Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 780382 | 9 pages | https://doi.org/10.1155/2012/780382

On the Existence of Variational Principles for a Class of the Evolutionary Differential-Difference Equations

Academic Editor: V. Stepanov
Received10 Mar 2011
Accepted18 Oct 2011
Published04 Jan 2012

Abstract

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.

1. Introduction

We consider the equation:𝑁(𝑢)=0,𝑢∈𝐷(𝑁),(1.1)

where 𝐷(𝑁) is a domain of the definition of the operator 𝑁∶𝐷(𝑁)⊆𝑈→𝑉, and 𝑈,𝑉 are normed linear spaces over the field of real numbers ℝ.

Later, we will assume that at every point 𝑢∈𝐷(𝑁), there exists the Gâteaux derivative ğ‘î…žğ‘¢ of 𝑁 defined by the formula (𝑑/𝑑𝜖)𝑁(𝑢+ğœ–â„Ž)|𝜖=0=𝛿𝑁(𝑢,ℎ)=ğ‘î…žğ‘¢â„Ž.

The operator 𝑁∶𝐷(𝑁)→𝑉 is said to be potential [1] on the set 𝐷(𝑁) relative to a given bilinear form Φ(⋅,⋅)∶𝑉×𝑈→𝑅, if there exists a functional 𝐹𝑁∶𝐷(𝐹𝑁)=𝐷(𝑁)→𝑅 such that𝛿𝐹𝑁[]𝑁𝑢,ℎ=Φ(𝑁(𝑢),ℎ),∀𝑢∈𝐷(𝑁),âˆ€â„Žâˆˆğ·î…žğ‘¢î€¸.(1.2)

In that case, we also say that the given equation admits the direct variational formulation.

A problem of the construction of the functional 𝐹𝑁 upon the given operator 𝑁 is known as the classical inverse problem of the calculus of variations [1]. 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 𝐷(𝑁) is a convex set, and we also need the following potentiality criterion [1]: Î¦î€·ğ‘î…žğ‘¢î€¸î€·ğ‘â„Ž,𝑔=Î¦î…žğ‘¢î€¸î€·ğ‘ğ‘”,ℎ,∀𝑢∈𝐷(𝑁),∀ℎ,ğ‘”âˆˆğ·î…žğ‘¢î€¸.(1.3) Under this condition, the potential 𝐹𝑁 is given by𝐹𝑁[𝑢]=10Φ𝑁𝑢0+𝜆𝑢−𝑢0,𝑢−𝑢0𝑑𝜆+const,(1.4)

where 𝑢0 is a fixed element of D(N).

The functional 𝐹𝑁 is called the potential of the operator 𝑁, and in turn the operator 𝑁 is called the gradient of the functional 𝐹𝑁.

2. Statement of the Problem

Let us consider the following differential-difference operator equation: 𝑁(𝑢)≡1𝜆=−1𝑃𝜆(𝑡)𝑢𝑡𝑡(𝑡+𝜆𝜏)+𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=0,𝑢∈𝐷(𝑁),𝑡∈0,𝑡1⊂ℝ.(2.1)

Here, 𝑃𝜆∶𝑈1→𝑉1(𝜆=−1,0,1) are linear operators depending on 𝑡;𝑄∶[𝑡0−𝜏,𝑡1+𝜏]×𝑈13→𝑉1 is in general a nonlinear operator;𝑁∶𝐷(𝑁)⊆𝑈→𝑉;𝑈=𝐶1([𝑡0−𝜏,𝑡1+𝜏];𝑈1);𝑉=𝐶([𝑡0−𝜏,𝑡1+𝜏];𝑉1), where 𝑈1,𝑉1 are real normed linear spaces, 𝑈1⊆𝑉1.

The domain of definition 𝐷(𝑁) is given by the equality: 𝐷(𝑁)=𝑢∈𝑈∶𝑢(𝑡)=𝜑1𝑡(𝑡),𝑡∈0−𝜏,𝑡0,𝑢(𝑡)=𝜑2𝑡(𝑡),𝑡∈1,𝑡1+𝜏,(2.2) where 𝜑𝑖,(𝑖=1,2) are given functions.

