On the group classification and conservation laws of the self-adjoint first order evolution equations

In this work we consider the problem on group classification and conservation laws of the general first order evolution equations. We obtain the subclasses of these general equations which are quasiself-adjoint and self-adjoint. By using the recent Ibragimov’s Theorem on conservation laws, we establish the conservation laws of the equations admiting self-adjoint equations. We illustrate our results applying them to the inviscid Burgers’ equation. In particular an infinite number of new symmetries of these equations are found and their corresponding conservation laws are established.

[1]  Fábio Antonio Dorini,et al.  Statistical moments of the solution of the random Burgers-Riemann problem , 2009, Math. Comput. Simul..

[2]  N. Ibragimov.  Transformation groups applied to mathematical physics , 1984 .

[3]  N. Ibragimov,et al.  Equivalence group of a fourth-order evolution equation unifying various non-linear models , 2008 .

[4]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[5]  R. K. Gazizov,et al.  Lie Symmetry Analysis of Differential Equations in Finance , 1998 .

[6]  R. Tracinà,et al.  On some differential invariants for a family of diffusion equations , 2007 .

[7]  Tarcísio M. Rocha Filho,et al.  Lie Symmetries of Fokker-Planck Equations with Logarithmic diffusion and Drift Terms , 2006 .

[8]  Classification of Similarity Solutions for Inviscid Burgers’ Equation , 2010 .

[9]  Igor Leite Freire,et al.  Conservations laws for critical Kohn-Laplace equations on the Heisenberg group , 2007, math/0703711.

[10]  C. Sarrico.  New solutions for the one-dimensional nonconservative inviscid Burgers equation☆ , 2006 .

[11]  S. El-Labany,et al.  Group classification and symmetry reduction of variable coefficient nonlinear diffusion-convection equation , 2002 .

[12]  Group Classification of the General Evolution Equation: Local and Quasilocal Symmetries , 2005, nlin/0510003.

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

[14]  Fábio Antonio Dorini,et al.  A finite volume method for the mean of the solution of the random transport equation , 2007, Appl. Math. Comput..

[15]  M. Sepanski.  Global actions of Lie symmetries for the nonlinear heat equation , 2009 .

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

[17]  M. Bruzón,et al.  Symmetry analysis and solutions for a family of Cahn-Hilliard equations , 2000 .

[18]  R. Zhdanov,et al.  Group classification of the general second-order evolution equation: semi-simple invariance groups , 2007 .

[19]  Emrullah Yaşar,et al.  On the conservation laws and invariant solutions of the mKdV equation , 2010 .

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

[21]  Fábio Antonio Dorini,et al.  Statistical moments of the random linear transport equation , 2008, J. Comput. Phys..

[22]  Changzheng Qu,et al.  Symmetries and solutions to the thin film equations , 2006 .

[23]  R. Z. Zhdanov,et al.  Group classification of heat conductivity equations with a nonlinear source , 1999 .

[24]  Nail H. Ibragimov,et al.  Quasi-self-adjoint differential equations , 2007 .

[25]  Yousef Hashem Zahran,et al.  Central ADER schemes for hyperbolic conservation laws , 2008 .

[26]  Meina Sun,et al.  Interactions of delta shock waves for the transport equations with split delta functions , 2009 .

[27]  E. Momoniat.  A mapping of a generalisation of a thin film equation on a moving substrate , 2007 .

[28]  Stephen C. Anco,et al.  New conservation laws obtained directly from symmetry action on a known conservation law , 2006 .

[29]  A. Samoilenko,et al.  Group Classification of Nonlinear Evolution Equations: I. Invariance Under Semisimple Local Transformation Groups , 2002 .

[30]  Roman O. Popovych,et al.  New results on group classification of nonlinear diffusion–convection equations , 2003 .

[31]  Mehdi Nadjafikhah.  Lie Symmetries of Inviscid Burgers’ Equation , 2009 .

[32]  Roman Cherniha,et al.  Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations , 2008 .

[33]  Nail H. Ibragimov,et al.  Self-adjoint sub-classes of generalized thin film equations , 2009 .

[34]  Wei Xu,et al.  Separation of variables and exact solutions to nonlinear diffusion equations with x-dependent convection and absorption ✩ , 2008 .

[35]  C. Sarrico.  Distributional products and global solutions for nonconservative inviscid Burgers equation , 2003 .

[36]  Michael Oberguggenberger,et al.  Interactions of delta shock waves in a strictly hyperbolic system of conservation laws , 2008 .