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August  2018, 11(4): 707-721. doi: 10.3934/dcdss.2018044

Conservation laws by symmetries and adjoint symmetries

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

2. 

Department of Mathematics and Statistics, University of South Florida Tampa, FL 33620, USA

3. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China

4. 

International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa

* Corresponding author: Wen-Xiu Ma

Received  January 2017 Published  November 2017

Conservation laws are fomulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not require the existence of a Lagrangian for a given system, and the presented examples include computations of conserved densities for the heat equation, Burgers' equation and the Korteweg-de Vries equation.

Citation: Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
References:
[1]

S. C. Anco and G. Bluman, Direct computation of conservation laws from filed equations, Phys. Rev. Lett., 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[2]

S. C. Anco and G. Bluman, Derivation of conservation laws from nonlocal symmetries of differential equations, J. Math. Phys., 37 (1996), 2361-2375.  doi: 10.1063/1.531515.  Google Scholar

[3]

G. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York-Berlin, 1989. doi: 10.1007/978-1-4757-4307-4.  Google Scholar

[4]

B. Fuchsstiener, Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal., 3 (1979), 849-862.  doi: 10.1016/0362-546X(79)90052-X.  Google Scholar

[5]

B. Fuchsstiener and A. C. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[6]

B. Fuchssteiner and W. X. Ma, An approach to master symmetries of lattice equations, in: Symmetries and Integrability of Difference Equations (Canterbury, 1996) (ed. P. A. Clarkson and F. W. Nijhoff), London Math. Soc. Lecture Note Ser., 255, Cambridge Univ. Press, Cambridge, 1999,247–260. doi: 10.1017/CBO9780511569432.020.  Google Scholar

[7]

N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, Holland, 1985. doi: 10.1007/978-94-009-5243-0.  Google Scholar

[8]

N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.  doi: 10.1016/j.jmaa.2006.10.078.  Google Scholar

[9]

N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., 44 (2011), 432002 (8pp). doi: 10.1088/1751-8113/44/43/432002.  Google Scholar

[10]

X. Y. Li and Q. L. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121 (2017), 123-137.  doi: 10.1016/j.geomphys.2017.07.010.  Google Scholar

[11]

X. Y. LiQ. L. ZhaoY. X. Li and H. H. Dong, Binary Bargmann symmetry constraint associated with 3× 3 discrete matrix spectral problem, J. Nonlinear Sci. Appl., 8 (2015), 496-506.   Google Scholar

[12]

Y. S. Li, Some algebraic properties of "C-integrable" nonlinear equations Ⅰ. Burgers equation and Calogero equation, Sci. China Ser. A, 33 (1990), 513-520.   Google Scholar

[13]

W. X. Ma and B. Fuchssteiner, Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations, J. Math. Phys., 40 (1999), 2400-2418.  doi: 10.1063/1.532872.  Google Scholar

[14]

W. X. Ma, K-symmetries and τ-symmetries of evolution equations and their Lie algebras, J. Phys. A: Math. Gen., 23 (1990), 2707-2716.  doi: 10.1088/0305-4470/23/13/011.  Google Scholar

[15]

W. X. Ma, New finite-dimensional integrable systems by symmetry constraint of the KdV equations, J. Phys. Soc. Jpn., 64 (1995), 1085-1091.  doi: 10.1143/JPSJ.64.1085.  Google Scholar

[16]

W. X. Ma and W. Strampp, An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems, Phys. Lett. A, 185 (1994), 277-286.  doi: 10.1016/0375-9601(94)90616-5.  Google Scholar

[17]

W. X. Ma, The algebraic structures of isospectral Lax operators and applications to integrable equations, J. Phys. A: Math. Gen., 25 (1992), 5329-5343.  doi: 10.1088/0305-4470/25/20/014.  Google Scholar

[18]

W. X. Ma, Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations, J. Math. Phys., 33 (1992), 2464-2476.  doi: 10.1063/1.529616.  Google Scholar

[19]

