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Rogue Wave Solutions Of Three Kinds Of Nonlinear Partial Diferential Equation

Posted on:2013-01-23Degree:MasterType:Thesis
Country:ChinaCandidate:S W XuFull Text:PDF
GTID:2230330362475515Subject:Applied Mathematics
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Rogue waves, which take place in a very short period of time, and the crest of which arevery high, are a kind of special waves. Rogue waves usually come without a sign. For instance,in a calm sea, a giant wave, which reach up to as high as dozens of meters, appears suddenly.In optics big bright spot can also appear, so mathematical physicists call them rogue waves.Recently mathematical physicists have conducted a lot of researches on rogue waves. Specially,the group of Professor Akhmediev obtain and analyse the rogue waves of nonlinear Schr¨odingerequation a litter fully. Recently, Professor M. S. Ruderman predict that there can be roguewaves of the Alfven waves in the sky. The Alfven waves can explained by the solutions ofDerivative Nonlinear Schr¨odinger (DNLS) equation and Variable Coefcient Derivative Non-linear Schr(o|¨)dinger (VCDNLS) equation. Moreover, considering from self-induced transparencyefect on the optical, and this efect can be described properly by the coupled system of nonlinearschrodinger (NLS) equation and maxwell-Bloch (MB) equation. In this paper, by the Darbouxtransformation, we got and classify the solutions of the DNLS equation, and in particular obtainthe traveling rational solution and rogue wave solutions by a Taylor series expansion about thebreather solutions. At the same time, considering the inhomogeneous conditions, the purposeto describe those physical problems needs the variable coefcient equations. So we consider allkinds of solutions of VCDNLS equation, and find reasonably the rogue wave solutions aboutthe explanation of the phenomenon. Then through a similar method, we find the rogue wavesolutions of the coupling system of nonlinear schr¨odinger equation and maxwell-Bloch equation(NLS-MB), and give dark rogue wave solutions.The thesis is arranged as follows:Chapter1. Reviews the history and development of rogue waves; introduce the developmentand research of the DNLS equation.Chapter2. The Darboux transformation and the determinant representation of the Dar-boux transformation of the DNLS equation is given.Chapter3. By the Darboux transformation, the bright soliton, dark soliton, breather solu-tion, rational traveling solution, rogue wave solutions of the DNLS equation are given explicitlyby diferent seed solutions.Chapter4. A general transformation which maps the variable coefcient derivative nonlin-ear Schr¨odinger equation to the DNLS equation is given with several arbitrary functions. Thesearbitrary functions provide a possibility to design an interesting integrable model. We obtain the rogue wave solutions of the VCDNLS equation through solutions of the derivative nonlinearSchr(o|¨)dinger equation.Chapter5. Using Darboux Transformation, with the aid of periodic seed solution, we findthe rogue wave solutions of NLS-MB system, and give dark rogue wave solutions.Chapter6. Conclusion and discussion.
Keywords/Search Tags:derivative nonlinear Schr(o|¨)dinger equation, variable coefcient derivative nonlinearSchr(o|¨)dinger equation, the coupling system of nonlinear schr(o|¨)dinger equation and maxwell-Blochequation, Darboux transformation, rogue wave
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