Yerel ve Lineer olmayan Kuple Dalga Denklemlerinin Genel Sınıfı için Yalnız Dalga Çözümleri

Year 2024, Volume: 12 Issue: 2, 947 - 956, 29.04.2024

Şenay Pasinlioğlu , Gülçin Mihriye Muslu

https://doi.org/10.29130/dubited.1249987

Cited By: 1

Abstract

Bu makalede çekirdek fonksiyonları ile konvolüsyon işlemini içeren, yerel ve doğrusal olmayan kuple dalga denklemlerinin genel bir sınıfını inceliyoruz. Çekirdek fonksiyonlarının uygun seçimleri için sistem, Toda kafes sistemi, kuple Boussinesq denklemleri gibi iyi bilinen doğrusal olmayan kuple dalga denklemleri haline gelir. Petviashvili yöntemi kullanılarak, sistemin yalnız dalga çözümleri için bir sayısal şema önerilmiştir. Farklı çekirdekler kullanılarak, sayısal yöntemin geçerliliği test edilmiştir.

Keywords

Kuple Boussinesq denklemler, Petviashvili iterasyon yöntemi, yalnız dalga çözümleri.

Solitary Wave Solutions to the General Class of Nonlocal Nonlinear Coupled Wave Equations

Abstract

In this paper, we study a general class of nonlocal nonlinear coupled wave equations that includes the convolution operation with kernel functions. For appropriate selections of the kernel functions, the system becomes well-known nonlinear coupled wave equations, for instance Toda lattice system, coupled improved Boussinesq equations. A numerical scheme is proposed for the solitary wave solutions of the system using the Pethiashvili method. Using the different kernels, the validity of the numerical method has been tested.

Keywords

Coupled Boussinesq equations, Petviashvili's iteration method, solitary wave solutions.

References

• [1] A.C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” Journal of Applied. Physics, vol. 54, pp. 4703–4710, 1983.
• [2] J.A.D. Wattis, “Solitary waves in a diatomic lattice: analytic approximations for a wide range of speeds by quasi-continuum methods,” Physics Letters A, vol. 284, pp. 16–22, 2001.
• [3] P.L. Christiansen, P.S. Lomdahl, V. Muto, “On a Toda lattice model with a transversal degree of freedom,” Nonlinearity, vol. 4, pp. 477–501, 1991.
• [4] K.R. Khusnutdinova, A.M. Samsonov, A.S. Zakharov, “Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures,” Physical Review E, vol. 79, Article ID 056606, 2009.
• [5] S.K. Turitsyn, “On a Toda lattice model with a transversal degree of freedom. Sufficient criterion of blow-up in the continuum limit,” Physics Letters A, vol. 267, pp. 173-267, 1993.
• [6] A. De Godefroy, “Blow up of solutions of a generalized Boussinesq equation,” IMA Journal of Applied Mathematics, vol. 60, pp. 123–138, 1998.
• [7] S. Wang, M. Li, “The Cauchy problem for coupled IMBq equations,” IMA Journal of Applied Mathematics, vol. 74, pp. 726–740, 2009.
• [8] M. Lazar, G.A. Maugin, and E.C. Aifantis, “On a theory of nonlocal elasticity of bi-Helmholtz type and some applications,” International Journal of Solids and Structures., 43, pp. 1404–1421, 2006.
• [9] P. Rosenau, “Dynamics of dense discrete systems,” Progress of Theoretical Physics, vol. 79, pp. 1028–1042, 1988.
• [10] N. Duruk, A. Erkip, and H.A. Erbay, “A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity,” IMA Journal of Applied Mathematics, vol. 74, pp. 97– 106, 2009.
• [11] N. Duruk, H.A. Erbay, A. Erkip, “Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity,” Nonlinearity, vol. 23, pp. 107-118, 2010.
• [12] N. Duruk, H.A. Erbay, A. Erkip, “Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations,” Journal of differential equations, vol. 250, pp.1448-1459, 2011.
• [13] V.I. Petviahvili, “Equation of an extraordinary soliton,” Plasma Physics., 2, pp. 469– 472, 1976.
• [14] D.E. Pelinovsky and Y.A. Stepanyants, “Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations,” SIAM Journal on Numerical Analysis. Vol. 42, pp. 1110– 1127, 2004.
• [15] M.J. Ablowitz, Z.H. Musslimani, “Spectral renormalization method for computing self-organized solutions to nonlinear systems,” Optics Letters, vol. 30, pp. 2140–2142, 2005.
• [16] G. Fibich, Y. Sivan, M. Weinstein, “Bound states of nonlinear Schr¨odinger equations with a periodic nonlinear microstructure,” Physica D, vol. 217, pp. 31–57, 2006.
• [17] T.I. Lakoba, J. Yang, “A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity,” Journal of Computational, vol. 226, pp. 1668–1692, 2007. [18] A. Duran, J. Alvarez, “Petviashvili type methods for traveling wave computations: I. Analysis of convergence,” Journal of Computational and Applied Mathematics, vol. 266, pp. 29–51, 2014.
• [19] G.M. Muslu, H. Borluk, “Numerical solution for a general class of nonlocal nonlinear wave equations arising in elasticity,” ZAMM - Journal of Applied Mathematics and Mechanics, vol. 97, no. 12, pp. 1600-1610, 2017.
• [20] A. Duran, “An efficient method to compute solitary wave solutions of fractional Korteweg-de Vries equations,” International Journal of Computer Mathematics, vol. 95, pp. 1362–1374, 2018.
• [21] V.A. Dougalis, A. Duran, D. Mitsotakis, “Numerical approximation to Benjamin type equations. Generation and stability of solitary waves,” Wave Motion, vol. 85, pp. 34–56, 2019.
• [22] D. Olson, S. Shukla, G. Simpson, D. Spirn, “Petviashvilli’s method for the Dirichlet problem,” Journal of Scientific Computing, vol. 66, pp. 296–320, 2016.
• [23] Z.H. Musslimani and J. Yang, “Self-trapping of light in a two-dimensional photonic lattice,” Journal of the Optical Society of America B, vol. 21, no. 5, pp. 973-981, 2004.
• [24] I.L. Bogolubsky, “Some examples of inelastic soliton interaction,” Computer Physics Communications, vol. 13, pp. 149–155, 1977.
• [25] R.L. Pego, P. Smereka and M.I. Weinstein, “Oscillatory instability of solitary waves in a continuum model of lattice vibrations,” Nonlinearity, vol. 8, pp. 921–941, 1995.