Group Analysis of Variable Coefficient Diffusion-Convection Equations. II. Contractions and Exact Solutions

Abstract

This is the second part of the series of papers on symmetry properties of a class of variable coefficient (1+1)-dimensional nonlinear diffusion-convection equations of general form $f(x)u_t=(g(x)A(u)u_x{)}_x+h(x)B(u)u_x$. At first, we review the results of Part 1 of the series on equivalence transformations and group classification of the class under consideration. Investigation of non-trivial limits of parameterized subclasses of equations from the given class, which generate contractions of the corresponding maximal Lie invariance algebras, leads to the natural notion of contractions of systems of differential equations. After a brief discussion on contractions of symmetries, equations and solutions in general case, such types of contractions are studied for diffusion--convection equations. A detailed symmetry analysis of an interesting equation from the class under consideration is performed. Exact solutions of some subclasses of the considered class are also given.

19 pages

References in corpus (5)

• Conditional Lie-Bäcklund symmetry and reduction of evolution equations.
• Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities
• A precise definition of reduction of partial differential equations
• Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation
• Group Analysis of Variable Coefficient Diffusion--Convection Equations. III. Conservation Laws

Cited by in corpus (10)

• Extended group analysis of variable coefficient reaction-diffusion equations with exponential nonlinearities
• Finding the sigma pole by analytic extrapolation of $\pi \pi$ scattering data
• Lie symmetries and exact solutions of variable coefficient mKdV equations: an equivalence based approach
• Enhanced group classification of Gardner equations with time-dependent coefficients
• Generalized conditional symmetries of evolution equations
• Group classification of variable coefficient quasilinear reaction-diffusion equations
• Group Analysis of Variable Coefficient Diffusion--Convection Equations. III. Conservation Laws
• Group Analysis of Variable Coefficient Diffusion-Convection Equations. IV. Potential Symmetries
• Exact Solutions of Diffusion-Convection Equations
• Group-theoretical analysis of variable coefficient nonlinear telegraph equations