Some invariant solutions and conservation laws of a type of long-water wave system
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We propose a generalized long-water wave system that reduces to the standard water wave system. We also obtain the Lax pair and symmetries of the generalized shallow-water wave system and single out some their similarity reductions, group-invariant solutions, and series solutions. We further investigate the corresponding self-adjointness and the conservation laws of the generalized system.
KeywordsSimilarity solution Conservation law Symmetry
PACS Codes05.45.Yv 02.30.Jr 02.30.Ik
2 Integrability of (2)
3 Similarity solutions and group-invariant solutions
4 The self-adjointness of system (2)
Ibragimov  introduced a few related notations of the strict self-adjointness, the nonlinear self-adjointness, and the quasiself-adjointness. Let us recall them.
5 Another expression of system (2) and some properties
In what follows, we consider the series solutions of (33).
6 Conservation laws
Anco and Bluman  proposed a method for constructing conservation laws of differential equations, which uses a formula directly generating the conservation laws and independent of the system having a Lagrangian formulation, in contrast to Noether’s theorem, which requires a Lagrangian. They adopted the linear equations and the adjoint equations of the original differential equations to study conservation laws. Essentially, the algorithm presented by Ibragimov et al. is the same as that of Anco and Bluman. Besides, Anco  also gave some comments on the work of Ibragimov.
In the paper, we have investigated various similarity reductions and exact solutions of the generalized BK system and various its conservation laws by the Lie group analysis. We have pointed out that the standard BK system is only a paticular case of the generalized BK system (2) when \(\alpha =-1\) and \(\beta =0\). In addition, Lou [11, 12] applied the symmetry group method to study some coherent solutions of nonlocal KdV systems and primary branch solutions of a first-order autonomous system. We hope to extend the methods to the systems presented in the paper in the forthcoming days. In addition, Ma  obtained some new conservation laws of some discrete evolution equation by symmetries and adjoint symmetries. Zhang, et al. [14, 15] considered symmetry properties of some fractional equations. Therefore there is an open problem how we can look for the fractional systems that correspond to the systems presented in the paper and how we can to solve them. Besides, Liu, Zhang, and Zhou  constructed the fractional Volterra hierarchy, gave a definition of the hierarchy in terms of Lax pair and Hamiltonian formalisms, and constructed its tau functions and multisoliton solutions. Bridgman, Hereman, Quispel, and Kamp  and El-Nabulsi  studied the peakon and Toda lattice. The approaches adopted in [16, 17, 18] can lead us to investigate some related properties of the generalized BK system presented in the paper. These questions will be discussed in the future.
The authors wish to thank the anonymous referees for their valuable suggestions.
Availability of data and materials
The authors declare that the study was realized in collaboration with the same responsibility. Both authors read and approved the final manuscript.
This work is supported by the Fundamental Research Funds for the Central University (No. 2017XKZD11).
The authors declare that they have no competing interests.
- 8.Ibagimov, N.H.: Nonlinear self-adjointness in constructing conservation laws pp. 1–104 (2011) arXiv:1109.1728vl [math-ph]
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