Second-Order Conditional Lie-Bäcklund Symmetries and Differential Constraints of Nonlinear Reaction-Diffusion Equations with Gradient-Dependent Diffusivity

The radially symmetric nonlinear reaction–diffusion equation with gradient-dependent diffusivity is investigated. We obtain conditions under which the equations admit second-order conditional Lie–Bäcklund symmetries and first-order Hamilton–Jacobi sign-invariants which preserve both signs (≥0 and ≤0) on the solution manifold. The corresponding reductions of the resulting equations are established due to the compatibility of the invariant surface conditions and the governing equations.

[1]  Stanislav Spichak,et al.  On the Poincare-Invariant Second-Order Partial Equations for a Spinor Field , 1996 .

[2]  Victor A. Galaktionov,et al.  Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications , 2004 .

[3]  J. Pascal,et al.  On some similarity solutions to shear flows of non-Newtonian power law fluids , 1995 .

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

[5]  Victor A. Galaktionov,et al.  A Stability Technique for Evolution Partial Differential Equations , 2009 .

[6]  J. Pascal,et al.  On some diffusive waves in non-linear heat conduction , 1993 .

[7]  V. Galaktionov,et al.  New explicit solutions of quasilinear heat equations with general first-order sign-invariants , 1996 .

[8]  Lina Ji,et al.  Conditional Lie Bäcklund symmetries and solutions to "n+1…-dimensional nonlinear diffusion equations , 2007 .

[9]  C. Qu,et al.  Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source , 2000 .

[10]  Artur Sergyeyev.  Constructing conditionally integrable evolution systems in (1 + 1) dimensions: a generalization of invariant modules approach , 2002 .

[11]  R. Zhdanov,et al.  Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries , 2001 .

[12]  R. Cherniha,et al.  Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications , 2017 .

[13]  R. Zhdanov.  Higher Conditional Symmetry and Reduction of Initial Value Problems , 2002 .

[14]  N. I. Serov,et al.  Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics , 1993 .

[15]  R. Cherniha,et al.  Lie symmetry properties of nonlinear reaction-diffusion equations with gradient-dependent diffusivity , 2016, Commun. Nonlinear Sci. Numer. Simul..

[16]  Liu,et al.  Nonlinear interaction of traveling waves of nonintegrable equations. , 1994, Physical review letters.

[17]  C. Qu,et al.  Conditional Lie–Bäcklund Symmetries and Invariant Subspaces to Nonlinear Diffusion Equations with Convection and Source , 2013 .

[18]  Shoufeng Shen,et al.  Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System , 2014 .

[19]  A. A. Samarskii,et al.  A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures , 1988 .

[20]  V. Galaktionov,et al.  Maximal sign-invariants of quasilinear parabolic equations with gradient diffusivity , 1998 .

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

[22]  B. H. Bradshaw-Hajek,et al.  Exact non-classical symmetry solutions of Arrhenius reaction–diffusion , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  J. Pascal,et al.  On some non-linear shear flows of non-Newtonian fluids , 1995 .

[24]  R. S. Mendes,et al.  Nonlinear diffusion equation, Tsallis formalism and exact solutions , 2005 .

[25]  Jianping Wang,et al.  Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems , 2015 .

[26]  Lina Ji,et al.  Conditional Lie Bäcklund Symmetries and Sign‐Invariants to Quasi‐Linear Diffusion Equations , 2007 .

[27]  R. Zhdanov,et al.  CONDITIONAL LIE-BACKLUND SYMMETRY AND REDUCTION OF EVOLUTION EQUATIONS , 1995 .

[28]  Peter J. Olver,et al.  Direct reduction and differential constraints , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

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

[30]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

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

[32]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .

[33]  S. Lou,et al.  New variable separation approach: application to nonlinear diffusion equations , 2002, nlin/0210017.

[34]  Changzheng Qu,et al.  Group Classification and Generalized Conditional Symmetry Reduction of the Nonlinear Diffusion–Convection Equation with a Nonlinear Source , 1997 .

[35]  Qu Changzheng,et al.  Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method , 1999 .

[36]  V. Galaktionov.  Quasilinear heat equations with first-order sign-invariants and new explicit solutions , 1994 .

[37]  R. Zhdanov,et al.  Non-classical reductions of initial-value problems for a class of nonlinear evolution equations , 2000 .

[38]  Lina Ji.  Conditional Lie–Bäcklund symmetries and invariant subspaces to nonlinear diffusion equations with source , 2012 .

[39]  S. Svirshchevskii.  Invariant Linear Spaces and Exact Solutions of Nonlinear Evolution Equations , 1996 .

[40]  C. Qu,et al.  Solutions and symmetry reductions of the n-dimensional non-linear convection―diffusion equations , 2010 .

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

[42]  Changzheng Qu,et al.  Invariant subspaces and conditional Lie-Bäcklund symmetries of inhomogeneous nonlinear diffusion equations , 2013 .

[43]  A. Jamiołkowski.  Book reviewApplications of Lie groups to differential equations : Peter J. Olver (School of Mathematics, University of Minnesota, Minneapolis, U.S.A): Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, XXVI+497pp. , 1989 .

[44]  Conditional Lie-B\"acklund symmetry and reduction of evolution equations. , 1995, solv-int/9505006.

[45]  C. Qu,et al.  On nonlinear diffusion equations with x-dependent convection and absorption , 2004 .

[46]  P. Broadbridge,et al.  Nonclassical symmetry analysis of nonlinear reaction-diffusion equations in two spatial dimensions , 1996 .