The Perturbation Theory For Dark Soliton And Its Application

Soliton theory is one of the most important branches of nonlinear science, which has been applied wider and wider to many fields of physics. The systems with exact soliton solutions are usually highly idealized. The equations governing the real physical systems differ from the standard soliton ones, and when the differences between two equations are always very small, one usually deals with the real physical equations slightly different from standard ones. Thus, the soliton perturbation theory is of importance for the research into more real nonlinear systems. In this thesis, we mainly study the self-defocusing nonlinear Schr(?)dinger equation with perturbations(PNLS~+E). First, PNLS~+E is dealt with by using the modified semi-direct method. Secondly, we develop an approach characterized by the novel construction of a complete set of basis, which is completely independent on the inverse scattering transformation(IST) and is applicable to both integrable and unintegrable systems. Comparing with other methods, our method is characterized by a clearer idea and a simpler mathematical calculation. Thirdly, our method is applied to investigate two physical problems about dark solitons propagation in the nonlinear optical fiber.The whole thesis consists of six chapters. In the first chapter, we briefly introduce the research progress, present situation and main methods of soliton theories and perturbation problems of bright and dark solitons. In the second chapter, we introduce the perturbation theory for dark solitons based on the squared Jost solutions, and make some improvements of it. Then we study three representations for such a dark soliton perturbation theory. We find that different, results(such as the speeds of soliton, first-order corrections)will be obtained by using different representations. However, it turns that the physical results with different forms essentially are equivalent with each other.In the third chapter, we develop a purely direct perturbation theory of dark solitons. Setting out from PNLS~+E and introducing a slow-time scale, we obtain the matrix form for a linear operator. We build a set of complete bases through resolving the eigenvalue problem of this linear operator and prove its completeness and orthogonality. The evolution equations of the soliton parameters are given from conditions eliminating secular terms, and then the first-order correction is eventually obtained.In the fourth chapter, we apply the purely direct perturbation method to two practical problems in the optical fibers. First, we research the combined effects of the linear gain and the two-photon absorption on dark solitons. Then, we research the higher-order self-defocusing nonlinear Schr(?)dinger equation(HONLS~+E)for femtosecond pules in the optical fiber, and examine the effects of the third-order dispersion(TOD), self-steepening(SS) and selffrequency shift(SFS)on dark solitons. We obtain not only some results consistent with some previous ones by others but also some new results never found.In the fifth chapter, we introduce some other works about soliton problems we have done. First, we briefly mention the development process of the theoretical research about Bose-Einstein condensation(BEC) and its experimental research, and starting from the Gross-Pitaevaskii(GP)equation satisfied by macro-wavefunction of the condensate body and using the variational principle as well as the skill in our study of BEG problem, we deduce a quasi-one-dimensional GP equation and then obtain a dark-soliton(solitary wave)solution, so as to discuss the effect of the transverse structure of the condensate atom cloud on dark solitons. Secondly, we obtain a new exact solution of the Sine-Gordon(SG)equation through using the mixture of the Hirota direct method and some small skills. Thirdly, we improve the direct perturbation method about the NLS equation for bright solitons, which makes the method simpler in mathematics, clearer in reason and more convenient in application.In the last chapter of this thesis, we give a simple summary to the above works and prospect the direct perturbation theory of dark solitons.