Newton’s method of constructing exact solutions to nonlinear differential equations and non-integrable evolution equations

A modification of the Newton’s power series method for solving nonlinear ordinary equations and non-integrable evolution equations is proposed. In the first stage of the method the first few terms of a power series for the sought dependent variable are determined. For this we use either the direct power series expansion in independent variable, followed by substitution into the equation, or the decomposition into functional series of perturbation method in powers of the formal parameter. In the second case, a sequential solutions of the equations of the perturbation method allows us to express the terms of the series in the form of increasing natural degrees of exponential solution of the linearized problem and obtain a power series after the corresponding replacement. In the second stage of the method we postulate that the resulting power series is the geometric. For most integrable equations the power series is unconditionally geometric, in other words, found terms of the series form a sequence of geometric progression. For many non-integrable equations, there are conditions linking the coefficients of the equation with the parameters of the sought solution, under which the terms of the series form a geometric progression. In these cases, the sum of a geometric progression is the exact solution to the original equation. It is shown that the denominator of the progression is represented by a polynomial, the degree of which is not less than the pole order of the solution. The effectiveness of the method is demonstrated on a third-order nonlinear ordinary differential equation and the family of generalized Kuramoto-Sivashinski evolution equations, for which the exact rational and solitary-wave solutions are found. The advantages and disadvantages of the proposed method in comparison with other known methods of solving nonlinear differential equations are given.