Conditional Lie Bäcklund Symmetries and Sign‐Invariants to Quasi‐Linear Diffusion Equations

Abstract

Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term u t = [u m (u x ) n ] x + P(u)u x + Q(u), where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bäcklund conditional symmetry with characteristic η = u xx + H (u)u 2x + G(u)(u x ) 2−n + F(u)u 1−n x and the Hamilton-Jacobi sign-invariant J = u t + A(u)u n+1x + B(u)u x + C(u) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

“…The homotopy model (8) can be freely chosen. Later for simplicity, the following simple homotopy model is exclusively taken…”

“…Usually, with a continuous differential equation, we can study its invariance, symmetry properties and similarity reductions by means of the Lie symmetry method [1,3]. In particular, for the mathematical models described by differential equations containing arbitrary elements (parameters or functions) which have been found experimentally and so are not strictly fixed, the symmetry approach allows one to simplify them which make the models admit a symmetry group with certain properties or the most extensive symmetry group [7][8][9].…”

Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems

“…It is important to present presumably the form of CLBS to study scalar diffusion equation. CLBSs related to separation of variables [50], sign-invariants [21] and invariant subspace [22,23] have been proved to be very effective to classify and seek for symmetry reductions of the considered equations. These form of CLBS can be extended to study diffusion systems and new results will be involved.…”

“…Indeed, since in principle, one can determine any solution by a suitably clever choice of CLBS unless one explicit know all possible solutions. An alternative tactic, which seems more practical, is to specify the CLBS by external considerations, for example, one might try CLBSs related to separation of variables [50], sign-invariants [21] and invariant subspaces [22,23] for scalar diffusion equations.…”

Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

“…The conditional Lie-Bäcklund symmetry (CLBS) method introduced by Zhdanov [9] and Fokas and Liu [10,11] firstly has been proved to be very powerful to classify equations or specify the functions appeared in the equations and construct the corresponding group invariant solutions. Furthermore, authors have shown that CLBS is closely related to the invariant subspace; namely, exact solutions defined on invariant subspaces for equations or their variant forms can be obtained by using the CLBS method [12][13][14][15][16][17][18][19][20][21][22][23][24].…”