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Conservation laws and symmetries of time-dependent generalized KdV equations
| a. | Department of Mathematics and Statistics, Brock University, St. Catharines, Canada |
| b. | Departamento de Matemáticas, Universidad de Cádiz, Polígono del Río San Pedro s/n 11510 Puerto Real, Cádiz, Spain |
A complete classification of low-order conservation laws is obtained for time-dependent generalized Korteweg-de Vries equations. Through the Hamiltonian structure of these equations, a corresponding classification of Hamiltonian symmetries is derived. The physical meaning of the conservation laws and the symmetries is discussed.
References:
| [1] |
S. C. Anco and G. Bluman,
Direct Construction of Conservation Laws from Field Equations, Phys. Rev. Lett., 78 (1997), 2869-2873.
doi: 10.1103/PhysRevLett.78.2869. |
| [2] |
S. C. Anco and G. Bluman,
Direct construction method for conservation laws of partial differential equations Ⅱ: General treatment, Euro. J. Appl. Math., 13 (2002), 567-585.
doi: 10.1017/S0956792501004661. |
| [3] |
S. C. Anco,
Generalization of Noether's theorem in modern form to non-variational partial differential equations, Recent progress and Modern Challenges in Applied Mathematics,
Modeling and Computational Science, 79 (2017), 119-182.
doi: 10.1007/978-1-4939-6969-2_5. |
| [4] |
S. C. Anco and M. L. Gandarias, Conservation laws and symmetries of a class of dispersive semilinear wave equations, in preparation, 2017. |
| [5] |
S. C. Anco and G. Bluman,
Direct construction method for conservation laws of partial differential equations Ⅰ: Examples of conservation law classifications, Euro. Jour. Appl. Math., 13 (2002), 545-566.
doi: 10.1017/S0956792501004661. |
| [6] |
I. Bakirtas and H. Demiray,
Weakly nonlinear waves in a tapered elastic tube filled with an inviscid fluid, Int. J. Nonlinear Mech., 40 (2005), 785-793.
doi: 10.1016/j.ijnonlinmec.2004.03.003. |
| [7] |
G. W. Bluman, A Cheviakov and S. C. Anco,
Applications of Symmetry Methods to Partial Differential Equations, New York: Springer, 2010.
doi: 10.1007/978-0-387-68028-6. |
| [8] |
R. C. Cascaval, Variable coefficient KdV equations and waves in elastic tubes, in Evolution Equations (eds. G.R. Goldstein, R. Nagel, S. Romanelli), 57-69, Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, 2003. |
| [9] |
H. Demiray, The effect of a bump on wave propagation in a fluid-filled elastic tube, Int. J. Eng. Sci., 42 (2004), 203-215; ibid, Weakly nonlinear waves in a linearly tapered elastic tube filled with a fluid, Math. Comput. Mod., 39 (2004), 151-162.
doi: 10.1016/S0020-7225(03)00284-2. |
| [10] |
A. G. Johnpillai, C. M. Khalique and A. Biswas,
Exact solutions of KdV equation with time-dependent coefficients, Applied Mathematics and Computation, 216 (2010), 3114-3119.
doi: 10.1016/j.amc.2010.03.133. |
| [11] |
T. Kakutani and H. Ono, J. Phys. Soc. Jpn., 26 (1969), 1305-1318. |
| [12] |
W.-X. Ma, R. K. Bullough, P. J. Caudrey and W. I. Fushchych,
Time-dependent symmetries of variable-coefficient evolution equations and graded Lie algebras, J. Phys. A: Math. Gen., 30 (1997), 5141-5149.
doi: 10.1088/0305-4470/30/14/023. |
| [13] |
W.-X. Ma and R. Zhou,
Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlin. Math. Phys., 9 (2002), 106-126.
doi: 10.2991/jnmp.2002.9.s1.10. |
| [14] |
M. Moulati and C. M. Khalique,
Group analysis of a generalized KdV equation, Appl. Math. Inf. Sci., 8 (2014), 2845-2848.
doi: 10.12785/amis/080620. |
| [15] |
V. Narayanamurti and C. M. Varma,
Nonlinear propagation of heat pulses in solids, Phys. Rev. Lett., 25 (1970), 1105-1108.
doi: 10.1103/PhysRevLett.25.1105. |
| [16] |
P. J. Olver,
Applications of Lie Groups to Differential Equations, Berlin: Springer, 1986.
doi: 10.1007/978-1-4684-0274-2. |
| [17] |
R. O. Popovych and A. Sergyeyev,
Conservation laws and normal forms of evolution equations, Phys. Lett. A, 374 (2010), 2210-2217.
doi: 10.1016/j.physleta.2010.03.033. |
| [18] |
F. D. Tappert and C. M. Varma,
Asymptotic theory of self-trapping of heat pulses in solids, Phys. Rev. Lett., 25 (1970), 1108-1111.
doi: 10.1103/PhysRevLett.25.1108. |
| [19] |
M. Wadati,
Wave propagation in nonlinear lattice, J. Phys. Soc. Japan, 38 (1975), 673-680.
