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Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations

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Abstract

Coupled nonlocal nonlinear wave equations describe the dynamics of wave systems with multiple interacting components in a wide range of physical applications. Investigating the long-time behavior of the solutions is crucial for understanding the stability, dynamics, and qualitative properties of these systems. In this study, solitary wave solutions to the coupled nonlocal nonlinear wave equations are generated numerically using the Petviashvili iteration method and the long-time behavior of the obtained solutions is investigated. In order to investigate the long-time behavior of the solutions obtained for the coupled nonlocal nonlinear wave equations, a Fourier pseudospectral method for space discretization and a fourth-order Runge–Kutta scheme for time discretization are proposed. Various numerical experiments are conducted to test the performance of the Petviashvili iteration method and the Fourier pseudo-spectral method. The proposed scheme can be used to solve the coupled nonlocal nonlinear system for arbitrary kernel functions representing the details of the atomic scale effects.

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References

  1. Duruk, N., Erbay, H.A., Erkip, A.: Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity. Nonlinearity 23(1), 107–118 (2010). https://doi.org/10.1088/0951-7715/23/1/006

    Article  MathSciNet  MATH  Google Scholar 

  2. Erbay, H.A., Erbay, S., Erkip, A.: On the convergence of the nonlocal nonlinear model to the classical elasticity equation. Physica D 427, 133010 (2021). https://doi.org/10.1016/j.physd.2021.133010

    Article  MathSciNet  MATH  Google Scholar 

  3. Erbay, H.A., Erbay, S., Erkip, A.: Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discret. Continuous Dyn. Syst. Ser. A 39(5), 2877–2891 (2019). https://doi.org/10.3934/dcds.2019119

    Article  MathSciNet  MATH  Google Scholar 

  4. Duruk, N., Erbay, H.A., Erkip, A.: Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations. J. Diff. Equ. 250(3), 1448–1459 (2011). https://doi.org/10.1016/j.jde.2010.09.002

    Article  MathSciNet  MATH  Google Scholar 

  5. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983). https://doi.org/10.1063/1.332803

    Article  MATH  Google Scholar 

  6. Wattis, J.A.D.: Solitary waves in a diatomic lattice: analytic approximations for a wide range of speeds by quasi-continuum methods. Phys. Lett. A 284(1), 16–22 (2001). https://doi.org/10.1016/S0375-9601(01)00277-8

    Article  MATH  Google Scholar 

  7. Christiansen, P.L., Lomdahl, P.S., Muto, V.: On a Toda lattice model with a transversal degree of freedom. Nonlinearity 4, 477–501 (1991). https://doi.org/10.1088/0951-7715/4/2/012

    Article  MathSciNet  MATH  Google Scholar 

  8. Khusnutdinova, K.R., Samsonov, A.M., Zakharov, A.S.: Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures. Phys. Rev. E 79, 056606 (2009). https://doi.org/10.1103/PhysRevE.79.056606

    Article  MathSciNet  MATH  Google Scholar 

  9. De Godefroy, A.: Blow up of solutions of a generalized Boussinesq equation. IMA J. Appl. Math. 60, 123–138 (1998). https://doi.org/10.1093/imamat/60.2.123

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, S., Li, M.: The Cauchy problem for coupled IMBq equations. IMA J. Appl. Math. 74(5), 726–740 (2009). https://doi.org/10.1093/imamat/hxp024

    Article  MathSciNet  MATH  Google Scholar 

  11. Turitsyn, S.K.: On a Toda lattice model with a transversal degree of freedom. Sufficient criterion of blow-up in the continuum limit. Phys. Lett. A 267(3), 173–267 (1993). https://doi.org/10.1016/0375-9601(93)90276-6

  12. Lazar, M., Maugin, G.A., Aifantis, E.C.: On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct. 43(6), 1404–1421 (2006). https://doi.org/10.1016/j.ijsolstr.2005.04.027

    Article  MathSciNet  MATH  Google Scholar 

  13. Duruk, N., Erkip, A., Erbay, H.A.: A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity. IMA J. Appl. Math. 74(1), 97–106 (2009). https://doi.org/10.1093/imamat/hxn020

    Article  MathSciNet  MATH  Google Scholar 

  14. Oruc, G., Muslu, G.M.: Existence and uniqueness of solutions to initial boundary value problem for the higher order Boussinesq equation. Nonlinear Anal. Real World Appl. 47, 436–445 (2019). https://doi.org/10.1016/j.nonrwa.2018.11.012

    Article  MathSciNet  MATH  Google Scholar 

  15. Oruc, G., Borluk, H., Muslu, G.M.: Higher order dispersive effects in regularized Boussinesq equation. Wave Motion 68, 272–282 (2017). https://doi.org/10.1016/j.wavemoti.2016.10.005

    Article  MathSciNet  MATH  Google Scholar 

  16. Canak, M.C., Muslu, G.M.: Error analysis of the exponential wave integrator sine pseudo-spectral method for the higher-order Boussinesq equation. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01763-6

    Article  MathSciNet  MATH  Google Scholar 

  17. Petviahvili, V.I.: Equation of an extraordinary soliton. Plasma Phys. 2, 469–472 (1976)

    MATH  Google Scholar 

  18. Pelinovsky, D.E., Stepanyants, Y.A.: Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations. SIAM J. Numer. Anal. 42(3), 1110–1127 (2004). https://doi.org/10.1137/S0036142902414232

    Article  MathSciNet  MATH  Google Scholar 

  19. Lakoba, T.I., Yang, J.: A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity. J. Comput. Phys. 226(2), 1668–1692 (2007). https://doi.org/10.1016/j.jcp.2007.06.009

    Article  MathSciNet  MATH  Google Scholar 

  20. Alvarez, J., Duran, A.: Petviashvili type methods for traveling wave computations: I. Analysis of convergence. J. Comput. Appl. Math. 266, 39–51 (2014). https://doi.org/10.1016/j.cam.2014.01.015

    Article  MathSciNet  MATH  Google Scholar 

  21. Alvarez, J., Duran, A.: Petviashvili type methods for traveling wave computations: II. Acceleration with vector extrapolation methods. Math. Comput. Simul. 123, 19–36 (2016). https://doi.org/10.1016/j.matcom.2015.10.015

    Article  MathSciNet  MATH  Google Scholar 

  22. Muslu, G.M., Borluk, H.: Numerical solution for a general class of nonlocal nonlinear wave equations arising in elasticity. Z. Angew. Math. Mech. 97(12), 1600–1610 (2017). https://doi.org/10.1002/zamm.201600023

    Article  MathSciNet  MATH  Google Scholar 

  23. Duran, A.: An efficient method to compute solitary wave solutions of fractional Kortewegde Vries equations. Int. J. Comput. Math. 95(6–7), 1362–1374 (2018). https://doi.org/10.1080/00207160.2017.1422732

    Article  MathSciNet  MATH  Google Scholar 

  24. Dougalis, V.A., Duran, A., Mitsotakis, D.: Numerical approximation to Benjamin type equations. Gener. Stab. Solitary Waves, Wave Motion 85, 34–56 (2019). https://doi.org/10.1016/j.wavemoti.2018.11.002

    Article  MATH  Google Scholar 

  25. Olson, D., Shukla, S., Simpson, G., Spirn, D.: Petviashvilli’s method for the Dirichlet problem. J. Sci. Comput. 66, 296–320 (2016). https://doi.org/10.1007/s10915-015-0023-6

    Article  MathSciNet  MATH  Google Scholar 

  26. Bona, J.L., Duran, A., Mitsotakis, D.: Solitary-wave solutions of Benjamin–Ono and other systems for internal waves. I. Approximations. Discrete Contin. Dynam. Syst. 41(1), 87–111 (2021). https://doi.org/10.3934/dcds.2020215

    Article  MathSciNet  MATH  Google Scholar 

  27. Pasinlioğlu, Ş, Muslu, G.M.: Solitary wave solutions to the general class of nonlocal nonlinear coupled wave equations. Düzce Univ. J. Sci. Technol. 12(2), 947–956 (2024). https://doi.org/10.29130/dubited.1249987

    Article  MATH  Google Scholar 

  28. Bogolubsky, I.L.: Some examples of inelastic soliton interaction. Comput. Phys. Commun. 13(3), 149–155 (1977). https://doi.org/10.1016/0010-4655(77)90009-1

    Article  MATH  Google Scholar 

  29. Borluk, H., Muslu, G.M.: A fourier Pseudospectral method for a generalized improved Boussinesq equation. Numer. Methods Part. Diff. Equ. 31(4), 995–1008 (2015). https://doi.org/10.1002/num.21928

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author very gratefully acknowledges the editor and the anonymous reviewers for their constructive comments and valuable suggestions, which improved the first draft of the paper.

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  1. Istanbul Technical University, Istanbul, Türkiye

    Şenay Pasinlioğlu

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Pasinlioğlu, Ş. Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations. Z. Angew. Math. Phys. 75, 209 (2024). https://doi.org/10.1007/s00033-024-02342-4

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