12. Bokolishvily I.B., Kaschenko S.A., Malinetskii G.G., et al. Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion // Journal of Nonlinear Science. 1994. Vol. 4, no. 1. P. 545–562. DOI: 10.1007/BF02430645.
  13. Kashchenko S.A. Quasinormal forms for parabolic equations with small diffusion. Dokl. Akad. Nauk., 1988, vol. 299, no. 5, pp. 1049–1052 (in Russian).
  14. Kaschenko S.A. Normalization in the systems with small diffusion // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 1996. Vol. 6, no. 6. P. 1093-1109. DOI: 10.1142/S021812749600059X.
  15. Grigorieva E.V., Haken H., Kashchenko S.A., et al. Travelling wave dynamics in a nonlinear interferometer with spatial field transformer in feedback // Physica D: Nonlinear Phenomena. 1999. Vol. 125, no. 1/2. P. 123–141. DOI: 10.1016/S0167-2789(98)00196-1.
  16. Kaschenko I.S., Kaschenko S.A. Local dynamics of the two-component singular perturbed systems of parabolic type // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2015. Vol. 25, no. 11. P. 1550142. DOI: 10.1142/S0218127415501424.
  17. Kashchenko I.S., Kashchenko S.A. Dynamics of the Kuramoto equation with spatially distributed control // Communications in Nonlinear Science and Numerical Simulation. 2016. May. Vol. 34. P. 123–129. DOI: 10.1016/j.cnsns.2015.10.011.
  18. Kaschenko S.A. Bifurcational features in systems of nonlinear parabolic equations with weak diffusion // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2005. Vol. 15, no. 11. P. 3595–3606. DOI: 10.1142/S0218127405014258.
  19. Courant R., Robbins H. What is Mathematics?: An Elementary Approach to Ideas and Methods / rev. by I. Stewart. 2nd ed. New York: Oxford University Press, 1996. 591 p.
  20. Kashchenko S.A. A study of the stability of solutions of linear parabolic equations with nearly constant coefficients and small diffusion. J. Sov. Math., 1992, vol. 60, no. 6, pp. 1742–1764. DOI: 10.1007/BF01102587.
  21. Kashchenko S.A. Normal form for the KdV–Burgers equation. Dokl. Math., 2016, vol. 93, no. 3, pp. 331–333. DOI: 10.1134/S1064562416030170.
Received: 
17.05.2018
Accepted: 
12.07.2018
Published: 
31.10.2018
Short text (in English):
(downloads: 84)