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Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov--Kuznetsov equations | | Ding-jiang Huang
; Nataliya M. Ivanova
; | | Date: |
6 Sep 2013 | | Abstract: | In this paper, we explain in more details the modern treatment of the problem
of group classification of (systems of) partial differential equations (PDEs)
from the algorithmic point of view. More precisely, we revise the classical
Lie--Ovsiannikov algorithm of construction of symmetries of differential
equations, describe the group classification algorithm and discuss the process
of reduction of (systems of) PDEs to (systems of) equations with smaller number
of independent variables in order to construct invariant solutions. The group
classification algorithm and reduction process are illustrated by the example
of the generalized Zakharov--Kuznetsov (GZK) equations of form
$u_t+(F(u))_{xxx}+(G(u))_{xyy}+(H(u))_x=0$. As a result, a complete group
classification of the GZK equations is performed and a number of new
interesting nonlinear invariant models which have non-trivial invariance
algebras are obtained. Lie symmetry reductions and exact solutions for two
important invariant models, i.e., the classical and modified
Zakharov--Kuznetsov equations, are constructed. The algorithmic framework for
group analysis of differential equations presented in this paper can also be
applied to other nonlinear PDEs. | | Source: | arXiv, 1309.1664 | | Services: | Forum | Review | PDF | Favorites | |
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