The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

Abstract

We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi-Pasta-Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlevé approach. We show that the equation does not pass the Painlevé test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth-order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.set of approaches was used to explain it (see for example [2][3][4][5][6][7]). The main ideas of these approaches and their results can be found in work [8].Usually α or β FPU model is considered in literature, but in paper [9] the α + β FPU model is investigated. In this paper, we also use the α + β model but we use the continuous limit approximation for the investigation.This approach was used for the first time by N. Zabusky and M. Kruskal in work [10], where the authors derived the Korteweg-de Vries (KdV) equation for the description of the α FPU model. This equation is integrable by the inverse scattering transform and has been widely studied by now ( [11, 12]). Zabusky and Kruskal explained the FPU paradox (recurrence of initial conditions) using the numerical simulation of the KdV equation. Another result of work [10] was an introduction of soliton. In article [13] the author took into account high order (in comparison with [10]) terms in the Taylor series expansions for the continuous limit approximation of the α FPU model. The generalized KdV equation was obtained for a more accurate description of wave processes in the FPU model. Some exact solutions of the derived equation were found . It is shown [13] that if one takes into account more terms in the Taylor series and acts in analogy to paper [10] then he can not derive an integrable equation for the description of wave processes in the FPU mass chain. Recurrence of initial statement is also unrepresentative for wave processes, described by the fifth-order equation with arbitrary initial conditions. In paper [14] wave processes for the generalized KdV equation are simulated numerically.This brings up the question, whether the situation is similar for the β and the α + β FPU models? It is known [11] that one can obtain the Gardner equation for the description of the α + β FPU model using the continuous limit. The Gardner equation is integrable by the inverse scatterring transform and can be reduced to the modified KdV equation [15]. If we take more terms in the Taylor series expansion, we can derive the fifth-order partial differential ...

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