| Abstract: |
| A powerful application of symmetries is finding symmetry-invariant solutions of nonlinear differential equations.
These solutions satisfy a reduced differential equation with one fewer independent variable.
It is well known that \cite{Sjo} a double reduction occurs
whenever the starting nonlinear differential equation possesses a conservation law
that is invariant with respect to the symmetry.
Recent work \cite{Anc2016,AncKar,AncGanRec2018,AncGan2019} has developed
a broad generalization of the double-reduction method
by considering the space of invariant conservation laws with respect to a given symmetry.
The generalization is able to reduce a nonlinear partial differential equation (PDE) in $n$ variables to an ODE with $m-n+2$ first integrals where $m$ is the dimension of the space of invariant conservation laws.
In the present talk,
we apply this general multi-reduction method to obtain travelling wave solutions of some physically interesting PDEs.
An interesting side result is that we show how conservation laws that explicitly contain the independent variables
can nevertheless be used under certain conditions to obtain a reduction.
\end{abstract}
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A. Sj\oberg,
Double reduction of PDEs from the association of symmetries with conservation laws with applications,
Appl. Math. Comput. 184 (2007), 608616.
\bibitem{Anc2016}
S.C. Anco,
Symmetry properties of conservation laws,
Int. J. Mod. Phys. B 30(28n29) (2016), 1640004.
\bibitem{AncKar}
S. C. Anco and A. Kara,
Symmetry invariance of conservation laws,
Euro. J. Appl. Math., 29(1) (2018), 78--117.
\bibitem{AncGanRec2018}
S. C. Anco, M. L. Gandarias, E. Recio,
Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions,
Theor. Math. Phys., 197(1) (2018), 1393--1411.
\bibitem{AncGan2019}
S. C. Anco, M. L. Gandarias,
Symmetry multi-reduction method for partial differential equations with conservation laws. arxiv/1910.10839
\end{thebibliography} |
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