We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(D(u)ux)x+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.

A solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q(u)=φ(x)+ψ(t) for some single variable functions, q, φand ψ. In this paper, the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source. We obtain a complete list of canonical forms for such equations which admit the functionally separable solutions. As a result, we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source. The behavior and blow-up properties of some solutions are described.

This paper discusses a class of (n+1)-dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non-Newtonian fluids. It is shown that some radially symmetric equations admit certain types of conditional Lie B

Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity, and projective geometries on the real line, and restricted conformal, conformal, and projective geometries in the plane. Together with Chou and Qu [Physica D 162 (2002), 9-33], we determine inextensible motions and their associated integrable equations in all Klein geometries in the plane. The relations between several pairs of these geometries provide a natural geometric explanation of the existence of transformations of Miura and Cole-Hopf type.

The motion of plane curves in Klein geometry is studied. It is shown that the KdV, Harry-Dym, Sawada-Kotera, Burgers, the defocusing mKdV hierarchies, the Camassa-Holm and the Kaup-Kupershmidt equation naturally arise from the motions of plane curves in SL(2)-, Sim(2)-, SA(2)- and A(2)-geometries. These local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area. Motions of curves in Euclidean, special linear and similarity geometries corresponding to the traveling wave solutions of the mKdV, KdV and Burgers equations are discussed.

In this paper, the potential symmetry method is developed to study systems of nonlinear diffusion equations. Potential variables of the systems are introduced through conservation laws; such conservation laws yield equivalent systemsauxiliary systems of PDEs with the given dependent and potential variables as new dependent variables. Lie point symmetries of the auxiliary systems which cannot be projected to the vector fields of the given dependent and independent variables yield potential symmetries of the systems. Classification for systems of nonlinear diffusion equations with two and three components is performed. Symmetry reductions associated with the potential symmetries are presented.

It is shown that many well-known integrable equations such as the KdV, Harry Dym, Sawada-Kotera hierarchies and the Kaup-Kupershmidt, Boussinesq, Tzitz eica, Hirota-Satsuma equations naturally arise from motions of space curves in affine and centro-affine geometries. The motions of curves corresponding to travelling waves of the KdV and Sawada-Kotera equations are constructed explicitly.

Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term ut=[um(ux)n]x+P(u)ux + Q(u), where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bǎcklund conditional symmetry with characteristic η=uxx+H(u)u2x+G(u)(ux)2-n+F(u)ux1-n and the Hamilton-Jacobi sign-invariant J=ut+A(u)uxn+1+B(u)ux+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.