Algebraization in Stability Problem for Stationary Waves of the Klein-Gordon Equation

Keywords: the Klein-Gordon equation, stationary waves stability, Ince algebraization

Abstract

Nonlinear traveling waves of the Klein-Gordon equation with cubic nonlinearity are considered. These waves are described by the nonlinear ordinary differential equation of the second order having the energy integral. Linearized equation for variation obtained for such waves is transformed to the ordinary one using separation of variables. Then so-called algebraization by Ince is used. Namely, a new independent variable associated with the solution under consideration is introduced to the equation in variations. Integral of energy for the stationary waves is used in this transformation. An advantage of this approach is that an analysis of the stability problem does no need to use the specific form of the solution under consideration. As a result of the algebraization, the equation in variations with variable in time coefficients is transformed to equation with singular points. Indices of the singularities are found. Necessary conditions of the waves stability are obtained. Solutions of the variational equation, corresponding to boundaries of the stability/instability regions in the system parameter space, are constructed in power series by the new independent variable. Infinite recurrent systems of linear homogeneous algebraic equations to determine coefficients of the series can be written. Non-trivial solutions of these systems can be obtained if their determinants are equal to zero. These determinants are calculated up to the fifth order inclusively, then relations connecting the system parameters and corresponding to boundaries of the stability/ instability regions in the system parameter place are obtained. Namely, the relation between parameters of anharmonicity and energy of the waves are constructed. Analytical results are illustrated by numerical simulation by using the Runge-Kutta procedure for some chosen parameters of the system. A correspondence of the numerical and analytical results is observed.

Downloads

Download data is not yet available.

References

G.B. Whitham, Linear and Nonlinear Waves, (Wiley, New York, 1999).

V. Benci and D. Fortunato, Varionational Methods in Nonlinear Field Equations. Springer Monographs in mathematics. (Springer, Switzerland, 2014).

A.D. Polyanin and V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, (Boca Raton, Chapman & Hall/CRC, 2004).

O. Waldron and R. A. Van Gorder, Physica Scripta, 92(10), 105001, 2017, https://doi.org/10.1088/1402-4896/aa86fa.

Е.I. Yakubovich, in: Нелинейные волны [Nonlinear waves], (Nauka, Moscow, 1979), 62-67. (in Russian)

N. Budinsky and T. Bountis, Physica D, 8(3), 445—452 (1983), https://doi.org/10.1016/0167-2789(83)90236-1.

A. Ghazaryan, S. Lafortune and V. Manukian, Philosophical Transactions of The Royal Society A. Mathematical Physical and Engineering Sciences, 376(2117), 20180001 (2018), https://doi.org/10.1098/rsta.2018.0001.

R.L. Ince, Ordinary Differential Equations. (Longmans Green, London, 1926).

Yu.V. Mikhlin and A.L Zhupiev, Int. J. of Non-Linear Mechanics, 32(2), 393-409, (1997), https://doi.org/10.1016/S0020-7462(96)00047 9.

Yu.V. Mikhlin, T.V. Shmatko and G.V. Manucharyan, Computer & Structures, 82(31), 2733–2742, 2004, https://doi.org/10.1016/j.compstruc.2004.03.082.

K.V. Avarmov, Yu.V. Mikhlin, Нелинейная динамика упругих систем. Т.1. Модели, методы, явления (Издание 2-е исправленное и дополненное) [Nonlinear Dynamics od Elastic Systems, V.1] (IKI, Moscow-Izhevsk, 2015). (in Russian)

A.F. Vakakis, L.I. Manevitch, Y.V. Mikhlin, V.N. Pilipchuk and A.A. Zevin, Normal Modes and Localization in Nonlinear Systems, (Wiley, New York, 1996).

Published
2019-07-29
Cited
How to Cite
Goloskubova, N., & Mikhlin, Y. (2019). Algebraization in Stability Problem for Stationary Waves of the Klein-Gordon Equation. East European Journal of Physics, (2), 5-10. https://doi.org/10.26565/2312-4334-2019-2-01