Under the solution of (2.1), we mean a function 𝑢∈𝐷(𝑁) satisfying the identity: 𝑁(𝑢)≡1𝜆=−1𝑃𝜆(𝑡)𝑢𝑡𝑡(𝑡+𝜆𝜏)+𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=0,∀𝑡∈0,𝑡1⊂ℝ.(2.3)

Let us give the following bilinear form: Φ(⋅,⋅)≡𝑡1𝑡0⟨⋅,⋅⟩𝑑𝑡∶𝑉×𝑈⟶ℝ,(2.4) where the bilinear mapping Φ1≡⟨⋅,⋅⟩ satisfies the following conditions: ⟨𝑣1,𝑣2⟩=⟨𝑣2,𝑣1⟩,∀𝑣1,𝑣2∈𝑉1,𝐷𝑡⟨𝑣,𝑔⟩=⟨𝐷𝑡𝑣,𝑔⟩+⟨𝑣,𝐷𝑡𝑔⟩,∀𝑣,𝑔∈𝑉.(2.5)

Our aim is to define the structure of operators 𝑃𝜆(𝜆=−1,0,1) and 𝑄 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 𝐷𝑡=𝑑/𝑑𝑡 is the antisymmetric operator on 𝐷(ğ‘î…žğ‘¢), that is, Î¦î€·ğ·ğ‘¡â„Ž1,ℎ2𝐷=âˆ’Î¦ğ‘¡â„Ž2,ℎ1,∀ℎ1,ℎ2î€·ğ‘âˆˆğ·î…žğ‘¢î€¸.(2.6)

3. Conditions of Potentiality and the Structure of (2.1) in the Case of Its Variationality

We denote by 𝐾∗ the operator adjoint to 𝐾.

Theorem 3.1. If 𝐷∗𝑡=−𝐷𝑡 on the set 𝐷(ğ‘î…žğ‘¢), then for the existence of the direct variational formulation for the operator (2.1) on the set 𝐷(𝑁) relative to (2.4), it is necessary and sufficient that the following conditions hold on the set 𝐷(ğ‘î…žğ‘¢): 𝑃−𝜆+𝑃∗𝜆|𝑡→𝑡−𝜆𝜏=0,𝜕𝑃∗𝜆||||𝜕𝑡𝑡→𝑡−𝜆𝜏+ğ‘„î…žğ‘¢(𝑡−𝜆𝜏)âˆ’ğ‘„î…žâˆ—ğ‘¢(𝑡+𝜆𝜏)||𝑡→𝑡−𝜆𝜏𝑡=0,𝜆=−1,0,1,∀𝑢∈𝐷(𝑁),∀𝑡∈0,𝑡1.(3.1)

