The Research On The Solutions And Their Properties To Some Differential-difference Equations In Nonlinear Mathematical Physics

With the progress of science and technology, nonlinear scientific research hasbeen rapid development. It has become an important symbol of the development ofmodern science. Nonlinear science involves many complex phenomena in nature andhas broad application prospect.One of the important achievements of nonlinear science is the soliton theory. Thesoliton theory is an important part of the application of mathematics and mathematicalphysics. It contains very rich in content and research methods. The soliton theory isstudied systematically by domestic and foreign scholars, especially in the past tenyears. The research team expands ceaselessly and achieves remarkable results. Solitontheory has successfully explained many physical, chemical, biological, geophysicalphenomena which are solved by the classical theory.The theory of solitons in optical soliton has always been the focus of research bythe scientists of Mathematical Physics. In nonlinear optics, the optical soliton is theterm used to describe the optical pulse propagation, but it is nonlinear wave equationswith localized traveling wave solution in Mathematics.Soliton theory has many applications in the nonlinear optics. Such as Bose-Einstein, Condensates (BECs), et al. Especially, the soliton dynamics and the stabilityanalysis in the nonlinear optics and BECs are very important for the solitoncommunication, BEC theory and its application. The research on these topics canprovide theoretical support for ultra-fast, big capacitance, long distance fibercommunication system, the applications and development of the nonlinear optics andBECs, and can be enrich and develop the theory and applications of the nonlinearscience in the mathematics.If the soliton is not stable in the process of communication, then either thetheoretical research or practical research will lost its meaning. Therefore, the study of the soliton stability analysis is very important.This paper will use perturbation method and two different extended (G’/G)-expansion method to the study the nonlinear differential-difference equation solitarywave solutions and their linear stability.Firstly, the exact solutions of the discrete complex cubic Ginzburg–Landauequation are derived using homogeneous balance principle and the (G’/G)-expansionmethod, and then linear stability of exact solutions is discussed. The extended (G’/G)-expansion method is improved. The application of the new method is put to the higherorder nonlinear differential difference equation. The QDNLS equation is studied,bright solitons, dark solitons, alternating phase solitons, trigonometric functionperiodic wave solutions and rational wave solutions with the arbitrary parameters areobtained. The linear stability of the bright soliton, dark soliton and rational wavesolution is analyzed using the perturbation method, and the conditions which stablesolitary wave solutions satisfy are presented. The stable solitary wave solutions ofQDNLS are useful to understand the complicated physical phenomena.In this paper, with the use of the (G’/G)-expansion method for solving nonlineardifferential-difference equation, the equation can be obtained bright soliton solutions,dark soliton solutions, triangle function periodic wave solution with more arbitraryparameters are derived. When some special values of these parameters are taken, theresults can be re-obtained in the articles published, and the conditions which stablesolitary wave solutions satisfy are presented.These exact solutions have important theoretical significance and scientific value.With the aids of the exact solutions, many phenomena described by the solitonequations can be understood well, such as the exciton motion in the molecular crystals,the propagation of discrete self-trapped beams in weakly-coupled nonlinear opticalwaveguides, the Bose-Einstein condensation in the periodical potential, the energystorage and transmission in the molecular chain in solid physics, nonlinear optics,condensed matter physics, biophysics, etc.