Group analysis of the Novikov equation
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Group analysis of the Novikov equation Yuri Bozhkov · Igor Leite Freire · Nail H. Ibragimov
Received: 15 May 2013 / Accepted: 20 June 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential equations, we find the conservation law corresponding to the dilation symmetry and show that other symmetries do not provide nontrivial conservation laws. Then we investigate the invariant solutions. Keywords Novikov equation · Lie symmetries · Strictly self-adjointness · Conservation laws · Invariant solutions Mathematics Subject Classification (2000)
76M60 · 58J70 · 35A30 · 70G65
Communicated by Eduardo Souza de Cursi. Y. Bozhkov Instituto de Matemática, Estatística e Computação Científica-IMECC, Universidade Estadual de Campinas-UNICAMP, Rua Sérgio Buarque de Holanda, 651, Campinas, SP 13083-859, Brasil e-mail: [email protected] I. L. Freire (B) Centro de Matemática, Computação e Cognição, Universidade Federal do ABC-UFABC, Rua Santa Adélia, 166, Bairro Bangu, Santo André, SP 09.210-170, Brasil e-mail: [email protected]; [email protected] N. H. Ibragimov Laboratory “Group Analysis of Mathematical Models in Natural and Engineering Sciences”, Ufa State Aviation Technical University, 450000 Ufa, Russia N. H. Ibragimov Department of Mathematics and Science, Blekinge Institute of Technology, 371 79 Karlskrona, Sweden e-mail: [email protected]; [email protected]
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1 Introduction In [4], Camassa and Holm derived a completely integrable dispersive nonlinear partial differential equation using an asymptotic expansion in the Hamiltonian for Euler’s equation concerned with shallow water regime. Such a derived equation u t + 2κu x − u x xt + 3uu x = 2u x u x x + uu x x x
(1)
is currently known as Camassa–Holm equation. Since their work, intense research dealing with integrable non-evolutionary partial differential equations of the type u t − u t x x = F(u, u x , u x x , u x x x , . . .)
(2)
has been carried out. Such equations are used in modeling shallow water waves. During sometime the Camassa–Holm equation was the only known example of integrable equation of the type (2) having solutions as a superposition of multipeakons, that is, peaked soliton solutions with discontinuous derivatives at the peaks. In [5], it was proved that an equation obtained previously by Degasperis and Procesi also had such property. In a recent communication [21], a classification of integrable equations of the type (2) with quadratic and cubic nonlinearities was carried out. In the same paper the new partial differential equation u t − u t x x + 4u 2 u x − 3uu x u x x − u 2 u x x x = 0
(3)
containing cubic nonlinearities was discovered by V. S. Novikov and is named after him. Equation (3) is considered as a type of generalization of (1). Since th
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