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. The order of the convergent is established to be not less than twice the pole order of the original equation’s solution. The effectiveness of the method is demonstrated on the solution of integrable 5th order equation of the Korteweg-de Vries family, 3rd order equation with 5 arbitrary constants, the Calogero-Degasperis-Fokas equation and the non-integrable Kuramoto-Sivashinsky equation. The analysis showed that in the case of integrable equations the continued fraction corresponding to the perturbation series terminates unconditionally, that is, the series is geometric or becomes so after regrouping the terms. For non-integrable equations the requirement of termination of the continued fraction that is equivalent to the geometricity of the perturbation series leads to the conditions on the original equation coefficients, which are necessary for the existence of exact soliton-like solutions. The advantages of the method, which can be easily implemented using any of the computer mathematics systems, include the ability to work with equations, the solution of which has a pole of zero, fractional or higher natural order.