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Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations

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Abstract

We investigate two Boussinesq equations where the fourth-order terms come with minus and plus signs. We show that the Boussinesq equation with minus fourth-order term gives multiple soliton solutions, whereas the model with the plus fourth-order term gives multiple complex soliton solutions. We show that the two models are characterized by real and complex dispersion relations, respectively. Moreover, we derive other solitonic, singular, and periodic solutions for each model.

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References

  1. Adhikari, D., Cao, C., Wu, J.: The 2D Boussinesq equations with vertical viscosity and vertical diffusivity. J. Differ. Equ. 249, 1078–1088 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biswas, A.: Solitary waves for power-law regularized long-wave equation and R(m, n) equation. Non. Dyn. 59, 423–426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biswas, A., Milovicb, D., Ranasinghec, A.: Solitary waves of Boussinesq equation in a power law media. Comm. Non. Sci. Numer. Simul. 14(11), 3738–3742 (2009)

    Article  Google Scholar 

  4. Biswas, A.: Solitary waves for power-law regularized long wave equation and R(m, n) equation. Nonlinear Dyn. 59(3), 423–426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biswas, A., Khalique, C.M.: Stationary solitons for nonlinear dispersive Schrodinger’s equation. Nonlinear Dyn. 63(4), 623–626 (2011)

    Article  MathSciNet  Google Scholar 

  6. Krishnan, E.V., Kumar, S., Biswas, A.: Solitons and other nonlinear waves of the Boussinesq equation. Non. Dyn. 70, 1213–1221 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small–amplitude long waves in nonlinear dispersive media. I: derivation and linear theory. J. Nonlinear Sci 12, 283–318 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long dun canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J Math Pures Appl 17, 55–108 (1872)

    MathSciNet  Google Scholar 

  9. Bryan, A.C., Haines, C.R., Stuart, A.E.: Complex solitons and poles of the sine-Gordon equation. Lett. Math. Phys. 2(6), 445–449 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Christov, C.I., Maugin, G.A., Velarde, M.G.: Well-posed Boussinesq paradigm with purely spatial higher-order derivatives. Phys. Rev. E 54(4), 3621–3638 (1996)

    Article  Google Scholar 

  11. Dehghan, M., Salehi, R.: A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation. Appl. Math. Model. 36, 1939–1956 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hereman, W., Nuseir, A.: Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simul. 43, 13–27 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  14. Hu, X.-B., Lou, S.-Y.: Nonlocal symmetries of nonlinear integrable models. Proc. Inst. Math. NAS Ukraine 30(1), 120–126 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Khastgir, S.P., Sasaki, R.: Instability of solitons in imaginary coupling affine Toda field theory. Prog. Theor. Phys. 95(3), 485–501 (1996)

    Article  MathSciNet  Google Scholar 

  16. Lu, K., Ma, W.X., Yu, J., Lin, F.H., Khalique, C.M.: Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model. Nonlinear Dyn. 82, 1211–1220 (2015)

    Article  MathSciNet  Google Scholar 

  17. Lu, X., Ma, W.X., Yu, J., Khalique, C.M.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrdinger equation. Commun. Nonlinear Sci. Numer. Simul. 31, 40–46 (2016)

    Article  MathSciNet  Google Scholar 

  18. Shi, C., Zhao, B., Ma, W.X.: Exact rational solutions to a Boussinesq-like equation in (1 + 1)-dimension. Appl. Math. Lett. 48, 170–176 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leblond, H., Mihalache, D.: Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523, 61–126 (2013)

  20. Leblond, H., Mihalache, D.: Few-optical-cycle solitons: modified Kortewegde Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models. Phys. Rev. A 79, 063835 (2009)

    Article  Google Scholar 

  21. Triki, H., Jovanoski, Z., Biswas, A.: Dynamics of two-layered shallow water waves with coupled KdV equations. Rom. Rep. Phys. 66, 251–261 (2014)

    Google Scholar 

  22. Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theorem. Springer and HEP, Berlin (2009)

    Book  MATH  Google Scholar 

  23. Wazwaz, A.M.: \(N\)-soliton solutions for the Vakhnenko equation and its generalized forms. Phys. Scr. 82, 065006 (2010)

    Article  MATH  Google Scholar 

  24. Wazwaz, A.M.: A new generalized fifth-order nonlinear integrable equation. Phys. Scr. 83, 035003 (2011)

    Article  MATH  Google Scholar 

  25. Wazwaz, A.M.: Distinct variants of the KdV equation with compact and noncompact structures. Appl. Math. Comput 150, 365–377 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Wazwaz, A.M.: New solitons and kinks solutions to the Sharma–Tasso–Olver equation. Appl. Math. Comput. 188, 1205–1213 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. (2015). doi:10.1007/s11071-015-2349-x

  28. Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3 + 1)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dynamics (2015). doi:10.1007/s11071-015-2427-0

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Saint Xavier University, Chicago, IL, 60655, USA

    Abdul-Majid Wazwaz

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  1. Abdul-Majid Wazwaz
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Correspondence to Abdul-Majid Wazwaz.

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Wazwaz, AM. Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations. Nonlinear Dyn 85, 731–737 (2016). https://doi.org/10.1007/s11071-016-2718-0

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  • DOI: https://doi.org/10.1007/s11071-016-2718-0

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