Geometric Structure and Some Exact Solutions of Plateau Equation

In this paper, we get the set of symmetry of Plateau differential equation. Using Lie symmetry method to obtain the classical symmetry operators. Also, we get one- dimensional optimal system of the Plateau equation and reduction Lie invariants, corresponding to infinitesimal symmetries.


INTRODUCTION
One of the most important discoveries of Sophus Lie, in differential equation is to show that, it is possible to transform non-linear conditions in a system, to linear conditions, by infinitesimal invariants, corresponding to the symmetry group generators, of the system [1]. In this article, our aim is to obtain a set of symmetries of Plateau equation [2,3]: Which it governs, under appropriate hypotheses the plane motion of a fluid. The function ‫ݑ‬ appearing in the equation is so-called kinetic potential [4] and the special case of the plateau Original Research Article equation may be used to derive a relation from the interfacial geometry of watting fluid in a capillary [5]. Minimal surfaces are defined as surfaces with zero mean curvature. The minimal surfaces PDE of the surface given by equation ‫ݑ‬ = ݂ሺ‫,ݔ‬ ‫ݕ‬ሻ has the Plateau equation form [6,7]. The classical Lie symmetries are obtained using the Lie symmetry method. This requires the utilization of computer softwares, because working with continuous groups has computations that follow from the algorithmic process. Having the symmetry group of a system of equations, has a lot of advantages, one of which is the classification of the solutions of the system. This classification is to consider, two solutions in one class if they can be converted to each other, by an element of the symmetry group. If we have an ordinary system, the symmetry group will help us to obtain the exact solution. If the equation is order one, it is possible to get the general solution, but it is not the case for PDE, unless the system is convertible to a linear system. Another application of the symmetry group is the probable reduction of the number of independent variables and the ideal condition is converting to ODE.

LIE SYMMETRY OF PLATEAU EQUATION
The method of determining the classical symmetries of a partial differential equation is standard and is described in [8,9,10]. To obtain the symmetry algebra of (1), we take an infinitesimal generator of symmetry algebra of the form: Using the invariant condition, i.e., applying the fourth prolongation ‫ݒ‬ ሺଶሻ to (1), the following system of 15 determining equation yields:

ሺ3ሻ
By solving the above system we will have the following theorem.
Theorem 2.1: The Lie group of point symmetries of the Plateau equation, has a Lie algebra generator in the form of the vector field ‫,ݒ‬ with the following functional coefficients.
Theorem 2.2: The infinitesimal generators from the Lie one-parameter group of the symmetries of the Plateau equation are as follows: These vector fields produce a Lie algebra space ࣡ with the following commutator

GROUP INVARIANT SOLUTIONS OF PLATEAU EQUATION
To obtain the group of transformations which are generated by infinitesimal generators ‫ݒ‬ , ݅ = 1, ⋯ ,7, we should solve the first order system involving first order equations in correspondence to each of the generators simultaneously. By solving this system, the one parameter group of ݃ ሺ‫ݏ‬ሻ: ‫ܯ‬ → ‫ܯ‬ generated by ‫ݒ‬ , ݅ = 1, ⋯ ,7 involved in theorem (2.2) is obtained in the following way; Therefore, we will have the following theorem:

OPTIMAL SYSTEM OF ONE-DIMENSIONAL SUBALGEBRAS OF PLATEAU EQUATION
Now we want to obtain one-dimensional optimal system of the Plateau equation using its symmetry group. The optimal system is in fact a standard method for the classification of one-dimensional sub-algebras in which each class involves conjugate equivalent members [11]. Also, they involve the group adjoint representation which establishes an equivalent relation among all conjugate sub-algebra elements. In fact, the classification problem for one-dimensional sub-algebra is the same as the problem of the classification of the representation of its adjoint orbits. In this way, the optimal system is constructed. The set of invariant solutions corresponding to a one-dimensional sub-algebra is a list of minimal solutions, where all the other invariant solutions can be obtained by transformations [12]. To calculate the adjoint representation, we consider the following Lie series: Where ܽ ∈ ℝ is arbitrary constant.

SIMILARITY REDUCTION OF PLATEAU EQUATION
The Plateau equation has been stated with the ሺ‫,ݔ‬ ‫;ݕ‬ ‫ݑ‬ሻ coordinate, but we are looking for a new coordinate that the equation will reduce, if we write it in the new coordinate. This new coordinate is obtained through ሺ‫,ݐ‬ ‫ݒ‬ሻ dependent invariant corresponding to the infinitesimal symmetry generator. If we state the Plateau equation with the new coordinate, using the chain rule, a reduced equation will result. Now we calculate the invariants corresponding to the symmetry generators existing in the optimal system. The first status for the one element of the optimal system is ‫ݒ‬ . It has the determining equation in the form , we can state the derivatives of ‫ݑ‬ with respect to ‫ݔ‬ and ‫,ݕ‬ in the form of ‫ݒ‬ and ‫ݐ‬ and the derivatives of ‫ݒ‬ with respect to ‫.ݐ‬ If we substitute it in the Plateau equation, the Plateau equation turns to an ordinary equation as: ሺ1 + ‫ݐ‬ ଶ + ‫ݒ‬ ଶ ሻ‫ݒ‬ ௧௧ = 0. In the Table below, the invariants corresponding to each of elements of symmetry group and optimal system have been calculated. By substituting ‫ݐ‬ and ‫ݒ‬ and the derivative of ‫ݒ‬ in respect to ‫ݐ‬ , in place of ‫,ݑ‬ ‫,ݔ‬ ‫ݕ‬ and the derivative of ‫ݑ‬ in respect to ‫,ݔ‬ ‫,ݕ‬ in the Plateau equation, it turns into an ODE.
In the above

CONCLUSION
In this paper we obtained the Lie point symmetries of the Plateau equation by using the Lie symmetry method. Also computed the one-dimensional optimal system. This led to reducing the Plateau equation to ODE's and computing the invariants of Plateau equation.