---

Discontinuity, Nonlinearity, and Complexity 6(2) (2017) 165--171 | DOI:10.5890/DNC.2017.06.004

Yuri Bozhkov$^{1}$, Valter Aparecido Silva Junior$^{2}$,$^{3}$

$^{1}$ Instituto de Matemática, Estatística e Computação Científica - IMECC, Universidade Estadual de Campinas - UNICAMP, 13083-859, Campinas/SP, Brasil

$^{2}$ Instituto Federal de Educação, Ciência e Tecnologia de São Paulo - IFSP, Acesso Dr. João Batista Merlin, s/no, Jardim Itália, 13872-551 - São João da Boa Vista - SP, Brasil

$^{3}$ Instituto de Física “Gleb Wataghin” - IFGW, Universidade Estadual de Campinas - UNICAMP, 13083-859 - Campinas - SP, Brasil

Download Full Text PDF

 

Abstract

We find the Lie point symmetries of the generalized two-component Hunter-Saxton system. Then we show that it is nonlinearly self-adjoint and establish the corresponding conservation laws using a recent theorem of Nail Ibragimov which enables one to determine conservation laws for problems without variational structure. Finally we obtain some invariant solutions.

Acknowledgments

We wish to thank Professor Nail Ibragimov for his useful comments on this work as well as for his firm encouragement. Yuri Bozhkov would also like to thank FAPESP, São Paulo, Brasil, for financial support.

References

1. [1]	Moon, B. (2013), Solitary wave solutions of the generalized two-component Hunter-Saxton system, Nonlinear Analysis, 89, 242-249.

2. [2]	Ivanov, R. (2009), Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46, 389-396.

3. [3]	Ibragimov, N.H. (2006), Integrating factors, adjoint equations and Lagrangians, J. Math. Anal. Appl., 318, 742-757.

4. [4]	Ibragimov, N.H. (2007), Quasi-self-adjoint differential equations, Archives of ALGA, 4, 55-60.

5. [5]	Ibragimov, N.H. (2007), A new conservation theorem, J. Math. Anal. Appl., 333, 311-328.

6. [6]	Ibragimov, N.H. (2011), Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., 44, 432002 (8pp).

7. [7]	Ibragimov, N.H. (2011), Nonlinear self-adjointness in constructing conservation laws, Archives of ALGA, 7/8, 1-90.

8. [8]	Ibragimov, N.H., Khamitova, R.S. and Valenti, A. (2011), Self-adjointness of a generalized Camassa-Holm equation, Appl. Math. Comput., 218, 2579-2583.

9. [9]	Gandarias, M.L. (2011),Weak self-adjoint differential equations, J. Phys. A: Math. Theor., 44, 262001 (6 pp).

10. [10]	Bluman, G.W. and Anco, S. (2002), Symmetry and Integration Methods for Differential Equations, Springer: New York.

11. [11]	Bluman, G.W. and Kumei, S. (1989), Symmetries and Differential Equations, Applied Mathematical Sciences, no. 81, Springer: New York.

12. [12]	Bluman, G.W., Cheviakov, A.F. and Anco, S. (2010), Applications of symmetry methods to partial differential equations, Springer: New York.

13. [13]	Ibragimov, N.H. (1983), Transformation groups in mathematical physics, Nauka: Moscow. English transl. (1985), Transformation groups applied to mathematical physics, Reidel: Dordrecht.

14. [14]	Ibragimov, N.H. (1994-1996), ed., CRC Handbook of Lie Group Analysis of Differential Equations, vols. 1-3, CRC Press: Boca Raton.

15. [15]	Olver, P.J. (1986), Applications of Lie groups to differential equations, Springer: New York.

16. [16]	Ovsyannikov, L.V. (1978), Group analysis of differential equations, Nauka: Moscow. English transl. (1982), Academic Press: New York.

17. [17]	Stephani, H. (1989), Differential equations: their solutions using symmetries, Cambridge Univ. Press: Cambridge.