Mathematical Modelling of Natural Phenomena

Research Article

On Numerical Solution of the Gardner–Ostrovsky Equation

M. A. Obregona1 and Y. A. Stepanyantsa2 c1

a1 E.T.S. Ingeniería Industrial, University of Malaga, Dr Ortiz Ramos s/n, 29071, Malaga, Spain

a2 Department of Mathematics and Computing, Faculty of Sciences, University of Southern Queensland, Toowoomba, Australia


A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovsky equation (ut + cux + α uux + α1u2ux + βuxxx)x = γu which is also known as the extended rotation-modified Korteweg–de Vries (KdV) equation. This equation is used for the description of internal oceanic waves affected by Earth’ rotation. Particular versions of this equation with zero some of coefficients, α, α1, β, or γ are also known in numerous applications. The proposed numerical scheme is a further development of the well-known finite-difference scheme earlier used for the solution of the KdV equation. The scheme is of the second order accuracy both on temporal and spatial variables. The stability analysis of the scheme is presented for infinitesimal perturbations. The conditions for the calculations with the appropriate accuracy have been found. Examples of calculations with the periodic boundary conditions are presented to illustrate the robustness of the proposed scheme.

(Online publication February 29 2012)

Key Words:

  • KdV equation;
  • Ostrovsky equation;
  • soliton;
  • terminal decay;
  • Petviashvili method;
  • numerical scheme

Mathematics Subject Classification:

  • 35Q51;
  • 35Q53;
  • 35Q90;
  • 37K40;
  • 65Y20;
  • 65Z05


c1 Corresponding author. E-mail: