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

Previous Article
Existence of chaos in weakly quasilinear systems
CPAA Home
This Issue
Next Article
Stability of linear differential equations with a distributed delay

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

Abstract
Related Papers

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.

Keywords: prolongation, connection, integral manifolds., Exterior differential system.

Mathematics Subject Classification: Primary: 35G20, 35Q53, 37J35; Secondary: 37K25, 37K35, 55R10, 53Z0.

Citation: Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & 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, (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. 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. doi: 10.1063/1.523056.
[5]	F. B. Estabrook, Moving frames and prolongation algebras,, J. Math. Phys., 23 (1982), 2071. 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., (1994).
[7]	P. Bracken, The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces,, in, (2010), 249.
[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. 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. 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. doi: 10.1155/IJMMS.2005.2159.
[11]	P. Olver, "Applications of Lie Groups to Differential Equations,", Springer-Verlag, (1993).
[12]	R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Advances in Appl. Mechanics, 31 (1994), 1. 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. 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. doi: 10.1103/PhysRevLett.71.1661.
[15]	J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. A: Math. Gen., 38 (2005), 869.
[16]	A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle,, J. Nonlin. Sci., 16 (2006), 109. 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. 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. doi: 10.1016/S0375-9601(99)00466-1.
[19]	F. B. Estabrook and H. D. Wahlquist, Prolongation structures, connection theory and Bäcklund transformation,, in, (1978).

show all references

References:

[1]	A. C. Newell, "Solitons in Mathematics and Physics,", SIAM, (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. 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. doi: 10.1063/1.523056.
[5]	F. B. Estabrook, Moving frames and prolongation algebras,, J. Math. Phys., 23 (1982), 2071. 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., (1994).
[7]	P. Bracken, The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces,, in, (2010), 249.
[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. 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. 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. doi: 10.1155/IJMMS.2005.2159.
[11]	P. Olver, "Applications of Lie Groups to Differential Equations,", Springer-Verlag, (1993).
[12]	R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Advances in Appl. Mechanics, 31 (1994), 1. 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. 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. doi: 10.1103/PhysRevLett.71.1661.
[15]	J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. A: Math. Gen., 38 (2005), 869.
[16]	A. Constantin and B. Kolev, Integrability of invariant metrics on the diffeomorphism group of the circle,, J. Nonlin. Sci., 16 (2006), 109. 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. 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. doi: 10.1016/S0375-9601(99)00466-1.
[19]	F. B. Estabrook and H. D. Wahlquist, Prolongation structures, connection theory and Bäcklund transformation,, in, (1978).

[1]	Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006
[2]	Santiago Capriotti. Dirac constraints in field theory and exterior differential systems. Journal of Geometric Mechanics, 2010, 2 (1) : 1-50. doi: 10.3934/jgm.2010.2.1
[3]	Marta Strani. Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1653-1667. doi: 10.3934/cpaa.2014.13.1653
[4]	Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
[5]	Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[6]	Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[7]	Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1
[8]	Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
[9]	Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054
[10]	Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349
[11]	Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038
[12]	Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems & Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23
[13]	Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[14]	Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
[15]	C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002
[16]	Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155
[17]	Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[18]	Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167
[19]	Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[20]	Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147

2017 Impact Factor: 0.884

American Institute of Mathematical Sciences