Methods for exact solutions of nonlinear ordinary differential equations\

TL;DR

This paper surveys algorithmic methods for obtaining exact closed-form solutions of nonlinear autonomous ODEs that are polynomial in the derivatives, distinguishing necessary (global) strategies from ad hoc tricks. It relies on Eremenko's theorem to classify ODEs whose meromorphic solutions are elliptic or degenerate elliptic and outlines a finite, constructive pipeline (Briot-Bouquet, Hermite, subequation methods) to extract all such solutions within the privileged class. The authors illustrate the approach with concrete physics-driven examples (notably a fourth-order NLS dispersion and CGL5 variants), deriving new elliptic pulse solutions and detailing multiple routes to their explicit forms. They further discuss extensions to nonmeromorphic closed forms via Abel subequations and Lambert-function representations, expanding the scope beyond meromorphic solutions and opening avenues for richer exact solution families.

Abstract

In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by Eremenko, which can be turned into algorithms, thus avoiding \textit{ad hoc} assumptions, able to provide \textit{all} (as opposed to some) solutions in a precise class. To illustrate these methods, we present some new such exact solutions, physically relevent.

Figures (1)

• Figure 1: Raman scattering, periodic solution $u(x-x_0)$, $b_4=-1, b_2=1, \gamma=5, \omega=-21/5, a=1, g_2=24/5, g_3=-4/5$. The shift $x_0$ is a half-period of $\wp$: $x_0=1.337-1.198 i$, and