American Institute of Mathematical Sciences

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

Connections of zero curvature and applications to nonlinear partial differential equations

Paul Bracken 1,

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.
Keywords: lax pair, structure equations, Connection, Bäcklund., bundle, curvature.
Mathematics Subject Classification: Primary: 53Z05, 57R99; Secondary: 55R10, 53B5.
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.  Google Scholar

[2]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, Studies in Applied Mathematics, (1981).   Google Scholar

[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.  Google Scholar

[4]

P. Bracken, A geometric interpretation of prolongation by means of connections,, J. Math. Phys., 51 (2010).  doi: 10.1063/1.3504172.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. S. Chern and K. Tenenblat, Pseudospherical Surfaces and Evolution Equations,, Studies in Applied Mathematics, 74 (1986), 55.   Google Scholar

[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.  Google Scholar

[11]

F. B. Estabrook, Moving frames and prolongation algebras,, J. Math. Phys., 23 (1982), 2071.  doi: 10.1063/1.525248.  Google Scholar

[12]

F. B. Estabrook, Bäcklund Transformations the Inverse Scattering Method, Solitons and Their Applications,, Lecture Notes in Mathematics, (1976), 12.   Google Scholar

[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.  Google Scholar

[14]

E. van Groesen and E. M. Jager, Mathematical Structures in Continuous Dynamical Systems,, Studies in Math. Physics, (1994).   Google Scholar

[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.  Google Scholar

[16]

R. Hermann, The Geometry of Nonlinear Differential Equations, Bäcklund Transformations and Solitons,, Vol. XII, (1976).   Google Scholar

[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.  Google Scholar

[18]

P. W. Michor, Topics in Differential Geometry,, Graduate Studies in Mathematics, (2008).   Google Scholar

[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.  Google Scholar

[20]

C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511606359.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, Studies in Applied Mathematics, (1981).   Google Scholar

[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.  Google Scholar

[4]

P. Bracken, A geometric interpretation of prolongation by means of connections,, J. Math. Phys., 51 (2010).  doi: 10.1063/1.3504172.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. S. Chern and K. Tenenblat, Pseudospherical Surfaces and Evolution Equations,, Studies in Applied Mathematics, 74 (1986), 55.   Google Scholar

[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.  Google Scholar

[11]

F. B. Estabrook, Moving frames and prolongation algebras,, J. Math. Phys., 23 (1982), 2071.  doi: 10.1063/1.525248.  Google Scholar

[12]

F. B. Estabrook, Bäcklund Transformations the Inverse Scattering Method, Solitons and Their Applications,, Lecture Notes in Mathematics, (1976), 12.   Google Scholar

[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.  Google Scholar

[14]

E. van Groesen and E. M. Jager, Mathematical Structures in Continuous Dynamical Systems,, Studies in Math. Physics, (1994).   Google Scholar

[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.  Google Scholar

[16]

R. Hermann, The Geometry of Nonlinear Differential Equations, Bäcklund Transformations and Solitons,, Vol. XII, (1976).   Google Scholar

[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.  Google Scholar

[18]

P. W. Michor, Topics in Differential Geometry,, Graduate Studies in Mathematics, (2008).   Google Scholar

[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.  Google Scholar

[20]

C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511606359.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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