A Numerical Study on the Stability of Solitary Wave Solutions of the Coupled Improved Boussinesq Equations

Year 2025, Volume: 15 Issue: 3, 1089 - 1099, 01.09.2025

Şenay Pasinlioğlu

https://doi.org/10.21597/jist.1640298

Abstract

In this study, the time evolution of solitary wave solutions to the coupled improved Boussinesq equations, and their stability properties under small perturbations are numerically investigated. Examining the long-term behavior of solitary waves is of great importance for understanding nonlinear wave dynamics. For this purpose, the dynamics of solitary wave solutions are examined using a numerical scheme that combines the Fourier pseudo-spectral method for spatial discritization and the fourth-order Runge–Kutta method for time discritization. Several numerical experiments are carried out to demonstrate the accuracy and efficiency of the proposed method in both time and space. The long-time behavior of the waves with initially applied small perturbations is observed, and their stability is examined. The obtained results indicate that the solitary wave solutions of the coupled improved Boussinesq equations are stable under small perturbations.

Keywords

Coupled improved Boussinesq equations , Time evolution , Stability

Kuple Düzgünleştirilmiş Boussinesq Denklemlerinin Yalnız Dalga Çözümlerinin Kararlılığı Üzerine Sayısal Bir Çalışma

Abstract

Bu çalışmada, kuple düzgünleştirilmiş Boussinesq denklemleri için yalnız (soliter) dalga çözümlerinin zaman evrimi ve küçük pertürbasyonlar altındaki kararlılık özellikleri sayısal olarak incelenmiştir. Yalnız dalgaların uzun zaman davranışlarını incelemek, doğrusal olmayan dalga dinamiklerini anlamak bakımından büyük önem taşımaktadır. Bu amaçla, uzay ayrıklaştırması için Fourier sözde (psödo)-spektral yöntemi ve zaman ayrıklaştırması için dördüncü mertebeden Runge-Kutta yöntemini birleştiren bir sayısal şema kullanılarak yalnız dalga çözümlerinin dinamikleri araştırılmıştır. Önerilen yöntemin hem zaman hem de uzaydaki doğruluğunu ve etkinliğini göstermek için çeşitli sayısal deneyler gerçekleştirilmiştir. Başlangıçta uygulanan küçük pertürbasyonlar ile dalgaların uzun zaman davranışları gözlemlenmiş ve kararlılıkları incelenmiştir. Elde edilen sonuçlar, kuple düzgünleştirilmiş Boussinesq denklemlerinin yalnız dalga çözümlerinin küçük pertürbasyonlar altında kararlı olduğunu göstermektedir.

Keywords

Kuple düzgünleştirilmiş Boussinesq denklemleri , Zaman evrimi , Kararlılık

References

• Bona, J., Durán, A., & Mitsotakis, D. (2023). Solitary-wave solutions of Benjamin–Ono and other systems for internal waves: II. Dynamics. Water Waves, 5, 161–190. doi:10.1007/s42286-023-00076-w
• Chen, G., Guo, H., & Zhang, H. (2009). Global existence of solutions of Cauchy problem for generalized system of nonlinear evolution equations arising from DNA. Journal of Mathematical Physics, 50(8), 083514. doi:10.1063/1.3191683
• Chen, G., & Zhang, H. (2004). Initial boundary value problem for a system of generalized IMBq equations. Mathematical Methods in the Applied Sciences, 27(5), 497–518. doi:10.1002/mma.444
• Christiansen, P. L., Lomdahl, P. S., & Muto, V. (1991). On a Toda lattice model with a transversal degree of freedom. Nonlinearity, 4(2), 477–501. doi:10.1088/0951-7715/4/2/012
• De Godefroy, A. (1998). Blow up of solutions of a generalized Boussinesq equation. IMA Journal of Applied Mathematics (Institute of Mathematics & Its Applications), 60(2), 123–138. doi:10.1093/imamat/60.2.123
• Dougalis, V. A., Durán, A., López-Marcos, M. A., & Mitsotakis, D. E. (2007). A numerical study of the stability of solitary waves of the Bona–smith family of Boussinesq systems. Journal of Nonlinear Science, 17(6), 569–607. doi:10.1007/s00332-007-9004-8
• Dougalis, Vassilios A., Duran, A., & Saridaki, L. (2023). On the numerical approximation of Boussinesq/Boussinesq systems for internal waves. Numerical Methods for Partial Differential Equations, 39(5), 3677–3704. doi:10.1002/num.23021
• Gozukizil, O. F., & Akcagil, S. (2014). Travelling wave solutions for the coupled IBq equations by using the tanh-coth method. Journal of Applied Mathematics, 2014, 1–14. doi:10.1155/2014/486269
• Grimshaw, R. H. J., Khusnutdinova, K. R., & Moore, K. R. (2017). Radiating solitary waves in coupled Boussinesq equations. IMA Journal of Applied Mathematics (Institute of Mathematics & Its Applications), 82(4), 802–820. doi:10.1093/imamat/hxx014
• Guo, H., & Chen, G. (2013). A note on the Cauchy problem for coupled imbq equations. Acta Mathematica Scientia. Series B. English Edition, 33(2), 375–392. doi:10.1016/s0252-9602(13)60005-3
• Khusnutdinova, K. R., Samsonov, A. M., & Zakharov, A. S. (2009). Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 79(5 Pt 2), 056606. doi:10.1103/PhysRevE.79.056606
• Li, Y. A. (2002). Hamiltonian structure and linear stability of solitary waves of the Green-naghdi equations. Journal of Nonlinear Mathematical Physics, 9(Supplement 1), 99-105. doi:10.2991/jnmp.2002.9.s1.9
• Pasinlioğlu, Ş. (2024). Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations. Zeitschrift Fuer Angewandte Mathematik Und Physik, 75(6). doi:10.1007/s00033-024-02342-4
• Pego, R. L., Smereka, P., & Weinstein, M. I. (1995). Oscillatory instability of solitary waves in a continuum model of lattice vibrations. Nonlinearity, 8(6), 921–941. doi:10.1088/0951-7715/8/6/003
• Turitsyn, S. K. (1993). On a Toda lattice model with a transversal degree of freedom. Sufficient criterion of blow-up in the continuum limit. Physics Letters. A, 173(3), 267–269. doi:10.1016/0375-9601(93)90276-6
• Wang, S., & Li, M. (2009). The Cauchy problem for coupled IMBq equations. IMA Journal of Applied Mathematics (Institute of Mathematics & Its Applications), 74(5), 726–740. doi:10.1093/imamat/hxp024
• Wang, Y., & Tian, N. (2019). On the Cauchy problem for IMBq system arising from DNA. Acta Mathematica Scientia. Series B. English Edition, 39(4), 1136–1148. doi:10.1007/s10473-019-0416-y Wattis, J. A. D. (2001). Solitary waves in a diatomic lattice: analytic approximations for a wide range of speeds by quasi-continuum methods. Physics Letters. A, 284(1), 16–22. doi:10.1016/s0375-9601(01)00277-8