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Connections of zero curvature and applications to nonlinear partial differential equations
1. | Department of Mathematics, University of Texas, Edinburg, TX, 78540, United States |
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), 125-127.
doi: 10.1103/PhysRevLett.31.125. |
[2] |
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Studies in Applied Mathematics, Philadelphia, PA, 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), 29-60.
doi: 10.1007/s10440-012-9716-0. |
[4] |
P. Bracken, A geometric interpretation of prolongation by means of connections, J. Math. Phys., 51 (2010), 113502, 6pp.
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), 247-263.
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), 501-503.
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), 157-163.
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), 55-83. |
[10] |
F. B. Estabrook and H. D. Wahlquist, Prolongation structures of nonlinear evolution equations II, J. Math. Phys., 17 (1976), 1293-1297.
doi: 10.1063/1.523056. |
[11] |
F. B. Estabrook, Moving frames and prolongation algebras, J. Math. Phys., 23 (1982), 2071-2076.
doi: 10.1063/1.525248. |
[12] |
F. B. Estabrook, Bäcklund Transformations the Inverse Scattering Method, Solitons and Their Applications, Lecture Notes in Mathematics, ed. R. Miura, Springer, Berlin, 515, (1976), 12-24. |
[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), 263-274.
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, Vol. 6, North Holland, Amsterdam, II, Ch. 6, 1994. |
[15] |
R. Hermann, Pseudodifferentials of Estabrook and Wahlquist, the geometry of solutions and the theory of connections, Phys. Rev. Letts., 36 (1976), 835-836.
doi: 10.1103/PhysRevLett.36.835. |
[16] |
R. Hermann, The Geometry of Nonlinear Differential Equations, Bäcklund Transformations and Solitons, Vol. XII, A, Math. Sci. Press, Brookline, MA, 1976. |
[17] |
J. Krasilshchik and A. Verbovetsky, Geometry of jet spaces and integrable systems, J. Geom. and Physics, 61 (2011), 1633-1674.
doi: 10.1016/j.geomphys.2010.10.012. |
[18] |
P. W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, AMS, Providence, RI, 2008. |
[19] |
E. G. Reyes, Pseudo-spherical surfaces and integrability of evolution equations, Russian J. of Diff. Equations, 147 (1998), 195-230.
doi: 10.1006/jdeq.1998.3430. |
[20] |
C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, Cambridge University Press, Cambridge, 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), 195-210.
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), 1685-1699.
doi: 10.1007/BF02365073. |
[23] |
H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, Journal of Math. Phys., 16 (1975), 1-7.
doi: 10.1063/1.522396. |
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), 125-127.
doi: 10.1103/PhysRevLett.31.125. |
[2] |
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Studies in Applied Mathematics, Philadelphia, PA, 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), 29-60.
doi: 10.1007/s10440-012-9716-0. |
[4] |
P. Bracken, A geometric interpretation of prolongation by means of connections, J. Math. Phys., 51 (2010), 113502, 6pp.
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), 247-263.
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), 501-503.
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), 157-163.
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), 55-83. |
[10] |
F. B. Estabrook and H. D. Wahlquist, Prolongation structures of nonlinear evolution equations II, J. Math. Phys., 17 (1976), 1293-1297.
doi: 10.1063/1.523056. |
[11] |
F. B. Estabrook, Moving frames and prolongation algebras, J. Math. Phys., 23 (1982), 2071-2076.
doi: 10.1063/1.525248. |
[12] |
F. B. Estabrook, Bäcklund Transformations the Inverse Scattering Method, Solitons and Their Applications, Lecture Notes in Mathematics, ed. R. Miura, Springer, Berlin, 515, (1976), 12-24. |
[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), 263-274.
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, Vol. 6, North Holland, Amsterdam, II, Ch. 6, 1994. |
[15] |
R. Hermann, Pseudodifferentials of Estabrook and Wahlquist, the geometry of solutions and the theory of connections, Phys. Rev. Letts., 36 (1976), 835-836.
doi: 10.1103/PhysRevLett.36.835. |
[16] |
R. Hermann, The Geometry of Nonlinear Differential Equations, Bäcklund Transformations and Solitons, Vol. XII, A, Math. Sci. Press, Brookline, MA, 1976. |
[17] |
J. Krasilshchik and A. Verbovetsky, Geometry of jet spaces and integrable systems, J. Geom. and Physics, 61 (2011), 1633-1674.
doi: 10.1016/j.geomphys.2010.10.012. |
[18] |
P. W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, AMS, Providence, RI, 2008. |
[19] |
E. G. Reyes, Pseudo-spherical surfaces and integrability of evolution equations, Russian J. of Diff. Equations, 147 (1998), 195-230.
doi: 10.1006/jdeq.1998.3430. |
[20] |
C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, Cambridge University Press, Cambridge, 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), 195-210.
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), 1685-1699.
doi: 10.1007/BF02365073. |
[23] |
H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, Journal of Math. Phys., 16 (1975), 1-7.
doi: 10.1063/1.522396. |
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