Abstract
We study the physical and collision properties of the combined KdV–mKdV solitons given by the Gardner equation which possess solitary wave solution characterized by sech function. A collision of the two solitary waves produces 2-soliton solution. We make a physical form of the 2-soliton solution where the fast soliton moves with speed c 1 and the slow soliton moves with speed c 2. In the collision described by the 2-soliton solution, the solitary waves preserve their shapes and speeds, but get a shift in position where the fast soliton overtakes the slow soliton if their speeds have same direction, and two solitons cross head-on if their speeds have opposite direction. For a collision there exist three different types of interactions which depend on the relative ratio \(c_1/c_2\) of speeds and the relative orientation of the two solitary waves.
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References
Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)
Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1989)
Fu, Z., Liu, S., Liu, S.: New kinds of solutions to Gardner equation. Chaos, Solitons & Fractals 20(2), 301–309 (2004)
Slyunaev, A.V., Pelinovski, E.N.: Dynamics of large-amplitude solitons. J. Exp. Theor. Phys. 89(1), 173–181 (1999)
Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations. J. Math. Phys. 28(9), 2094–2101 (1987)
Hirota, R.: Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27(18), 1192–1194 (1971)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
Wazwaz, A.M.: New solitons and kink solutions for the Gardner equation. Commun. Nonlinear Sci. Numer. Simul. 12(8), 1395–1404 (2007)
Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijing (2009)
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Mia, A. (2015). Collision Effects of Solitary Waves for the Gardner Equation. In: Cojocaru, M., Kotsireas, I., Makarov, R., Melnik, R., Shodiev, H. (eds) Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. Springer Proceedings in Mathematics & Statistics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-12307-3_44
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DOI: https://doi.org/10.1007/978-3-319-12307-3_44
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