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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

Stephen Anco a, , Maria Rosa b, and  Maria Luz Gandarias b,,

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.

Keywords: Conservation law, symmetry, generalized KdV equation.
Mathematics Subject Classification: 37K05, 76M60, 35Q53.
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. 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.  Google Scholar

[11]

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

[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.  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. 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.  Google Scholar

[11]

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

[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.  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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