W. X. Ma, A method of zero curvature representation for constructing symmetry algebras of integrable systems, in: Proceedings of the 21st International Conference on the Differential Geometry Methods in Theoretical Physics (ed. M. L. Ge), World Scientific, Singapore, 1993,535–538. Google Scholar

[20]

W. X. Ma, Conservation laws of discrete evolution equations by symmetries and adjoint symmetries, Symmetry, 7 (2015), 714-725.  doi: 10.3390/sym7020714.  Google Scholar

[21]

W. X. Ma and R. G. Zhou, Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlinear Math. Phys., 9 (2002), 106-126.  doi: 10.2991/jnmp.2002.9.s1.10.  Google Scholar

[22]

P. Morando and S. Pasquero, The symmetry in the structure of dynamical and adjoint symmetries of second-order differential equations, J. Phys. A: Math. Gen., 28 (1995), 1943-1955.  doi: 10.1088/0305-4470/28/7/016.  Google Scholar

[23]

C. S. Morawetz, Variations on conservation laws for the wave equation, Bull. Amer. Math. Soc. (N.S.), 37 (2000), 141-154.  doi: 10.1090/S0273-0979-00-00857-0.  Google Scholar

[24]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107, Springer-Verlag, New York-Berlin, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[25]

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18 (1977), 1212-1215.  doi: 10.1063/1.523393.  Google Scholar

[26]

W. Sarlet and A. Ramos, Adjoint symmetries, separability, and volume forms, J. Math. Phys., 41 (2000), 2877-2888.  doi: 10.1063/1.533277.  Google Scholar

[27]

C. Tian, New strong symmetry, symmetries and Lie algebra of Burgers' equation, Sci. Sinica Ser. A, 31 (1988), 141-151.   Google Scholar

[28]

G. Z. Tu and M. Z. Qin, The invariant groups and conservation laws of nonlinear evolution equations--an approach of symmetric function, Sci. Sinica, 24 (1981), 13-26.   Google Scholar

[29]

X. R. Wang, X. E. Zhang and P. Y. Zhao, Binary nonlinearization for AKNS-KN coupling system, Abstr. Appl. Anal., 2014 (2014), Art. ID 253102, 12 pp. doi: 10.1155/2014/253102.  Google Scholar

show all references

References:
[1]

S. C. Anco and G. Bluman, Direct computation of conservation laws from filed equations, Phys. Rev. Lett., 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[2]

S. C. Anco and G. Bluman, Derivation of conservation laws from nonlocal symmetries of differential equations, J. Math. Phys., 37 (1996), 2361-2375.  doi: 10.1063/1.531515.  Google Scholar

[3]

G. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York-Berlin, 1989. doi: 10.1007/978-1-4757-4307-4.  Google Scholar

[4]

B. Fuchsstiener, Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal., 3 (1979), 849-862.  doi: 10.1016/0362-546X(79)90052-X.  Google Scholar

[5]

B. Fuchsstiener and A. C. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[6]

B. Fuchssteiner and W. X. Ma, An approach to master symmetries of lattice equations, in: Symmetries and Integrability of Difference Equations (Canterbury, 1996) (ed. P. A. Clarkson and F. W. Nijhoff), London Math. Soc. Lecture Note Ser., 255, Cambridge Univ. Press, Cambridge, 1999,247–260. doi: 10.1017/CBO9780511569432.020.  Google Scholar

[7]

N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, Holland, 1985. doi: 10.1007/978-94-009-5243-0.  Google Scholar

[8]

N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.  doi: 10.1016/j.jmaa.2006.10.078.  Google Scholar

[9]

N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., 44 (2011), 432002 (8pp). doi: 10.1088/1751-8113/44/43/432002.  Google Scholar

[10]

X. Y. Li and Q. L. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121 (2017), 123-137.  doi: 10.1016/j.geomphys.2017.07.010.  Google Scholar

[11]

