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Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 35--47 | DOI:10.5890/DNC.2017.03.004

Georgy Omel’yanov

Department of Mathematics, University of Sonora, Hermosillo, Sonora, Mexico

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Abstract

We consider an approach which allows to describe uniformly in time the process of collision of solitary waves. Next we apply it to the KdV-type equation with nonlinearity u4 for three interacting waves assuming that all wave trajectories intersect at the same point. The constructed asymptotic solution satisfies the equation in a weak sense and it can be treated as a classical asymptotics in the sense that it satisfies some conservation and balance laws associated with the gKdV-4 equation. Results of numerical simulation confirm the theoretical conclusion about the elastic type of the wave interaction.

Acknowledgments

The research was supported by SEP-CONACYT under grant 178690 (Mexico).

References

1. [1]	Schamel, H. (1973), A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons, J. Plasma Physics, 9, 377-387.

2. [2]	Kimiaki, Konno and Ichikawa, Yoshi H. (1974), A modified Korteweg de Vries equation for ion acoustic waves, J. Phys. Soc. Jpn. 37, 1631-1636

3. [3]	Rahman, O., Bhuyan, M., Haider, M., and Islam, J. (2014), Dust-acoustic solitary waves in an unmagnetized dusty plasma with arbitrarily charged dust fluid and trapped ion distribution, International Journal of Astronomy and Astrophysics, 4(1), DOI: 10.4236/ijaa.2014.41011

4. [4]	Ostrovsky, L. and Shrira, V. (1976), Instability and self-refraction of solitons, Sov. Phys. JETP, 44(4), 738-743.

5. [5]	Bona, J.L., Souganidis, P.E., and Strauss, W. (1987), Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A, 411(1841), 395-412.

6. [6]	Gorshkov, K.A. and Ostrovsky, L.A. (1981), Interaction of solitons in nonintegrable systems: direct perturbation method and applications, Physica D, 3, 428-438.

7. [7]	Ostrovsky, L.A. and Potapov, A.I. (1999), Modulated Waves: Theory and Applications, Johns Hopkins University Press, Baltimore, London.

8. [8]	Ostrovsky, L.A. and Gorshkov, K.A. (2000), Perturbation theories for nonlinear waves. In: Christiansen P., Soerensen M. (eds.) Nonlinear science at the down at the XXI century, Amsterdam, Elsevier, 47-65.

9. [9]	Ostrovsky, L. (2015), Asymptotic perturbation theory of waves, Imperial College Press, London.

10. [10]	Danilov, V. and Shelkovich, V. (1997), Generalized solutions of nonlinear differential equations and the Maslov algebras of distributions, Integral Transformations and Special Functions, 6, 137-146.

11. [11]	Danilov, V. and Omel’yanov, G. (2003), Weak asymptotics method and the interaction of infinitely narrow deltasolitons, Nonlinear Analysis: Theory, Methods and Applications, 54, 773-799.

12. [12]	Danilov, V. and Shelkovich, V. (2001), Propagation and interaction of shock waves of quasilinear equations, Nonlinear Studies, 8(1), 135-169.

13. [13]	Danilov, V., Omel’yanov, G., and Shelkovich, V. (2003)Weak asymptotics method and interaction of nonlinear waves, in: M.V. Karasev (Ed.), Asymptotic methods for wave and quantum problems, AMS Trans., Ser. 2, AMS: Providence, RI, 208, 33-164.

14. [14]	Danilov, V. and Shelkovich, V. (2005), Dynamics of propagation and interaction of delta-shock waves in conservation law systems, Journal of Differential Equations, 211(2), 333-381.

15. [15]	Panov, E. and Shelkovich, V. (2006), δ'-shock waves as a new type of solutions to systems of conservation laws, Journal of Differential Equations, 228(1), 49-86.

16. [16]	Kulagin, D. and Omel’yanov, G. (2006), Interaction of kinks for semilinear wave equations with a small parameter. Nonlinear Analysis, 65(2), 347-378.

17. [17]	Danilov, V. (2007),Weak asymptotic solution of phase-field system in the case of confluence of free boundaries in the Stefan problem with underheating. European Journal of Applied Mathematics, 18(05), 537-569.

18. [18]	Garcia,M. and Omel’yanov,G. (2009),Kink-antikink interaction for semilinear wave equations with a small parameter, Electron. J. Diff. Eqns., 2009(45), 1-26.

19. [19]	Omel’yanov,G. (2011), Uniform in time description of singularities interaction and the uniqueness problem, Nonlinear Phenomena in Complex Systems, 14(1), 38-48.

20. [20]	Omel’yanov, G. (2011), About the stability problem for strictly hyperbolic systems of conservation laws, Rend. Sem. Mat. Univ. Politec. Torino, 69(4), 377-392.

21. [21]	Danilov, V. and Mitrovic, D. (2011), Shock wave formation process for a multidimensional scalar conservation law, Quart. Appl. Math., 69(4), 613-634.

22. [22]	Garcia, M. and Omel’yanov, G. (2012), Interaction of solitary waves for the generalized KdV equation, Communications in Nonlinear Science and Numerical Simulation, 17(8), 3204-3218.

23. [23]	Kalisch, H. and Mitrovic, D. (2012), Singular solutions of a fully nonlinear 2×2 system of conservation laws, Proceedings of the Edinburgh Mathematical Society II, 55, 711-729.

24. [24]	Omel’yanov, G. and Valdez-Grijalva, M. (2014), Asymptotics for a C1-version of the KdV equation, Nonlinear Phenomena in Complex Systems, 17(2), 106-115.

25. [25]	Omel’yanov, G. (2015), Soliton-type asymptotics for non-integrable equations: a survey, Mathematical Methods in The Applied Sciences, 38(10), 2062-2071.

26. [26]	Maslov, V., and Omel’yanov, G. (1981), Asymptotic soliton-form solutions of equations with small dispersion. Russian Math. Surveys. 36, 73-149.

27. [27]	Maslov, V.P. and Omel’yanov, G.A. (2001), Geometric asymptotics for nonlinear PDE, AMS, MMONO Providence, RI, 202.