ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

nonlinear evolution equations

Continued fractions, the perturbation method and exact solutions to nonlinear evolution equations

A new method is proposed in which constructing exact solutions to nonlinear evolution equations is based on successive applying the perturbation method and apparatus of the continued fractions. It is shown that exact solitary-wave solutions arise in the limiting case as the sum of geometric series of the perturbation method based on the linearized problem. It is demonstrated that the continued fraction corresponding to the perturbation series, terminates to a convergent giving an expression for the desired exact soliton-like solution.

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 equa- tions 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.