# American Institute of Mathematical Sciences

2014, 7(6): 1165-1179. doi: 10.3934/dcdss.2014.7.1165

## Connections of zero curvature and applications to nonlinear partial differential equations

 1 Department of Mathematics, University of Texas, Edinburg, TX, 78540, United States

Received  January 2013 Revised  September 2013 Published  June 2014

A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations. It is discussed how Lax pairs and Bäcklund tranformations can be formulated for such equations that occur as zero curvature terms.
Citation: Paul Bracken. Connections of zero curvature and applications to nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1165-1179. doi: 10.3934/dcdss.2014.7.1165
##### References:
 [1] M. J. Ablowitz, D. K. Kaup, A. C. Newell and H. Segur, Nonlinear evolution equations of physical significance,, Phys. Rev. Letts., 31 (1973), 125. doi: 10.1103/PhysRevLett.31.125. [2] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, Studies in Applied Mathematics, (1981). [3] I. M. Anderson and M. E. Fels, Symmetry reduction of exterior differential systems and Bäcklund transformations for PDE in the plane,, Acta Appl. Math., 120 (2012), 29. doi: 10.1007/s10440-012-9716-0. [4] P. Bracken, A geometric interpretation of prolongation by means of connections,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3504172. [5] P. Bracken, Exterior differential systems prolongations and applications to a study of two nonlinear partial differential equations,, Acta Appl. Math., 113 (2011), 247. doi: 10.1007/s10440-010-9597-z. [6] P. Bracken, Integrable systems of partial differential systems determined by structure equations and lax pair,, Phys. Letts. A, 374 (2010), 501. doi: 10.1016/j.physleta.2009.11.042. [7] P. Bracken, Connections defining representations of zero curvature and their lax and Bäcklund mappings,, J. of Geometry and Physics, 70 (2013), 157. doi: 10.1016/j.geomphys.2013.03.024. [8] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt and P. A. Griffiths, Exterior Differential Systems,, Springer-Verlag, (1991). doi: 10.1007/978-1-4613-9714-4. [9] S. S. Chern and K. Tenenblat, Pseudospherical Surfaces and Evolution Equations,, Studies in Applied Mathematics, 74 (1986), 55. [10] F. B. Estabrook and H. D. Wahlquist, Prolongation structures of nonlinear evolution equations II,, J. Math. Phys., 17 (1976), 1293. doi: 10.1063/1.523056. [11] F. B. Estabrook, Moving frames and prolongation algebras,, J. Math. Phys., 23 (1982), 2071. doi: 10.1063/1.525248. [12] F. B. Estabrook, Bäcklund Transformations the Inverse Scattering Method, Solitons and Their Applications,, Lecture Notes in Mathematics, (1976), 12. [13] F. B. Estabrook and H. D. Wahlquist, Classical geometries defined by exterior differential systems on higher frame bundles,, Classical and Quantum Gravity, 6 (1989), 263. doi: 10.1088/0264-9381/6/3/008. [14] E. van Groesen and E. M. Jager, Mathematical Structures in Continuous Dynamical Systems,, Studies in Math. Physics, (1994). [15] R. Hermann, Pseudodifferentials of Estabrook and Wahlquist, the geometry of solutions and the theory of connections,, Phys. Rev. Letts., 36 (1976), 835. doi: 10.1103/PhysRevLett.36.835. [16] R. Hermann, The Geometry of Nonlinear Differential Equations, Bäcklund Transformations and Solitons,, Vol. XII, (1976). [17] J. Krasilshchik and A. Verbovetsky, Geometry of jet spaces and integrable systems,, J. Geom. and Physics, 61 (2011), 1633. doi: 10.1016/j.geomphys.2010.10.012. [18] P. W. Michor, Topics in Differential Geometry,, Graduate Studies in Mathematics, (2008). [19] E. G. Reyes, Pseudo-spherical surfaces and integrability of evolution equations,, Russian J. of Diff. Equations, 147 (1998), 195. doi: 10.1006/jdeq.1998.3430. [20] C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511606359. [21] A. K. Rybnikov, Connections defining representations of zero curvature and the solitons of sine-Gordon and Korteweg-de Vries equations,, Russian J. of Math. Phys., 18 (2011), 195. doi: 10.1134/S1061920811020087. [22] A. K. Rybnikov, Equations of the inverse problem, Bäcklund transformations and the theory of connections,, J. of Math Sciences, 94 (1999), 1685. doi: 10.1007/BF02365073. [23] H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations,, Journal of Math. Phys., 16 (1975), 1. doi: 10.1063/1.522396.

