The investigation of resonant interaction of some mathematical model of nonlinear waves

The dissertation focuses of the resonant interaction of the some mathematical model of nonlinear waves. Them described by differential equations with partial derivatives when nonlinearity is proportional to small parameter ε. Solving such tasks for applied specific methods the averaging and two-scale method, because in case of small parameter ε direct expansions have the so-called secular members εt; for this reason, asymptotic approximations are applied in short intervals of time only. A study on periodic tasks is compounded by the emergence of resonances, and therefore the systems must be averaged using a specific method accordingly to characteristics, when undetected functions – the estimates of the task averaged.

Of the work the following tasks are set: to investigate specific asymptotic analysis methods of differential equations; to validate the method of averaging according to characteristics in the case of quasilinear hyperbolic systems; to use the internal averaging method to solving practical tasks: of nonlinear oscillations of the absolutely elastic weightless string, one dimensional gas dynamics system.

The dissertation consists of an introduction, three chapters, four addends, main conclusions, bibliography and the list of the author’s publications. Introductory chapter covers the problem formulation and the object of research, the topicality of the thesis, the aims and objectives of the dissertation, the methodology of research, scientific novelty and the practical value of the achieved results. The defended thesis and structure of the dissertation are given in this chapter.

The first chapter presents problems, where with confront due soluting by differential equations and partial derivatives when nonlinear members included in the equations are proportional to small parameter ε. In second chapter the presence of the estimate of quasilinear hyperbolic systems and proximity to the exact estimate have been validated. The third chapter presents of resonant interaction of the some examples of mathematical model.

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DOI: https://doi.org/10.20334/2281-M