X. Y. LiQ. L. ZhaoY. X. Li and H. H. Dong, Binary Bargmann symmetry constraint associated with 3× 3 discrete matrix spectral problem, J. Nonlinear Sci. Appl., 8 (2015), 496-506.   Google Scholar

[12]

Y. S. Li, Some algebraic properties of "C-integrable" nonlinear equations Ⅰ. Burgers equation and Calogero equation, Sci. China Ser. A, 33 (1990), 513-520.   Google Scholar

[13]

W. X. Ma and B. Fuchssteiner, Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations, J. Math. Phys., 40 (1999), 2400-2418.  doi: 10.1063/1.532872.  Google Scholar

[14]

W. X. Ma, K-symmetries and τ-symmetries of evolution equations and their Lie algebras, J. Phys. A: Math. Gen., 23 (1990), 2707-2716.  doi: 10.1088/0305-4470/23/13/011.  Google Scholar

[15]

W. X. Ma, New finite-dimensional integrable systems by symmetry constraint of the KdV equations, J. Phys. Soc. Jpn., 64 (1995), 1085-1091.  doi: 10.1143/JPSJ.64.1085.  Google Scholar

[16]

W. X. Ma and W. Strampp, An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems, Phys. Lett. A, 185 (1994), 277-286.  doi: 10.1016/0375-9601(94)90616-5.  Google Scholar

[17]

W. X. Ma, The algebraic structures of isospectral Lax operators and applications to integrable equations, J. Phys. A: Math. Gen., 25 (1992), 5329-5343.  doi: 10.1088/0305-4470/25/20/014.  Google Scholar

[18]

W. X. Ma, Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations, J. Math. Phys., 33 (1992), 2464-2476.  doi: 10.1063/1.529616.  Google Scholar

[19]

W. X. Ma, A method of zero curvature representation for constructing symmetry algebras of integrable systems, in: Proceedings of the 21st International Conference on the Differential Geometry Methods in Theoretical Physics (ed. M. L. Ge), World Scientific, Singapore, 1993,535–538. Google Scholar

[20]

W. X. Ma, Conservation laws of discrete evolution equations by symmetries and adjoint symmetries, Symmetry, 7 (2015), 714-725.  doi: 10.3390/sym7020714.  Google Scholar

[21]

W. X. Ma and R. G. Zhou, Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlinear Math. Phys., 9 (2002), 106-126.  doi: 10.2991/jnmp.2002.9.s1.10.  Google Scholar

[22]

P. Morando and S. Pasquero, The symmetry in the structure of dynamical and adjoint symmetries of second-order differential equations, J. Phys. A: Math. Gen., 28 (1995), 1943-1955.  doi: 10.1088/0305-4470/28/7/016.  Google Scholar

[23]

C. S. Morawetz, Variations on conservation laws for the wave equation, Bull. Amer. Math. Soc. (N.S.), 37 (2000), 141-154.  doi: 10.1090/S0273-0979-00-00857-0.  Google Scholar

[24]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107, Springer-Verlag, New York-Berlin, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[25]

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18 (1977), 1212-1215.  doi: 10.1063/1.523393.  Google Scholar

[26]

W. Sarlet and A. Ramos, Adjoint symmetries, separability, and volume forms, J. Math. Phys., 41 (2000), 2877-2888.  doi: 10.1063/1.533277.  Google Scholar

[27]

C. Tian, New strong symmetry, symmetries and Lie algebra of Burgers' equation, Sci. Sinica Ser. A, 31 (1988), 141-151.   Google Scholar

[28]

G. Z. Tu and M. Z. Qin, The invariant groups and conservation laws of nonlinear evolution equations--an approach of symmetric function, Sci. Sinica, 24 (1981), 13-26.   Google Scholar

[29]

X. R. Wang, X. E. Zhang and P. Y. Zhao, Binary nonlinearization for AKNS-KN coupling system, Abstr. Appl. Anal., 2014 (2014), Art. ID 253102, 12 pp. doi: 10.1155/2014/253102.  Google Scholar

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