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Weak Multi-Phase Asymptotics for Nonintegrable Equations

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Abstract

We describe an approach that allows us to construct multi-soliton asymptotic solutions for essentially nonintegrable equations and analyze the type of wave collision. The general idea is demonstrated by the example of a generalization of Korteweg-de Vries equation with small dispersion in the cases of two and three waves. We also discuss the phenomenon of nonuniqueness that occurs in the framework of the weak description of a multi-soliton interaction.

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References

  1. A. V. Faminskii, “Cauchy Problem for the Korteweg-de Vries Equation and Its Generalizations”, J. Sov. Math., 50:1 (1990), 1381–1420.

    Article  MathSciNet  Google Scholar 

  2. J. L. Bona, P. E. Souganidis, and W. Strauss, “Stability and Instability of Solitary Waves of Korteweg-de Vries Type”, Proc. Roy. Soc. London Ser. A, 411:1841 (1987), 395–412.

    ADS  MathSciNet  MATH  Google Scholar 

  3. H. Schamel, “A Modified Korteweg-de Vries Equation for Ion Acoustic Waves Due to Resonant Electrons”, J. Plasma Phys., 9 (1973), 377–387.

    Article  ADS  Google Scholar 

  4. K. Konno and Y. H. Ichikawa, “A Modified Korteweg de Vries Equation for Ion Acoustic Waves”, J. Phys. Soc. Japan, 37 (1974), 1631–1636.

    Article  ADS  Google Scholar 

  5. O. Rahman et al, “Dust-Acoustic Solitary Waves in an Unmagnetized Dusty Plasma with Arbitrarily Charged Dust Fluid and Trapped Ion Distribution”, Int. J. Astron. Astrophys., 4:1 (2014).

    Google Scholar 

  6. M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.

    Book  Google Scholar 

  7. V. Karpman and E. Maslov, “Perturbation Theory for Solitons”, J. Exp. Theor. Phys., 46:2 (1977), 281–291.

    ADS  MathSciNet  Google Scholar 

  8. D. Kaup and A. Newell, “Solitons as Particles, Oscillators and in Slowly Changing Media: a Singular Perturbation Theory”, Proc. R. Soc. London Ser. A., 361 (1978), 413–446.

    Article  ADS  Google Scholar 

  9. K. A. Gorshkov and L. A. Ostrovsky, “Interaction of Solitons in Nonintegrable Systems: Direct Perturbation Method and Applications”, Phys. D, 3 (1981), 428–438.

    Article  Google Scholar 

  10. L. Ostrovsky, Asymptotic Perturbation Theory of Waves, Imperial College Press, London, 2015.

    MATH  Google Scholar 

  11. R. Dingwall et al,, “Non-Integrable Dynamics of Matter-Wave Solitons in a Density-Dependent Gauge Theory”, New J. Phys., 20:4 (2018), 1–20.

    Google Scholar 

  12. V. Danilov and V. Shelkovich, “Generalized Solutions of Nonlinear Differential Equations and the Maslov Algebras of Distributions”, Integral Transforms Spec. Funct., 6 (1997), 137–146.

    MathSciNet  Google Scholar 

  13. V. Danilov and G. Omel’yanov, “Weak Asymptotics Method and the Interaction of Infinitely Narrow Delta-Solitons”, Nonlinear Anal Theory Methods Appl., 54 (2003), 773–799.

    Article  Google Scholar 

  14. V. Danilov, G. Omel’yanov and V. Shelkovich, “Weak Asymptotics Method and Interaction of Nonlinear Waves”, M.V. Karasev (Ed.) Asymptotic Methods for Wave and Quantum Problems, Amer. Math. Soc. Transl. Ser., Providence, RI, 2003, 33–165.

    MathSciNet  MATH  Google Scholar 

  15. D. Kulagin and G. Omel’yanov, “Asymptotics of Kink-Kink Interaction for Sine-Gordon Type Equations”, Math. Notes, 75:4 (2004), 563–567.

    Article  MathSciNet  Google Scholar 

  16. V. Danilov and V. Shelkovich, “Dynamics of Propagation and Interaction of Delta-Shock Waves in Conservation Law Systems”, Differ. Equ., 211:2 (2005), 333–381.