doi: 10.1143/JPSJ.38.673. |
| [20] |
N. J. Zabusky, A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, in Proc. Symp. Nonlinear Partial Differential Equations (ed. W. Ames), 223-258, Academic Press, 1967. |
show all references
References:
| [1] |
S. C. Anco and G. Bluman,
Direct Construction of Conservation Laws from Field Equations, Phys. Rev. Lett., 78 (1997), 2869-2873.
doi: 10.1103/PhysRevLett.78.2869. |
| [2] |
S. C. Anco and G. Bluman,
Direct construction method for conservation laws of partial differential equations Ⅱ: General treatment, Euro. J. Appl. Math., 13 (2002), 567-585.
doi: 10.1017/S0956792501004661. |
| [3] |
S. C. Anco,
Generalization of Noether's theorem in modern form to non-variational partial differential equations, Recent progress and Modern Challenges in Applied Mathematics,
Modeling and Computational Science, 79 (2017), 119-182.
doi: 10.1007/978-1-4939-6969-2_5. |
| [4] |
S. C. Anco and M. L. Gandarias, Conservation laws and symmetries of a class of dispersive semilinear wave equations, in preparation, 2017. |
| [5] |
S. C. Anco and G. Bluman,
Direct construction method for conservation laws of partial differential equations Ⅰ: Examples of conservation law classifications, Euro. Jour. Appl. Math., 13 (2002), 545-566.
doi: 10.1017/S0956792501004661. |
| [6] |
I. Bakirtas and H. Demiray,
Weakly nonlinear waves in a tapered elastic tube filled with an inviscid fluid, Int. J. Nonlinear Mech., 40 (2005), 785-793.
doi: 10.1016/j.ijnonlinmec.2004.03.003. |
| [7] |
G. W. Bluman, A Cheviakov and S. C. Anco,
Applications of Symmetry Methods to Partial Differential Equations, New York: Springer, 2010.
doi: 10.1007/978-0-387-68028-6. |
| [8] |
R. C. Cascaval, Variable coefficient KdV equations and waves in elastic tubes, in Evolution Equations (eds. G.R. Goldstein, R. Nagel, S. Romanelli), 57-69, Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, 2003. |
| [9] |
H. Demiray, The effect of a bump on wave propagation in a fluid-filled elastic tube, Int. J. Eng. Sci., 42 (2004), 203-215; ibid, Weakly nonlinear waves in a linearly tapered elastic tube filled with a fluid, Math. Comput. Mod., 39 (2004), 151-162.
doi: 10.1016/S0020-7225(03)00284-2. |
| [10] |
A. G. Johnpillai, C. M. Khalique and A. Biswas,
Exact solutions of KdV equation with time-dependent coefficients, Applied Mathematics and Computation, 216 (2010), 3114-3119.
doi: 10.1016/j.amc.2010.03.133. |
| [11] |
T. Kakutani and H. Ono, J. Phys. Soc. Jpn., 26 (1969), 1305-1318. |
| [12] |
W.-X. Ma, R. K. Bullough, P. J. Caudrey and W. I. Fushchych,
Time-dependent symmetries of variable-coefficient evolution equations and graded Lie algebras, J. Phys. A: Math. Gen., 30 (1997), 5141-5149.
doi: 10.1088/0305-4470/30/14/023. |
| [13] |
W.-X. Ma and R. Zhou,
Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlin. Math. Phys., 9 (2002), 106-126.
doi: 10.2991/jnmp.2002.9.s1.10. |
| [14] |
M. Moulati and C. M. Khalique,
Group analysis of a generalized KdV equation, Appl. Math. Inf. Sci., 8 (2014), 2845-2848.
doi: 10.12785/amis/080620. |
| [15] |
V. Narayanamurti and C. M. Varma,
Nonlinear propagation of heat pulses in solids, Phys. Rev. Lett., 25 (1970), 1105-1108.
doi: 10.1103/PhysRevLett.25.1105. |
| [16] |
P. J. Olver,
Applications of Lie Groups to Differential Equations, Berlin: Springer, 1986.
doi: 10.1007/978-1-4684-0274-2. |
| [17] |
R. O. Popovych and A. Sergyeyev,
Conservation laws and normal forms of evolution equations, Phys. Lett. A, 374 (2010), 2210-2217.
doi: 10.1016/j.physleta.2010.03.033. |
| [18] |
F. D. Tappert and C. M. Varma,
Asymptotic theory of self-trapping of heat pulses in solids, Phys. Rev. Lett., 25 (1970), 1108-1111.
doi: 10.1103/PhysRevLett.25.1108. |
| [19] |
M. Wadati,
Wave propagation in nonlinear lattice, J. Phys. Soc. Japan, 38 (1975), 673-680.
doi: 10.1143/JPSJ.38.673. |
| [20] |
N. J. Zabusky, A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, in Proc. Symp. Nonlinear Partial Differential Equations (ed. W. Ames), 223-258, Academic Press, 1967. |
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