Symmetry analysis of a system of modified shallow-water equations

We revise the symmetry analysis of a modified system of one-dimensional shallow-water equations (MSWE) recently considered by Raja Sekhar and Sharma [Commun. Nonlinear Sci. Numer. Simulat. 20 (2012) 630-636]. Only a finite dimensional subalgebra of the maximal Lie invariance algebra of the MSWE, which in fact is infinite dimensional, was found in the aforementioned paper. The MSWE can be linearized using a hodograph transformation. An optimal list of inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra is constructed and used for Lie reductions. Non-Lie solutions are found from solutions of the linearized MSWE.

[1]  Nikolai A. Kudryashov,et al.  Seven common errors in finding exact solutions of nonlinear differential equations , 2009, 1011.4268.

[2]  K. T. Vu,et al.  Finding higher symmetries of differential equations using the MAPLE package DESOLVII , 2012, Comput. Phys. Commun..

[3]  C. Rogers,et al.  Group theoretical analysis of a rotating shallow liquid in a rigid container , 1989 .

[4]  Complete classification of stationary flows with constant total pressure of ideal incompressible infinitely conducting fluid , 2011, 1111.7119.

[5]  B. Huard.  Conditionally invariant solutions of the rotating shallow water wave equations , 2010, 1002.3669.

[6]  Elsa Dos Santos Cardoso-Bihlo,et al.  Complete point symmetry group of the barotropic vorticity equation on a rotating sphere , 2012, 1206.6919.

[7]  Elsa Dos Santos Cardoso-Bihlo,et al.  Complete group classification of a class of nonlinear wave equations , 2011, 1106.4801.

[8]  R. Popovych,et al.  Lie symmetries and exact solutions of barotropic vorticity equation , 2009, 0902.4099.

[9]  Peter E. Hydon,et al.  Symmetry Methods for Differential Equations: Preface , 2000 .

[10]  Alexander Bihlo,et al.  Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures , 2009, 0909.1957.

[11]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[12]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[13]  How to construct the discrete symmetries of partial differential equations , 2000, European Journal of Applied Mathematics.

[14]  O. Hijazi.  Application of symmetry methods to partial differential equations , 2013 .

[15]  Sergey V. Meleshko,et al.  Methods for Constructing Exact Solutions of Partial Differential Equations: Mathematical and Analytical Techniques with Applications to Engineering , 2005 .

[16]  Roman O. Popovych,et al.  Invariant Discretization Schemes for the Shallow-Water Equations , 2012, SIAM J. Sci. Comput..

[17]  O. Vaneeva,et al.  More common errors in finding exact solutions of nonlinear differential equations: Part I , 2009, 0911.1848.

[18]  Victor A. Galaktionov,et al.  Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics , 2006 .

[19]  Vanessa Hertzog,et al.  Applications Of Group Theoretical Methods In Hydrodynamics , 2016 .

[20]  Elsa Dos Santos Cardoso-Bihlo,et al.  Enhanced preliminary group classification of a class of generalized diffusion equations , 2010, 1012.0297.

[21]  Jiří Patera,et al.  Subalgebras of real three‐ and four‐dimensional Lie algebras , 1977 .

[22]  R. Popovych,et al.  Point symmetry group of the barotropic vorticity equation , 2010, 1009.1523.

[23]  G. Bluman,et al.  Applications of Symmetry Methods to Partial Differential Equations , 2009 .

[24]  A. Chesnokov.  Symmetries and exact solutions of the rotating shallow-water equations , 2008, European Journal of Applied Mathematics.

[25]  Elsa Dos Santos Cardoso-Bihlo,et al.  Complete point symmetry group of vorticity equation on rotating sphere , 2012 .

[26]  S. Meleshko.  A Particular Class of Partially Invariant Solutions of the Navier—Stokes Equations , 2004 .

[27]  G. Popovich,et al.  Flows of an incompressible fluid with linear vorticity , 1999 .

[28]  R. Popovych.  On Lie Reduction of the Navier-Stokes Equations , 1995 .

[29]  R. Popovych,et al.  Symmetry reduction and exact solutions of the Navier-Stokes equations. I , 1994, math-ph/0207016.

[30]  Roman O. Popovych,et al.  Realizations of real low-dimensional Lie algebras , 2003 .

[31]  V. Sharma,et al.  Similarity analysis of modified shallow water equations and evolution of weak waves , 2012 .

[32]  Halyna V. Popovych.  Lie, Partially Invariant, and Nonclassical Submodels of Euler Equations , 2002 .

[33]  J H Nixon,et al.  Application of Lie groups and differentiable manifolds to general methods for simplifying systems of partial differential equations , 1991 .

[34]  S. Meleshko,et al.  Rotationally Invariant and Partially Invariant Flows of a Viscous Incompressible Fluid and a Viscous Gas , 2002 .

[35]  A. Aksenov,et al.  CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 2. Applications in Engineering and Physical Sciences , 1995 .

[36]  R. Popovych,et al.  Symmetry reduction and exact solutions of the Navier-Stokes equations. I , 1994, math-ph/0207016.