AIMS Mathematics, 2017, 2(4): 610-621. doi: 10.3934/Math.2017.4.610

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A geometric formulation of Lax integrability for nonlinear equationsin two independent variables

Department of Mathematics, University of Texas, TX 78539-2999 Edinburg, USA

A geometric formulation of Lax integrability is introduced which makes use of a Pfaffianformulation of Lax integrability. The Frobenius theorem gives a necessary and sufficient conditionfor the complete integrability of a distribution, andprovides a powerful way to study nonlinear evolution equations. This permits an examination of the relationbetween complete integrability and Lax integrability. The prolongation method is formulated in this contextand gauge transformations can be examined in terms ofdifferential forms as well as the Frobenius theorem.
  Article Metrics


1. W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, NY, 1975.

2. P. Bracken, Integrability and Prolongation Structure for a Generalized Korteweg-de Vries Equation, Pacific J. Math., 2, (2009), 293-302.

3. P. Bracken, A Geometric Interpretation of Prolongation by Means of Connections, J. Math. Phys., 51, (2010), 113502.

4. P. Bracken, Geometric Approaches to Produce Prolongations for Nonlinear Partial Differential Equations, Int. J. Geom. Methods M., 10, (2013), 1350002.

5. P. Bracken, An Exterior Differential System for a Generalized Korteweg-de Vries Equation and its Associated Integrability, Acta Appl. Math., 95, (2007), 223-231.

6. S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry, World Scientific, Singpore, 1999.

7. C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Muira, A method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, (1967), 1095-1097.

8. P. D. Lax, Integrals of Nonlinear Equations of Evolution and Solitary Waves, Commun. Pur. Appl. Math., 21 (1968), 467-490.    

9. J. M. Lee, Manifolds and Differential Geometry, AMS Graduate Studies in Mathematics, vol. 107, Providence, RI, 2009.

10. C.-Q. Su, Y. Tian Gao, X. Yu, L. Xue and Yu-Jia Shen, Exterior differential expression of the (1+1)-dimensional nonlinear evolution equation with Lax integrability, J. Math. Anal. Appl., 435, (2016), 735-745.

11. H. D. Wahlquist and F. B. Estabrook, Bäcklund Transformation for Solutions of the Korteweg-de Vries Equation, Phys. Rev. Lett., 31 (1973), 1386-1390.    

12. H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16, (1975), 1-7.

13. H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 17, (1976), 1293-1297.

Copyright Info: © 2017, Paul Bracken, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved