Analysis of a class of potential Korteweg-de Vries-like equations

Abstract

We analyze a class of third-order evolution equations, i.e. u t = f(x, u x , u xx ) u xxx +g(x, u x , u xx ) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries.

On the method of preliminary group classification applied to the nonlinear heat equation ut=f(x,ux)uxx+g(x,ux)

“…In its most general form, the nonlinear heat equation, viz, is of considerable interest in mathematical physics . Various forms of this equation have been analyzed and successfully used to model physical situations involving diffusion in a wide range of fields . Specific forms of Equation are also utilized in plasma physics and metallurgy problems .…”

On a generalized variable‐coefficient Gardner equation with linear damping and dissipative terms

“…For instance, it can be used to determine exact solutions, to reduce PDEs or to construct conservation laws. ()…”

“…For instance, it can be used to determine exact solutions, to reduce PDEs or to construct conservation laws. [13][14][15][16][17][18][19][20][21][22] The problem lies in the fact that the analysis of variable-coefficient equations is often difficult. Equivalence transformations fit perfectly into the study of variable-coefficient PDEs.…”