Lie symmetry analysis of the Heisenberg equation

Abstract The Lie symmetry analysis is performed on the Heisenberg equation from the statistical physics. Its Lie point symmetries and optimal system of one-dimensional subalgebras are determined. The similarity reductions and invariant solutions are obtained. Using the multipliers, some conservation laws are obtained. We prove that this equation is nonlinearly self-adjoint. The conservation laws associated with symmetries of this equation are constructed by means of Ibragimov’s method.

[1]  Gang-Wei Wang,et al.  Symmetry analysis and rogue wave solutions for the (2+1)-dimensional nonlinear Schrödinger equation with variable coefficients , 2016, Appl. Math. Lett..

[2]  Bo Han,et al.  On optimal system, exact solutions and conservation laws of the Broer-Kaup system , 2015 .

[3]  Nail H. Ibragimov,et al.  Nonlinear self-adjointness in constructing conservation laws , 2011, 1109.1728.

[4]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[5]  Mehdi Nadjafikhah,et al.  Symmetry analysis and conservation laws of the geodesic equations for the Reissner-Nordström de Sitter black hole with a global monopole , 2012 .

[6]  S. I. Svinolupov,et al.  Deformations of triple-Jordan systems and integrable equations , 1996 .

[7]  G. Bluman,et al.  Symmetry and Integration Methods for Differential Equations , 2002 .

[8]  M. B. Sheftel.  Lie groups and differential equations: symmetries, conservation laws, and exact solutions of mathematical models in physics , 1997 .

[9]  Sergey V. Meleshko,et al.  Symmetries of Integro-Differential Equations: With Applications in Mechanics and Plasma Physics , 2010 .

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

[11]  Muhammad Alim Abdulwahhab,et al.  Optimal system and exact solutions for the generalized system of 2-dimensional Burgers equations with infinite Reynolds number , 2015, Commun. Nonlinear Sci. Numer. Simul..

[12]  Giuseppe Saccomandi,et al.  Ordinary differential equations described by their Lie symmetry algebra , 2014, 1409.0762.

[13]  D. P. Mason,et al.  Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics , 2008, Appl. Math. Comput..

[14]  Nail H. Ibragimov,et al.  Nonlinear self-adjointness and conservation laws , 2011, 1107.4877.

[15]  Jean-François Ganghoffer,et al.  Similarity and Symmetry Methods , 2014 .

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

[17]  Zhi-Yong Zhang,et al.  On the existence of conservation law multiplier for partial differential equations , 2015, Commun. Nonlinear Sci. Numer. Simul..

[18]  Nail H. Ibragimov,et al.  Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras , 2016, Appl. Math. Lett..

[19]  Rahila Naz,et al.  Conservation laws for some compacton equations using the multiplier approach , 2012, Appl. Math. Lett..

[20]  Alexei F. Cheviakov,et al.  GeM software package for computation of symmetries and conservation laws of differential equations , 2007, Comput. Phys. Commun..

[21]  Mehdi Nadjafikhah,et al.  Lie symmetries and conservation laws of the Hirota–Ramani equation , 2011, 1111.5178.

[22]  Nail H. Ibragimov,et al.  Group analysis of kinetic equations in a non-linear thermal transport problem , 2015 .

[23]  N. Ibragimov.  A new conservation theorem , 2007 .