    Article  ADS  Google Scholar 

  17. M. Garcia and G. Omel’yanov, “Kink-Antikink Interaction for Semilinear Wave Equations with a Small Parameter”, Electron. J. Differential Equations, 45 (2009), 1–26.

    Article  MathSciNet  Google Scholar 

  18. V. Danilov and D. Mitrovic, “Shock Wave Formation Process for a Multidimensional Scalar Conservation Law”, Quart. Appl. Math., 69:4 (2011), 613–634.

    Article  MathSciNet  Google Scholar 

  19. H. Kalisch and D. Mitrovic, “Singular Solutions of a Fully Nonlinear \(2\times2\) System of Conservation Laws”, Proc. Edinb. Math. Soc. 2, Cambridge Univ. Press, Cambridge, 2012, 711–729.

    Article  MathSciNet  Google Scholar 

  20. M. Garcia and G. Omel’yanov, “Interaction of Solitary Waves for the Generalized KdV Equation”, Commun. Nonlinear Sci., 17:8 (2012), 3204–3218.

    Article  MathSciNet  Google Scholar 

  21. G. Omel’yanov and I. Segundo Caballero, “Interaction of Solitons for Sine-Gordon-Type Equations”, J. Math., (2013), 1–8.

    Article  MathSciNet  Google Scholar 

  22. M. Garcia and G. Omel’yanov, “Interaction of Solitons and the Effect of Radiation for the Generalized KdV Equation”, Commun. Nonlinear Sci., 19:8 (2014), 2724–2733.

    Article  MathSciNet  Google Scholar 

  23. G. Omel’yanov and M. Valdez-Grijalva, “Asymptotics for a \(C^1\)-Version of the KdV Equation”, Nonlinear Phenom. Complex Syst., 17:2 (2014), 106–115.

    MathSciNet  MATH  Google Scholar 

  24. G. Omel’yanov, “Soliton-Type Asymptotics for Non-Integrable Equations: a Survey”, Math. Methods Appl. Sci., 38:10 (2015), 2062–2071.

    Article  ADS  MathSciNet  Google Scholar 

  25. G. Omel’yanov, “Propagation and Interaction of Solitons for Nonintegrable Equations”, Russ. J. Math. Phys., 23 (2016), 225–243.

    Article  MathSciNet  Google Scholar 

  26. E. Abreu, M. Colombeau, and E. Panov, “Weak Asymptotic Methods for Scalar Equations and Systems”, J. Math. Anal. Appl., 444:2 (2016), 1203–1232.

    Article  MathSciNet  Google Scholar 

  27. M. Colombeau, “Asymptotic Study of the Initial Value Problem to a Standard One Pressure Model of Multifluid Flows in Nondivergence Form”, Differ. Equations, 260:1 (2016), 197–217.

    Article  MathSciNet  Google Scholar 

  28. G. Omel’yanov, “Multi-Soliton Collision for Essentially Nonintegrable Equations”, M. Oberguggenberger at al (Eds.) Generalized Functions and Fourier Analysis, Birkhäuser, Basel, 2017, 153–170.

    MathSciNet  MATH  Google Scholar 

  29. G. Omel’yanov, “Collision of Solitons in Non-Integrable Versions of the Degasperis-Procesi Model”, Chaos Solitons Fractals, 136 (2020).

    Article  MathSciNet  Google Scholar 

  30. V. Maslov and G. Omel’yanov, “Asymptotic Soliton-Form Solutions of Equations with Small Dispersion”, Russ. Math. Surv., 36 (1981), 73–149.

    Article  Google Scholar 

  31. V. G. Danilov, V. P. Maslov, and K. A. Volosov, Mathematical Modeling of Heat and Mass Transfer Processes, Kluwer, Dordrecht, 1995.

    Book  Google Scholar 

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Authors and Affiliations

  1. University of Sonora, Sonora, Mexico

    G. A. Omel’yanov

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  1. G. A. Omel’yanov
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Correspondence to G. A. Omel’yanov.

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Omel’yanov, G.A. Weak Multi-Phase Asymptotics for Nonintegrable Equations. Russ. J. Math. Phys. 28, 84–95 (2021). https://doi.org/10.1134/S106192082101009X

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  • DOI: https://doi.org/10.1134/S106192082101009X

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