The perturbation method and exact solutions of nonlinear dynamics equations for media with microstructure

It is shown that exact soliton-like solutions of nonlinear evolution equations can be obtained by the direct perturbation method on the basis of the solution of the linearized equation. The solutions are the sums of the perturbation series calculated under the assumption that the series is geometric. The criterion for geometricity of the perturbation series is the equality of the sequential diagonal Pade approximants, the minimum order of which is determined by the order of the sought solution’s pole, obtained by analyzing the leading terms of the equation. Computational features of the method are demonstrated on the example of solving the Korteweg-de Vries equation. The system of equations for the sought functions of the perturbation series is given, the transformation of the perturbation series into a power series is demonstrated. It is shown that there exists a sequence of coinciding Pade approximants, the minimum order of which matches the order of the pole of the sought solution. Using the proposed computational method, classes of exact soliton-like solutions to a non-integrable fourth-order equation with an arbitrary degree of nonlinearity, simulating the propagation of nonlinear waves in granular media, are constructed. Classes of exact solutions to a generalized non-integrable equation of the sixth order with cubic nonlinearity are presented. The relationship between the coefficients of the sixth-order equation that is necessary for the existence of the exact soliton-like solutions is revealed. It is shown that in media with soft nonlinearity the exact solution has the form of the kink. In the case of hard nonlinearity of the media, a solitary wave has the form of a classical soliton. For effective use of the method it is necessary that the perturbation series contains all natural degrees of the series variable and the series characterizes the function with integer order of pole. For equations which have a pole of fractional orders, procedures for converting the power series to the required form are proposed.