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##### References:
 [1] M. J. Ablowitz, D. K. Kaup, A. C. Newell and H. Segur, Nonlinear evolution equations of physical significance,, Phys. Rev. Letts., 31 (1973), 125. doi: 10.1103/PhysRevLett.31.125. [2] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, Studies in Applied Mathematics, (1981). [3] I. M. Anderson and M. E. Fels, Symmetry reduction of exterior differential systems and Bäcklund transformations for PDE in the plane,, Acta Appl. Math., 120 (2012), 29. doi: 10.1007/s10440-012-9716-0. [4] P. Bracken, A geometric interpretation of prolongation by means of connections,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3504172. [5] P. Bracken, Exterior differential systems prolongations and applications to a study of two nonlinear partial differential equations,, Acta Appl. Math., 113 (2011), 247. doi: 10.1007/s10440-010-9597-z. [6] P. Bracken, Integrable systems of partial differential systems determined by structure equations and lax pair,, Phys. Letts. A, 374 (2010), 501. doi: 10.1016/j.physleta.2009.11.042. [7] P. Bracken, Connections defining representations of zero curvature and their lax and Bäcklund mappings,, J. of Geometry and Physics, 70 (2013), 157. doi: 10.1016/j.geomphys.2013.03.024. [8] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt and P. A. Griffiths, Exterior Differential Systems,, Springer-Verlag, (1991). doi: 10.1007/978-1-4613-9714-4. [9] S. S. Chern and K. Tenenblat, Pseudospherical Surfaces and Evolution Equations,, Studies in Applied Mathematics, 74 (1986), 55. [10] F. B. Estabrook and H. D. Wahlquist, Prolongation structures of nonlinear evolution equations II,, J. Math. Phys., 17 (1976), 1293. doi: 10.1063/1.523056. [11] F. B. Estabrook, Moving frames and prolongation algebras,, J. Math. Phys., 23 (1982), 2071. doi: 10.1063/1.525248. [12] F. B. Estabrook, Bäcklund Transformations the Inverse Scattering Method, Solitons and Their Applications,, Lecture Notes in Mathematics, (1976), 12. [13] F. B. Estabrook and H. D. Wahlquist, Classical geometries defined by exterior differential systems on higher frame bundles,, Classical and Quantum Gravity, 6 (1989), 263. doi: 10.1088/0264-9381/6/3/008. [14] E. van Groesen and E. M. Jager, Mathematical Structures in Continuous Dynamical Systems,, Studies in Math. Physics, (1994). [15] R. Hermann, Pseudodifferentials of Estabrook and Wahlquist, the geometry of solutions and the theory of connections,, Phys. Rev. Letts., 36 (1976), 835. doi: 10.1103/PhysRevLett.36.835. [16] R. Hermann, The Geometry of Nonlinear Differential Equations, Bäcklund Transformations and Solitons,, Vol. XII, (1976). [17] J. Krasilshchik and A. Verbovetsky, Geometry of jet spaces and integrable systems,, J. Geom. and Physics, 61 (2011), 1633. doi: 10.1016/j.geomphys.2010.10.012. [18] P. W. Michor, Topics in Differential Geometry,, Graduate Studies in Mathematics, (2008). [19] E. G. Reyes, Pseudo-spherical surfaces and integrability of evolution equations,, Russian J. of Diff. Equations, 147 (1998), 195. doi: 10.1006/jdeq.1998.3430. [20] C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511606359. [21] A. K. Rybnikov, Connections defining representations of zero curvature and the solitons of sine-Gordon and Korteweg-de Vries equations,, Russian J. of Math. Phys., 18 (2011), 195. doi: 10.1134/S1061920811020087. [22] A. K. Rybnikov, Equations of the inverse problem, Bäcklund transformations and the theory of connections,, J. of Math Sciences, 94 (1999), 1685. doi: 10.1007/BF02365073. [23] H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations,, Journal of Math. Phys., 16 (1975), 1. doi: 10.1063/1.522396.
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