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August  2018, 11(4): 607-615. doi: 10.3934/dcdss.2018035

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

* Corresponding author: M. L. Gandarias

Received  January 2017 Revised  May 2017 Published  November 2017

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.

Citation: Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. C. Anco and M. L. Gandarias, Conservation laws and symmetries of a class of dispersive semilinear wave equations, in preparation, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

A. G. JohnpillaiC. 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.  Google Scholar

[11]

T. Kakutani and H. Ono, J. Phys. Soc. Jpn., 26 (1969), 1305-1318. Google Scholar

[12]

W.-X. MaR. K. BulloughP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. J. Olver, Applications of Lie Groups to Differential Equations, Berlin: Springer, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Wadati, Wave propagation in nonlinear lattice, J. Phys. Soc. Japan, 38 (1975), 673-680.  doi: 10.1143/JPSJ.38.673.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. C. Anco and M. L. Gandarias, Conservation laws and symmetries of a class of dispersive semilinear wave equations, in preparation, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

A. G. JohnpillaiC. 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.  Google Scholar

[11]

T. Kakutani and H. Ono, J. Phys. Soc. Jpn., 26 (1969), 1305-1318. Google Scholar

[12]

W.-X. MaR. K. BulloughP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. J. Olver, Applications of Lie Groups to Differential Equations, Berlin: Springer, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Wadati, Wave propagation in nonlinear lattice, J. Phys. Soc. Japan, 38 (1975), 673-680.  doi: 10.1143/JPSJ.38.673.  Google Scholar

[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. Google Scholar

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