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AIMS Mathematics

2017, Volume 2, Issue 4: 610-621. doi: 10.3934/Math.2017.4.610
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Research article

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

  • Department of Mathematics, University of Texas, TX 78539-2999 Edinburg, USA
  • Received: 20 October 2017 Accepted: 31 October 2017 Published: 06 November 2017

  • A geometric formulation of Lax integrability is introduced which makes use of a Pfaffan formulation of Lax integrability. The Frobenius theorem gives a necessary and suffcient condition for the complete integrability of a distribution, and provides a powerful way to study nonlinear evolution equations. This permits an examination of the relation between complete integrability and Lax integrability. The prolongation method is formulated in this context and gauge transformations can be examined in terms of differential forms as well as the Frobenius theorem.

    Citation: Paul Bracken. A geometric formulation of Lax integrability for nonlinear equationsin two independent variables[J]. AIMS Mathematics, 2017, 2(4): 610-621. doi: 10.3934/Math.2017.4.610

    Related Papers:

  • A geometric formulation of Lax integrability is introduced which makes use of a Pfaffan formulation of Lax integrability. The Frobenius theorem gives a necessary and suffcient condition for the complete integrability of a distribution, and provides a powerful way to study nonlinear evolution equations. This permits an examination of the relation between complete integrability and Lax integrability. The prolongation method is formulated in this context and gauge transformations can be examined in terms of differential forms as well as the Frobenius theorem.


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    [4] P. Bracken, Geometric Approaches to Produce Prolongations for Nonlinear Partial Differential Equations, Int. J. Geom. Methods M., 10, (2013), 1350002.
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    [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.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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