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A numerical study of compactons

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  • Ismail, M.S.
  • Taha, T.R.

Abstract

The Korteweg–de Vries equation has been generalized by Rosenau and Hyman [Compactons: Solitons with finite wavelength, Phys. Rev. Lett. 70(5) (1993) 564] to a class of partial differential equations that has soliton solutions with compact support (compactons). Compactons are solitary waves with the remarkable soliton property that after colliding with other compactons, they re-emerge with the same coherent shape [Rosenau and Hyman, Compactons: Solitons with finite wave length, Phys. Rev. Lett. 70(5) (1993) 564]. In this paper finite difference and finite element methods have been developed to study these types of equations. The analytical solutions and conserved quantities are used to assess the accuracy of these methods. A single compacton as well as the interaction of compactons have been studied. The numerical results have shown that these compactons exhibit true soliton behavior.

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  • Ismail, M.S. & Taha, T.R., 1998. "A numerical study of compactons," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 47(6), pages 519-530.
    • Handle: RePEc:eee:matcom:v:47:y:1998:i:6:p:519-530
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    References listed on IDEAS

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    1. Thiab R. Taha, 1994. "NUMERICAL SIMULATION OF THE KdV-MKdV EQUATION," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 407-410.
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    Cited by:

    1. Wazwaz, A.M., 2001. "A study of nonlinear dispersive equations with solitary-wave solutions having compact support," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 56(3), pages 269-276.
    2. Wazwaz, A.M., 2002. "General compactons solutions and solitary patterns solutions for modified nonlinear dispersive equations mK(n,n) in higher dimensional spaces," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 59(6), pages 519-531.
    3. Wazwaz, Abdul-Majid, 2003. "An analytic study of compactons structures in a class of nonlinear dispersive equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 63(1), pages 35-44.
    4. Yadong, Shang, 2005. "Explicit and exact special solutions for BBM-like B(m,n) equations with fully nonlinear dispersion," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1083-1091.
    5. Odibat, Zaid M., 2009. "Exact solitary solutions for variants of the KdV equations with fractional time derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1264-1270.
    6. Yin, Jun & Lai, Shaoyong & Qing, Yin, 2009. "Exact solutions to a nonlinear dispersive model with variable coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1249-1254.
    7. Chen, Yong & Li, Biao & Zhang, Hongqing, 2004. "New exact solutions for modified nonlinear dispersive equations mK(m,n) in higher dimensions spaces," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(5), pages 549-559.
    8. Kuru, S., 2009. "Compactons and kink-like solutions of BBM-like equations by means of factorization," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 626-633.
    9. Xu, Chuanhai & Tian, Lixin, 2009. "The bifurcation and peakon for K(2,2) equation with osmosis dispersion," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 893-901.
    10. Wazwaz, Abdul-Majid, 2005. "Generalized forms of the phi-four equation with compactons, solitons and periodic solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(5), pages 580-588.
    11. Wazwaz, Abdul-Majid & Taha, Thiab, 2003. "Compact and noncompact structures in a class of nonlinearly dispersive equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(1), pages 171-189.
    12. Rus, Francisco & Villatoro, Francisco R., 2007. "Padé numerical method for the Rosenau–Hyman compacton equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 76(1), pages 188-192.
    13. Ahmat, Muyassar & Qiu, Jianxian, 2023. "Direct WENO scheme for dispersion-type equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 216-229.
    14. Garralon-López, Rubén & Rus, Francisco & Villatoro, Francisco R., 2023. "Robustness of the absolute Rosenau–Hyman |K|(p,p) equation with non-integer p," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).

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