Equations with the Fermi-Pasta-Ulam and dislocations nonlinearity

Issue. The class of Fermi–Pasta–Ulam equations and equations describing dislocations are investigated. Being a bright representative of integrable equations, they are of interest both in theoretical constructions and in applied research. Investigation methods. In the present work, a model combining these two equations is considered, and local dynamic properties of solutions are investigated. An important feature of the model is the fact that the infinite set of characteristic numbers of the equation linearized at zero consists of purely imaginary values. Thus, the critical case of infinite dimension is realized in the problem on the stability of the zero solution. In this case a special asymptotic method for construction of the so-called normalized equations is used. Using such equations, we determine the main part of the solutions of the original equation, after that we can investigate the asymptotic behavior using perturbation theory methods. Results. All solutions are naturally divided into two classes: regular solutions that smoothly depend on a small parameter entering the equation, and irregular ones, which are a superposition of functions that oscillate rapidly on a spatial variable. For each class of solutions, areas of such changes in the parameters of the equation are distinguished in which the main parts are described by different normalized equations. Sufficiently wide classes of such equations are presented, which include, for example, the families of the Schrödinger, Korteweg–de Vries, and other equations. The problem of determining such a set of parameters of the original equation for which the nonlinearity of dislocations and the nonlinearity of the FPU are comparable «in force» is considered, i.e. none of them can be neglected in the first approximation. Discussion. It is interesting to note that for regular and irregular solutions the areas of parameters in which nonlinearities are comparable are different. In the second case the corresponding region is much wider. The article consists of two chapters. In the first chapter, normalized equations for regular solutions are constructed, and in the second, for irregular ones. In turn, the first chapter is divided into three parts, in each part different normalized equations are constructed (depending on the values of the parameters).