This research deals with the study of various nonlinear wave processes in dispersive media by means of asymptotic methods developed upon existing exact methods in application to non-integrable systems. The aim of the research is to analyse wave models possessing solitary solutions and establish common features in the description of such solutions and classical particles. The new model equations have been derived for the description of long transverse waves propagating in the generalized atomic chain. The mathematical analogy between the model equations describing internal waves in stratified fluid (the Korteweg–de Vries and Gardner–Ostrovsky equations) and waves in discrete chain models (the generalized sine-Gordon–Toda model or Frenkel–Kontorova model) have been established. Chain models are described by sets of ODEs which can be readily solved with a high accuracy by existing well-developed solvers in mathematical software. The research includes solutions to important wave problems by means of approximate asymptotic and numerical methods. Results obtained provide an insight in understanding of details of nonlinear wave propagation in continuous and discrete media. An effective numerical code has been developed for the modeling of nonlinear phenomena both in continuous media and in the discrete models of interacting oscillators
