Other Articles by Authors

Anita Kirichuka
Felix Sadyrbaev

Authors and WSEAS

Anita Kirichuka
Felix Sadyrbaev

WSEAS Transactions on Mathematics

Print ISSN: 1109-2769
E-ISSN: 2224-2880

Volume 17, 2018

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Volume 17, 2018

Remark on Boundary Value Problems Arising in Ginzburg-Landau Theory

AUTHORS: Anita Kirichuka, Felix Sadyrbaev

Download as PDF

The equation x 00 = −a(x − x 3 ) (i) is considered together with the boundary conditions x 0 (0) = 0, x0 (1) = 0 (ii), x0 (0) = 0, x0 (T) = 0 (iii). The exact number of solutions for the boundary value problems (BVP) (i), (ii) and (i), (iii) is given. The problem of finding the initial values x0 = x(0) of solutions to the problem (i), (iii) is solved also.

KEYWORDS: Boundary value problem, Jacobian elliptic functions, cubic nonlinearity, phase trajectory, multiplicity of solutions


[1] J. V. Armitage, W. F. Eberlein, Elliptic Functions, Cambridge University Press, 2006.

[2] I. Aranson, The World of the Complex Ginzburg-Landau Equation, Rev. Mod. Phys. Vol. 74, 99, 2002. Available at https://doi.org/10.1103/RevModPhys.74.99.

[3] M. Dobkevich, F. Sadyrbaev, N. Sveikate, and I. Yermachenko, On Types of Solutions of the Second Order Nonlinear Boundary Value Problems, Abstract and Applied Analysis, Vol. 2014, Spec. Is. (2013), Article ID 594931, 9 pages.

[4] M. Dobkevich, F.Sadyrbaev, Types and Multiplicity of Solutions to Sturm - Liouville Boundary Value Problem, Mathematical Modelling and Analysis, Vol. 20, 2015 - Issue 1,1-8.

[5] M. Dobkevich, F.Sadyrbaev, On Different Type Solutions of Boundary Value Problems, Mathematical Modelling and Analysis, Vol. 21, 2016 - Issue 5, 659-667.

[6] V. L. Ginzburg, Nobel Lecture: On superconductivity and superfluidity (what I have and have not managed to do) as well as on the physical minimum at the beginning of the XXI century, Rev. Mod. Phys., Vol. 76, No. 3, 2004, pp. 981– 998.

[7] K. Johannessen, A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions, International Journal of Differential Equations, Vol. 2012, Article ID 412569, 9 pages, http://dx.doi.org/10.1155/2012/412569.

[8] A. Kirichuka, F. Sadyrbaev, Multiple Positive Solutions for the Dirichlet Boundary Value Problems by Phase Plane Analysis, Abstract and Applied Analysis, Vol. 2015(2015), http://www.hindawi.com/journals/aaa/2015/ 302185/.

[9] N. B. Konyukhova, A. A. Sheina, On an Auxiliary Nonlinear Boundary Value Problem in the Ginzburg Landau Theory of Superconductivity and its Multiple Solution, RUDN Journal of Mathematics, Information Sciences and Physics, Vol. 3, pp. 5–20, 2016. Available at http://journals.rudn.ru/miph/article/view/13385.

[10] L. M. Milne-Thomson, Handbook of Mathematical Functions, Chapter 16. Jacobian Elliptic Functions and Theta Functions, Dover Publications, New York, NY, USA, 1972, Edited by: M. Abramowitz and I. A. Stegun.

[11] Wim van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D (NorthHolland), Vol. 56 (1992), pp. 303–367.

[12] Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh, Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial, Communications on Pure and Applied Analysis, Vol. 2016, 15(4): 1497-1514, doi: 10.3934/cpaa.2016.15.1497

[13] S. Ogorodnikova, F. Sadyrbaev, Multiple solutions of nonlinear boundary value problems with oscillatory solutions, Mathematical modelling and analysis, Vol. 11, N 4, pp. 413 − 426, 2006.

[14] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, (1940, 1996), ISBN 0-521-58807-3.

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 17, 2018, Art. #36, pp. 290-295

Copyright Β© 2018 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0

Bulletin Board


The editorial board is accepting papers.

WSEAS Main Site