Exact solutions of differential equations: renormalization group based polynomial scheme

Abstract A polynomial scheme is proposed here to compute exact solutions of nonlinear partial differential equations (NPDEs) based on a series expansions of solutions and a renormalization group (RG) related resummation. The most salient feature of the current approach is that only linear algebraic equations need to be solved to implement the resummation for closed-form exact solution and parameter dependence, which does not require any sophisticated analysis like Cole–Hopf transformation or Painlevé test. New exact solutions of typical NPDEs are computed with this novel method, including one- and two-soliton (solitary wave) solutions, periodic solutions of exponential or elliptic function type. Moreover, exact reduced equations may also be conveniently computed for further analysis.