September  2011, 10(5): 1345-1360. doi: 10.3934/cpaa.2011.10.1345

Exterior differential systems and prolongations for three important nonlinear partial differential equations

1. 

Department of Mathematics, University of Texas, Edinburg, TX 78539, United States

Received  February 2009 Revised  August 2010 Published  April 2011

Partial differential systems which have applications to water waves will be formulated as exterior differential systems. A prolongation structure is determined for each of the equations. The formalism for studying prolongations is reviewed and the prolongation equations are solved for each equation. One of these differential systems includes the Camassa-Holm and Degasperis-Procesi equations as special cases. The formulation of conservation laws for each of the systems introduced is discussed and a single example for each is given. It is shown how a Bäcklund transformation for the last case can be obtained using the prolongation results.
Citation: Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
References:
[1]

A. C. Newell, "Solitons in Mathematics and Physics," SIAM, Philadelphia, 1985.

[2]

M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering," Cambridge University Press, 1991.

[3]

H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16 (1975), 1-7. doi: 10.1063/1.522396.

[4]

H. D. Wahlquist and F. B. Estabrook, Prolongation Structures of Nonlinear Evolution Equations II, J. Math. Phys., 17 (1976), 1293-1297. doi: 10.1063/1.523056.

[5]

F. B. Estabrook, Moving frames and prolongation algebras, J. Math. Phys., 23 (1982), 2071-2076. doi: 10.1-63/1101.525248.

[6]

E. van Groesen and E. M. de Jager, "Mathematical Structures in Continuous Dynamical Systems," Studies in Math. Phys., vol. 6, North Holland, 1994.

[7]

P. Bracken, The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces, in "Solitary Waves in Fluid Media" (C. David and Z. Feng eds. ), Bentham Science Publishers, pgs. 249-295, (2010).

[8]

E. M. de Jager and S. Spannenburg, Prolongation structures and Bäcklund transformations for the matrix Korteweg-de Vries and Boomeron equation, J. Phys. A: Math. Gen., 18 (1985), 2177-2189. doi: 10.1088/0305-4470/18/12/015.

[9]

P. Bracken, An exterior differential system for a generalized Korteweg-de Vries equation and its associated integrability, Acta Applicandae Mathematicae, 95 (2007), 223-231. doi: 10.1007/s10440-007-9086-1.

[10]

P. Bracken, Symmetry properties of a generalized Korteweg-de Vries equation and some explicit solutions, Int. J. Math. and Math. Sciences, 13 (2005), 2159-2173. doi: 10.1155/IJMMS.2005.2159.

[11]

P. Olver, "Applications of Lie Groups to Differential Equations," Springer-Verlag, New York, 1993.

[12]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Advances in Appl. Mechanics, 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[13]

E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169.

[14]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letts., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[15]

J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A: Math. Gen., 38 (2005), 869-880.

[16]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlin. Sci., 16 (2006), 109-122. doi: 10.1007s00332-005-0707-4.

[17]

A. Hone and J. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems, 19 (2003), 129-145. doi: 10.1088/0266-5611/19/1/307.

[18]

M. Fisher and J. Schiff, The Camassa-Holm equation: conserved quantities and the initial value problem, Phys. Lett., A 259 (1999), 371-376. doi: 10.1016/S0375-9601(99)00466-1.

[19]

F. B. Estabrook and H. D. Wahlquist, Prolongation structures, connection theory and Bäcklund transformation, in "Nonlinear Evolution Equations Solvable by the Spectral Transform" (F. Calogero ed.), Pitman, (1978).

show all references

References:
[1]

A. C. Newell, "Solitons in Mathematics and Physics," SIAM, Philadelphia, 1985.

[2]

M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering," Cambridge University Press, 1991.

[3]

H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16 (1975), 1-7. doi: 10.1063/1.522396.

[4]

H. D. Wahlquist and F. B. Estabrook, Prolongation Structures of Nonlinear Evolution Equations II, J. Math. Phys., 17 (1976), 1293-1297. doi: 10.1063/1.523056.

[5]

F. B. Estabrook, Moving frames and prolongation algebras, J. Math. Phys., 23 (1982), 2071-2076. doi: 10.1-63/1101.525248.

[6]

E. van Groesen and E. M. de Jager, "Mathematical Structures in Continuous Dynamical Systems," Studies in Math. Phys., vol. 6, North Holland, 1994.

[7]

P. Bracken, The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces, in "Solitary Waves in Fluid Media" (C. David and Z. Feng eds. ), Bentham Science Publishers, pgs. 249-295, (2010).

[8]

E. M. de Jager and S. Spannenburg, Prolongation structures and Bäcklund transformations for the matrix Korteweg-de Vries and Boomeron equation, J. Phys. A: Math. Gen., 18 (1985), 2177-2189. doi: 10.1088/0305-4470/18/12/015.

[9]

P. Bracken, An exterior differential system for a generalized Korteweg-de Vries equation and its associated integrability, Acta Applicandae Mathematicae, 95 (2007), 223-231. doi: 10.1007/s10440-007-9086-1.

[10]

P. Bracken, Symmetry properties of a generalized Korteweg-de Vries equation and some explicit solutions, Int. J. Math. and Math. Sciences, 13 (2005), 2159-2173. doi: 10.1155/IJMMS.2005.2159.

[11]

P. Olver, "Applications of Lie Groups to Differential Equations," Springer-Verlag, New York, 1993.

[12]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Advances in Appl. Mechanics, 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[13]

E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., 59 (2002), 117-131. doi: 10.1023/A:1014933316169.

[14]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letts., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[15]

J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A: Math. Gen., 38 (2005), 869-880.

[16]

A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle, J. Nonlin. Sci., 16 (2006), 109-122. doi: 10.1007s00332-005-0707-4.

[17]

A. Hone and J. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems, 19 (2003), 129-145. doi: 10.1088/0266-5611/19/1/307.

[18]

M. Fisher and J. Schiff, The Camassa-Holm equation: conserved quantities and the initial value problem, Phys. Lett., A 259 (1999), 371-376. doi: 10.1016/S0375-9601(99)00466-1.

[19]

F. B. Estabrook and H. D. Wahlquist, Prolongation structures, connection theory and Bäcklund transformation, in "Nonlinear Evolution Equations Solvable by the Spectral Transform" (F. Calogero ed.), Pitman, (1978).

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