Symmetries and solutions to the thin film equations

Abstract

We study symmetries and solutions of the generalized fourth-order nonlinear partial differential equations which arise from studies of thin liquid films. It is shown that the equations admit extended scaling and rotation symmetries and a class of generalized conditional symmetries for certain coefficient functions. Exact solutions associated to the symmetries are obtained.

Equivalence group of a fourth-order evolution equation unifying various non-linear models

“…(1) considered recently by Qu [1] encapsulates various particular mathematical models discussed intensely in the literature (see [1] and the references therein). The following cases are of great interest due to significant applications.…”

Invariant sets and solutions to the two-dimensional nonlinear reaction–diffusion equations

“…If c 4 < 0 or (2c 1 c 4 )/(2c 1 + c 3 ) > 0, then the point P 2 is unstable for t → ∞. If c 4 > 0 and (2c 1 c 4 )/(2c 1 + c 3 ) < 0, then P 2 is a stable fixed point for t → ∞, which implies that f (t) → 0, g(t) → p 22 and h(t) → p 23 as t → ∞. …”

“…The approach has been used to construct exact solutions to several nonlinear evolution equations. Further extension to the scaling and rotation groups were introduced by Qu and Estevez [11,22], where the invariant set is governed by…”

The Periodic Solitary Wave Solutions for the (2 + 1)-Dimensional Fifth-Order KdV Equation

“…The researches on the explicit analytic solutions for the soliton equations can help understand the nonlinear dynamics better. With the development of soliton theory, there are many systematic approaches solving different kinds of soliton solutions, such as the inverse scattering transformation [1] [2], the Darboux transformation [3], the variable seperation method [4], the bilinear method and so on [5]- [7]. Among those methods, the bilinear method is a powerful and direct approach to find soliton solutions for the nonlinear partial differential equations.…”