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Classical Solutions for the Coupled System gKdV Equations

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Abstract

In this article we investigate the coupled system of gKdV equations. A new topological approach is applied to prove the existence of at least one classical solution and at least two nonnegative classical solutions. The arguments are based upon recent theoretical results.

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REFERENCES

  1. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Nonlinear-evolution equations of physical significane,” Phys. Rev. Lett. 31, 125–127 (1973). https://doi.org/10.1103/PhysRevLett.31.125

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Alarcon, J. Angulo, and J. F. Montenegro, “Stability and instability of solitary waves for a nonlinear dispersive system,” Nonlinear Anal.: Theory, Methods Appl. 36, 1015–1035 (1999). https://doi.org/10.1016/S0362-546X(97)00724-4

    Article  MATH  Google Scholar 

  3. M. Panthee and M. Scialom, “On the Cauchy problem for a coupled system of KdV equations: Critical case,” Adv. Differ. Equations 13, 1–26 (2008). https://doi.org/10.57262/ade/1355867358

    Article  MathSciNet  MATH  Google Scholar 

  4. J. F. B. Montenegro, “Sistemas de equações de evolução não-lineares: estudo local, global e estabilidade de ondas solitárias,” PhD Thesis (Inst. de Math. Pura e Aplicada, Rio de Janeiro, 1995).

  5. X. Carvajal and M. Panthee, “Sharp well-posedness for a coupled system of mKdV-type equations,” J. Evol. Equations 19, 1167–1197 (2019). https://doi.org/10.1007/s00028-019-00508-6

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Angulo, J. L. Bona, F. Linares, and M. Scialom, “Scaling, stability and singularities for nonlinear, dispersive wave equations: The critical case,” Nonlinearity 15, 759–786 (2002). https://doi.org/10.1088/0951-7715/15/3/315

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Hakkaev and K. Kirchev, “Stability of solitary wave solutions of nonlinear dispersive system in a critical case,” Nonlinear Anal.: Theory, Methods Appl. 67, 2890–2899 (2007). https://doi.org/10.1016/j.na.2006.09.047

    Article  MATH  Google Scholar 

  8. X. Carvajal, L. Esquivel, and R. Santos, “On local well-posedness and ill-posedness results for a coupled system of mkdv type equations,” Discrete Continuous Dyn. Syst. 41, 2699–2723 (2021). https://doi.org/10.3934/dcds.2020382

    Article  MathSciNet  MATH  Google Scholar 

  9. A. J. Majda and J. A. Biello, “The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves,” J. Atmos. Sci. 60, 1809–1821 (2003). https://doi.org/10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2

    Article  MathSciNet  Google Scholar 

  10. T. Oh, “Diophantine conditions in well-posedness theory of coupled KdV-type sytsems: Local theory,” Int. Math. Res. Notes 2009, 3516–3556 (2009). https://doi.org/10.1093/imrn/rnp063

    Article  MATH  Google Scholar 

  11. J. Banaś, “On measures of noncompactness in Banach spaces,” Comm. Math. Univ. Carolinae 21, 131–143 (1980).

    MathSciNet  MATH  Google Scholar 

  12. P. Drábek and Ja. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, Birkhäuser Advanced Texts Basler Lehrbücher (Birkhäuser, Basel, 2013). https://doi.org/10.1007/978-3-0348-0387-8

  13. S. Djebali and K. Membarki, “Fixed point index theory for perturbation of expansive mappings by k-set contractions,” Topol. Methods Nonlinear Anal. 52, 613–640 (2019). https://doi.org/10.12775/TMNA.2019.055

    Article  MathSciNet  MATH  Google Scholar 

  14. A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equaions (CRC Press, New York, 2008). https://doi.org/10.1201/9781420010558

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Differential Equations, Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria

    S. G. Georgiev

  2. Department of Mathematics, Higher School of Technological Education, 21000, Skikda, Algeria

    A. Boukarou

  3. Department of Mathematics, College of Sciences and Arts, Qassim University, 55555, Ar-Rass, Saudi Arabia

    Kh. Zennir

Authors
  1. S. G. Georgiev
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  2. A. Boukarou
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  3. Kh. Zennir
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Corresponding authors

Correspondence to S. G. Georgiev, A. Boukarou or Kh. Zennir.

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Georgiev, S.G., Boukarou, A. & Zennir, K. Classical Solutions for the Coupled System gKdV Equations. Russ Math. 66, 1–15 (2022). https://doi.org/10.3103/S1066369X22120052

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  • DOI: https://doi.org/10.3103/S1066369X22120052

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