Research Article | Open Access
On the Existence of Variational Principles for a Class of the Evolutionary Differential-Difference Equations
Necessary and sufficient conditions for the existence of variational principles for a given wide class of evolutionary differential-difference operator equation are obtained. The theoretic results are illustrated by two examples.
We consider the equation:
where is a domain of the definition of the operator , and are normed linear spaces over the field of real numbers .
Later, we will assume that at every point , there exists the Gâteaux derivative of defined by the formula .
The operator is said to be potential  on the set relative to a given bilinear form , if there exists a functional such that
In that case, we also say that the given equation admits the direct variational formulation.
A problem of the construction of the functional upon the given operator is known as the classical inverse problem of the calculus of variations . Note that practically no one has been solving inverse problem of the calculus of variations for partial differential-difference operators until recently [2–4]. Let us also note that for a wide classes of partial differential equations, there has been developped the problem of recongnition of variationality upon the structure of corresponding operators [5, 6]. There is a theoretical and practical interst in the extention of these results on partial differential-difference equations [7–9].
For what follows, we suppose that is a convex set, and we also need the following potentiality criterion : Under this condition, the potential is given by
where is a fixed element of D(N).
The functional is called the potential of the operator , and in turn the operator is called the gradient of the functional .
2. Statement of the Problem
Let us consider the following differential-difference operator equation:
Here, are linear operators depending on is in general a nonlinear operator;, where are real normed linear spaces, .
The domain of definition is given by the equality: where are given functions.
Under the solution of (2.1), we mean a function satisfying the identity:
Let us give the following bilinear form: where the bilinear mapping satisfies the following conditions:
Our aim is to define the structure of operators and under which (2.1) allows the solution of the inverse problem of the calculus of variations relative to the bilinear form (2.4) such that is the antisymmetric operator on , that is,
3. Conditions of Potentiality and the Structure of (2.1) in the Case of Its Variationality
We denote by the operator adjoint to .
Theorem 3.1. If on the set , then for the existence of the direct variational formulation for the operator (2.1) on the set relative to (2.4), it is necessary and sufficient that the following conditions hold on the set :
Proof. Taking into account formula (2.1), we get The criterion of potentiality takes the following form: or Bearing into account the condition on the set , from (3.4), we get or Thus, condition (3.4) can be reduced to the following form: This equality is fulfilled identically if and only if for all . Thus, it is necessary and sufficient that conditions (3.1) hold.
Proof. If on the set and conditions (3.1) are held, then according to Theorem 3.1, operator (2.1) is potential on the set relative to a given bilinear form (2.4).
Let us consider the following functional: It is easy to check that Then functional (3.10) is a potential of evolutionary operator (2.1).
If on , then
This theorem shows the structure of the given kind of differential-difference operator, which admits the solution of the inverse problem of the calculus of variations.
Example 4.1. Let us consider the evolutionary differential-difference equation with partial derivatives in the following form:
is a bounded domain in with piecewise smooth boundary .
The domain of definition is given by the equality: where , are given functions.
We investigate the existence of variational principle for (4.1) relative to a given bilinear form (2.4).
For (4.1), we get
Necessary and sufficient conditions of potentiality take the form:
From that, we come to the following:
That is true if and only if
Under the fulfilment of that conditions, the corresponding functional is given by
Let us consider an example when this criterion of potentiality fails.
Example 4.2. Consider the equation:
Let us note that this equation is a Korteweg-de Vries’ equation .
We denote It is easy to be convinced that operator (4.8) is not potential on set (4.9) relative to the bilinear form: Here, .
We define the integrating operator as .
Then, the operator,
is potential on set (4.9) relative to bilinear form (4.10). The corresponding functional has the form:
Indeed, using (4.12), we find that
From the condition , for all , we obtain this equation is equivalent to (4.8).
Let us note that the formula defines the first integral of (4.8).
The work of V. M. Sovchin was partially supported by the Russian Foundation for Basic Research (Projects 09-01-00093 and 09-01-00586).
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Copyright © 2012 I. A. Kolesnikova and V. M. Savchin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.