Solitary Waves In One-dimensional Nonlinear Lattice Models

Soliton is a typical and important nonlinear phenomenon. The theory of solitonshas important application in many fields ranging from condensed matter physics andplasma physics to biophysics, optical fibers and geological systems. In1988, by usingthe rotating wave approximation and Green function method, Sievers and Takenofirstly showed that there exist intrinsic localized modes in the perfect lattices resultingfrom the interplay between discreteness and nonlinearity. Since then, a vast amount ofstudies has been poured to nonlinear localized modes in discrete nonlinear lattices. Inthe past few decades, people have found the classical solitary wave solutions andquantum solitary wave solutions in a large number of one-dimensional nonlinearcrystal lattices and achieved great progress, made important contributions to enrichingtheories of nonlinear physics and condensed matter physics, provided important newideas for designs of new materials, however, there is much work to be done in thisfield, such as quantum energy levels of solitary waves, quantum magnetic momentscarried by solitary waves, influences of quantum properties of solitary waves onphysical characteristics of systems, establishments of the semiclassical theory andquantum theory of interactions between the time-varying electromagnetic field andsolitary waves and various dynamical characteristics of this interaction and so on. Weanticipate that physical phenomena in nonlinear systems with solitary waveexcitations are very wonderful, interesting and attractive.In this paper, we will study solitary waves and quantum properties inone-dimensional nonlinear crystal lattices.In chapter one, we briefly introduce the basic theoretical concept of nonlinearscience, soliton, the history of the study of solitons, three nonlinear equations withsoliton solutions, present status and significance of the study of solitons.In chapter two, by using the number state method and the simplified method ofquasidiscreteness multiple scales, we have studied quantum solitary wave solutions ina quasi-one-dimensional molecular crystal model. In this model, there are both traveling and stationary quantum solitary waves. With the help of the obtainedquantum solitary wave solution, the energy levels of the quantum solitary wave havebeen investigated further, it is shown that the energy of the quantum solitary wave areof quantization, in this case, we possibly observe quantum thermal conduction in thematerial.In chapter three, using a simplified method of quasidiscreteness multiple scales,we have studied solitary wave solutions in a discrete nonlinear lattice with a nonlinearsubstrate potential, it is shown that the nonlinear substrate potential influences kineticproperties of the solitary wave, such as frequencies of the carrier wave, the groupvelocity, the vibration amplitude. On the other hand, we have calculated numericallynonlinear kinetic equations, and find that approximate solutions obtained with thehelp of multiple-scales methods are in good agreement with the results of numericalsimulations of computers.Finally, we give summary and prospect of our work.