Darboux transformation for soliton solutions of the modified Kadomtsev-Petviashvili-II equation

Yıl 2018, Cilt: 3 Sayı: 3, 28 - 36, 31.12.2018

Mohamed R. Ali

Öz

Soliton solutions as far as hyperbolic cosines to the modified Kadomtsev–Petviashvili II equation are displayed. The
behaviour of each line soliton in the far region can be characterized analytically. It is revealed that under certain conditions, there may
appear an isolated bump in the interaction centre, which is much higher in peak amplitude than the surrounding line solitons, and the
more line solitons interact, the higher isolated bump will form. These results may provide a clue to generation of extreme
high-amplitude waves, in a reservoir of small waves, based on nonlinear interactions between the involved waves.

Anahtar Kelimeler

Darboux transformation, multisoliton solution, spectral problem, generalized Korteweg-de Vries equations

Kaynakça

• [1] J. S. Russell, Report on Waves, Report of the fourteenth meeting of the British Association for the Advancement of Science, York, September 1844.
• [2] V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9, 190 (1968).
• [3] A. Hasegawa and F. D. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys Lett. 23, 142 (1973).
• [4] L. F.Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers, Phys. Rev. Lett. 45, 1095 (1980).
• [5] Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, 2003.
• [6] B. A. Malomed, D. Mihalache, F.Wise, and L. Torner, Spatiotemporal optical solitons, J. Opt. B: Quantum Semiclass. Opt. 7, R53 (2005).
• [7] D. Mihalache, Linear and nonlinear light bullets: recent theoretical and experimental studies, Rom. J. Phys. 57, 352 (2012).
• [8] B. A.Malomed, Multidimensional solitons:Well-established results and novel findings, Eur. Phys. J. Spec. Top. 225, 2507 (2016).
• [9] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Formation of a matter-wave bright soliton, Science 296, 1290 (2002).
• [10] D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: from theory to experiments, J. Phys. A: Math. Theor. 43, 213001 (2010).
• [11] V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, BoseEinstein condensation: Twenty years after, Rom. Rep. Phys. 67, 5 (2015).
• [12] D. Mihalache, Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature, Rom. Rep. Phys. 69, 403 (2017).
• [13] M. J. Duff, R. R. Khuri, and J. X. Lu, String solitons, Phys. Rep. 259, 213 (1995).
• [14] Ph. Grelu and N. Akhmediev, Dissipative solitons for mode-locked lasers, Nature Photon. 6, 84 (2012).
• [15] H. A. Haus and W. S.Wong, Solitons in optical communications, Rev. Mod. Phys. 68, 423 (1996).
• [16] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for Solving the Korteweg–de Vries Equation, Phys. Rev. Lett. 19, 1095 (1967).
• [17] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.
• [18] A. B. Shabat, A. Gonzalez-L’opez, M.Ma’nas, L.Mart’inez Alonso, and M. A. Rodr´ıguez, New Trends in Integrability and Partial Solvability, Springer, Netherlands, 2004.
• [19] A. M. Wazwaz, Multi-front waves for extended form of modifed Kadomtsev-Petviashvili equation, Applied Mathematics and Mechanics. 32 (2011) 875–880.
• [20] HU Xiao-Rui, CHEN Yong, Binary Darboux Transformation for the Modified Kadomtsev–Petviashvili Equation, Chin.Phys.Lett. 25 (2008) 3840-3843.