×

On the number of solutions for a certain class of nonlinear second-order boundary-value problems. (English. Russian original) Zbl 1478.34027

J. Math. Sci., New York 257, No. 1, 31-40 (2021); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 160, 32-41 (2019).
In this paper, the authors study the number of solutions of a class of boundary-value problems for the nonlinear second order differential equations with quadratic nonlinearity and Neumann boundary conditions of the form \[ \begin{gathered} x''=-ax+bx^{2},\ a>0,\ b>0 \\ x'(0)=0,\ x'(T)=0, \ T>0. \end{gathered}\tag{\(*\)} \] The following results are obtained:
(i)
an estimate of the number of solutions of the Neumann problem (\(*\)).
(ii)
explicit formulas for solutions of the Cauchy problem are obtained in terms of the Jacobian elliptic functions.
(iii)
the equation for the initial values \(x_0\) of solutions of the Neumann problem (\(*\)) is derived.
(iv)
the results are tested and verified by an example.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ginzburg, VL, 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., 76, 3, 981-998 (2004) · doi:10.1103/RevModPhys.76.981
[2] Kirichuka, A., The number of solutions to the Neumann problem for the second order differential equation with cubic nonlinearity, Proc. IMCS Univ. Latv., 17, 44-51 (2017)
[3] Konyukhova, NB; Sheina, AA, On an auxiliary nonlinear boundary-value problem in the Ginzburg-Landau theory of superconductivity and its multiple solutions, Vestn. Ross. Univ. Druzhby Narodov. Ser. Mat. Inform. Fiz., 3, 5-20 (2016)
[4] Kuznetsov, AP; Kuznetsov, SP; Ryskin, NM, Nonlinear Oscillations [in Russian] (2002), Moscow: Fizmatlit, Moscow
[5] L. M. Milne-Thomson, “Jacobian Elliptic Functions and Theta Functions,” in: Handbook of Mathematical Functions, Chap. 16 (M. Abramowitz and I. A. Stegun, eds.), Dover Publications, New York (1972).
[6] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1927). · JFM 45.0433.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.