Something hidden in the Fourier series and its partial sum: Solitary wave solutions to polynomial nonlinear differential equations

Abstract

Transformation proposed in the author's recent work is utilized to obtain the polynomial and rational solitary wave solutions of polynomial nonlinear 1 + 1-dimension evolution equations, and thus some special solutions of polynomial nonlinear ODEs. In particular, several theorems are established to provide the necessary and sufficient conditions for the existence of these solutions. With the new transformation, polynomial solitary waves can be hidden in the form of a partial sum of the Fourier series of an intermediate variable (a curvilinear coordinate system), and therefore the balance between instabilities (nonlinear) and dispersive, dissipative, and other effects is well represented in a given polynomial nonlinear differential equation. A direct generalization of the method yields a solitary wave-like solution to the cylindrical KdV equation. In addition, we prove that the main transformation in this paper is equivalent to a simplified sine-Gordon equation. Finally, some new (complex) traveling wave solutions of the KdV equation, the Burgers equation and the modified KdV equation are obtained via our method. Among those new solutions, one of the KdV equation is shown taking similar form to the solitary wave solution of the nonlinear Schrödinger equation, which once again exposes the relationship between the two well-known nonlinear wave equations.