Discontinuity, Nonlinearity, and Complexity
Wave Collision for the gKdV4 equation. Asymptotic Approach
Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 3547  DOI:10.5890/DNC.2017.03.004
Georgy Omel’yanov
Department of Mathematics, University of Sonora, Hermosillo, Sonora, Mexico
Download Full Text PDF
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 KdVtype 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 gKdV4 equation. Results of numerical simulation confirm the theoretical conclusion about the elastic type of the wave interaction.
Acknowledgments
The research was supported by SEPCONACYT under grant 178690 (Mexico).
References

[1]  Schamel, H. (1973), A modified Kortewegde Vries equation for ion acoustic waves due to resonant electrons, J. Plasma Physics, 9, 377387. 

[2]  Kimiaki, Konno and Ichikawa, Yoshi H. (1974), A modified Korteweg de Vries equation for ion acoustic waves, J. Phys. Soc. Jpn. 37, 16311636 

[3]  Rahman, O., Bhuyan, M., Haider, M., and Islam, J. (2014), Dustacoustic 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]  Ostrovsky, L. and Shrira, V. (1976), Instability and selfrefraction of solitons, Sov. Phys. JETP, 44(4), 738743. 

[5]  Bona, J.L., Souganidis, P.E., and Strauss, W. (1987), Stability and instability of solitary waves of Kortewegde Vries type, Proc. Roy. Soc. London Ser. A, 411(1841), 395412. 

[6]  Gorshkov, K.A. and Ostrovsky, L.A. (1981), Interaction of solitons in nonintegrable systems: direct perturbation method and applications, Physica D, 3, 428438. 

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

[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, 4765. 

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

[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, 137146. 

[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, 773799. 

[12]  Danilov, V. and Shelkovich, V. (2001), Propagation and interaction of shock waves of quasilinear equations, Nonlinear Studies, 8(1), 135169. 

[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, 33164. 

[14]  Danilov, V. and Shelkovich, V. (2005), Dynamics of propagation and interaction of deltashock waves in conservation law systems, Journal of Differential Equations, 211(2), 333381. 

[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), 4986. 

[16]  Kulagin, D. and Omel’yanov, G. (2006), Interaction of kinks for semilinear wave equations with a small parameter. Nonlinear Analysis, 65(2), 347378. 

[17]  Danilov, V. (2007),Weak asymptotic solution of phasefield system in the case of confluence of free boundaries in the Stefan problem with underheating. European Journal of Applied Mathematics, 18(05), 537569. 

[18]  Garcia,M. and Omel’yanov,G. (2009),Kinkantikink interaction for semilinear wave equations with a small parameter, Electron. J. Diff. Eqns., 2009(45), 126. 

[19]  Omel’yanov,G. (2011), Uniform in time description of singularities interaction and the uniqueness problem, Nonlinear Phenomena in Complex Systems, 14(1), 3848. 

[20]  Omel’yanov, G. (2011), About the stability problem for strictly hyperbolic systems of conservation laws, Rend. Sem. Mat. Univ. Politec. Torino, 69(4), 377392. 

[21]  Danilov, V. and Mitrovic, D. (2011), Shock wave formation process for a multidimensional scalar conservation law, Quart. Appl. Math., 69(4), 613634. 

[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), 32043218. 

[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, 711729. 

[24]  Omel’yanov, G. and ValdezGrijalva, M. (2014), Asymptotics for a C1version of the KdV equation, Nonlinear Phenomena in Complex Systems, 17(2), 106115. 

[25]  Omel’yanov, G. (2015), Solitontype asymptotics for nonintegrable equations: a survey, Mathematical Methods in The Applied Sciences, 38(10), 20622071. 

[26]  Maslov, V., and Omel’yanov, G. (1981), Asymptotic solitonform solutions of equations with small dispersion. Russian Math. Surveys. 36, 73149. 

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