Symmetry Groups, Conservation Laws and Group– Invariant Solutions of the Membrane Shape Equation

Abstract

The six-parameter group of three dimensional Euclidean motions is recognized as the largest group of point transformations admitted by the membrane shape equation in Mongé representation. This equation describes the equilibrium shapes of biomembranes being the Euler-Lagrange equation associated with the Helfrich curvature energy functional under the constraints of fixed enclosed volume and membrane area. The conserved currents of six linearly independent conservation laws that correspond to the variational symmetries of the membrane shape equation and hold on its smooth solutions are obtained. All types of non-equivalent group-invariant solutions of the membrane shape equation are identified via an optimal system of onedimensional subalgebras of the symmetry algebra. The reduced equations determining these group-invariant solutions are derived. Special attention is paid to the translationally-invariant solutions of the membrane shape equation.

On the Translationally-invariant Solutions of the Membrane Shape Equation

“…Within the framework of the Helfrich spontaneous curvature model [3], the equilibrium shapes of a biomembrane, assumed as a bilayer of amphiphilic molecules (phospholipids, for instance), are described in terms of the mean H and Gaussian K curvatures of its middle-surface S by the membrane shape equation [7,8] 2k c ∆H + k c (2H + Ih) (2H 2 − IhH − 2K) − 2λH + p = 0 (1) where k c , Ih and λ are real constants representing the bending rigidity, spontaneous curvature and tensile stress of the membrane, respectively, while p is the osmotic pressure difference between the outer and inner media assumed to be a real constant too. Here, ∆ is the Laplace-Beltrami operator on the surface S. In a previous study by the present authors (see [10]), it is established that the six-parameter group of motions in the three dimensional Euclidean space is the largest group of point transformations admitted by the membrane shape equation in Mongé representation. In that work, all types of non-equivalent group-invariant solutions of this equation are identified via an optimal system of one-dimensional subalgebras of the symmetry algebra and the corresponding reduced equations are derived.…”

“…In that work, all types of non-equivalent group-invariant solutions of this equation are identified via an optimal system of one-dimensional subalgebras of the symmetry algebra and the corresponding reduced equations are derived. In [10], special attention is paid to the translationally-invariant solutions of the membrane shape equation assuming that the osmotic pressure difference p = 0 since the case p = 0 is thoroughly studied elsewhere (see [4,5,9,11]). All translationally-invariant solutions of the membrane shape equation that are expressed in elementary functions as well as a class of such solutions that are given in terms of elliptic functions are obtained in [10].…”

Traveling Wave Solutions of the Gardner Equation and Motion of Plane Curves Governed by the mKdV Flow

“…4-16] (see also [31,32]). In fact, it is relatively easy to analyse the cases in which the foregoing polynomial possesses multiple roots and to write down the corresponding general solutions, see [33].…”

“…(24) and dφ(ξ )/dξ =κ(ξ ). The analytic expressions for the aforementioned functionsκ(ξ ) andφ(ξ ) are known and can be found, for instance, in the recent papers [33,36,37] where the present authors have studied the differential equations (24) and (25) for the curvatureκ(ξ ) with the aim to achieve an analytic description of the equilibrium shapes of cylindrical lipid bilayer membranes, elastic rings and tubes under uniform hydrostatic pressure. A summary of these results is presented below.…”

Cylindrical Shapes of Helfrich Spontaneous-Curvature Model

“…In comparison with the fourth order nonlinear partial differential equation in the Mongé representation of the Helfrich equation (see e.g. [21]), the system of differential equations (5) is a simpler version of the Helfrich model.…”