Proof. Taking into account formula (2.1), we get ğ‘î…žğ‘¢â„Ž=1𝜆=−1ğ‘ƒğœ†â„Žğ‘¡(𝑡+𝜆𝜏)+1𝜆=−1ğ‘„î…žğ‘¢(𝑡+𝜆𝜏)ℎ(𝑡+𝜆𝜏).(3.2) The criterion of potentiality takes the following form: 𝑡1𝑡01𝜆=−1î‚†ğ‘ƒğœ†â„Žğ‘¡(𝑡+𝜆𝜏)+ğ‘„î…žğ‘¢(𝑡+𝜆𝜏)=ℎ(𝑡+𝜆𝜏),𝑔(𝑡)𝑑𝑡𝑡1𝑡01𝜆=−1𝑃𝜆𝑔𝑡(𝑡+𝜆𝜏)+ğ‘„î…žğ‘¢(𝑡+𝜆𝜏)𝑔(𝑡+𝜆𝜏),ℎ(𝑡)𝑑𝑡,(3.3) or 𝑡1𝑡01𝜆=−1î‚†ğ‘ƒğœ†â„Žğ‘¡(𝑡+𝜆𝜏)+ğ‘„î…žğ‘¢(𝑡+𝜆𝜏)−ℎ(𝑡+𝜆𝜏),𝑔(𝑡)1𝜆=−1𝑃𝜆𝑔𝑡(𝑡+𝜆𝜏)+ğ‘„î…žğ‘¢(𝑡+𝜆𝜏)𝑁𝑔(𝑡+𝜆𝜏),ℎ(𝑡)𝑑𝑡=0,∀𝑢∈𝐷(𝑁),∀𝑔,â„Žâˆˆğ·î…žğ‘¢î€¸.(3.4) Bearing into account the condition 𝐷∗𝑡=−𝐷𝑡 on the set 𝐷(ğ‘î…žğ‘¢), from (3.4), we get 𝑡1𝑡01𝜆=−1𝑃𝜆𝐷𝑡+ğ‘„î…žğ‘¢(𝑡+𝜆𝜏)−ℎ(𝑡+𝜆𝜏),𝑔(𝑡)1𝜆=−1î‚¬î‚ƒâˆ’ğ·ğ‘¡î€·ğ‘ƒâˆ—ğœ†î€¸â„Ž(𝑡)+ğ‘„î…žâˆ—ğ‘¢(𝑡+𝜆𝜏)ℎ(𝑡),𝑔(𝑡+𝜆𝜏)𝑑𝑡=0,(3.5) or 𝑡1𝑡01𝜆=−1𝑃−𝜆𝐷𝑡+ğ‘„î…žğ‘¢(𝑡−𝜆𝜏)−ℎ(𝑡−𝜆𝜏),𝑔(𝑡)1𝜆=−1−𝜕𝑃∗𝜆𝜕𝑡−𝑃∗𝜆𝐷𝑡+ğ‘„î…žğ‘¢âˆ—||||ğ‘¡â†’ğ‘¡âˆ’ğœ†ğœâ„Ž(𝑡−𝜆𝜏),𝑔(𝑡)𝑑𝑡=0.(3.6) Thus, condition (3.4) can be reduced to the following form: 𝑡1𝑡01𝜆=−1𝑃−𝜆𝐷𝑡+ğ‘„î…žğ‘¢(𝑡−𝜆𝜏)−ℎ(𝑡−𝜆𝜏)−𝜕𝑃∗𝜆𝜕𝑡−𝑃∗𝜆𝐷𝑡+ğ‘„î…žğ‘¢âˆ—î‚¶||||ğ‘¡â†’ğ‘¡âˆ’ğœ†ğœî„•î€·ğ‘â„Ž(𝑡−𝜆𝜏),𝑔(𝑡)𝑑𝑡=0,∀𝑢∈𝐷(𝑁),∀𝑔,â„Žâˆˆğ·î…žğ‘¢î€¸.(3.7) This equality is fulfilled identically if and only if 1𝜆=−1𝑃−𝜆+𝑃∗𝜆|𝑡→𝑡−𝜆𝜏𝐷𝑡+𝜕𝑃∗𝜆||||𝜕𝑡𝑡→𝑡−𝜆𝜏+ğ‘„î…žğ‘¢(𝑡−𝜆𝜏)âˆ’ğ‘„î…žâˆ—ğ‘¢(𝑡+𝜆𝜏)|ğ‘¡â†’ğ‘¡âˆ’ğœ†ğœîƒ­â‹…â„Ž(𝑡−𝜆𝜏)=0,(3.8) for all 𝑢∈𝐷(𝑁). Thus, it is necessary and sufficient that conditions (3.1) hold.

Theorem 3.2. Conditions (3.1) are held if and only if (2.1) has the following form: 𝑁1(𝑢)≡1𝜆=−1𝑅∗𝜆|𝑡→𝑡−𝜆𝜏−𝑅−𝜆𝑢𝑡(𝑡−𝜆𝜏)+gradΦ𝐵[𝑢]−1𝜆=−1𝜕𝑅𝜆𝑁𝜕𝑡𝑢(𝑡+𝜆𝜏)=0,∀𝑢∈𝐷1𝑡,𝑡∈0,𝑡1⊂ℝ.(3.9)
The operators 𝑅𝜆 and 𝐵 depend on 𝑃𝜆(𝑡) and 𝑄(𝑡,𝑢(𝑡+𝜆𝜏)).

Proof. If 𝐷∗𝑡=−𝐷𝑡 on the set 𝐷(ğ‘î…ž1𝑢) and conditions (3.1) are held, then according to Theorem 3.1, operator (2.1) is potential on the set 𝐷(𝑁) relative to a given bilinear form (2.4).
Let us consider the following functional:𝐹𝑁[𝑢]=𝑡1𝑡01𝜆=−1𝑅𝜆(𝑡)𝑢(𝑡+𝜆𝜏),𝑢𝑡[𝑢](𝑡)+𝐵𝑑𝑡+𝐹𝑁𝑢0.(3.10) It is easy to check that 𝛿𝐹𝑁[]=𝑢,â„Žğ‘¡1𝑡01𝜆=−1ğ‘…ğœ†â„Ž(𝑡+𝜆𝜏),𝑢𝑡+(𝑡)1𝜆=−1𝑅𝜆𝑢(𝑡+𝜆𝜏),â„Žğ‘¡î„•(𝑡)+⟨gradΦ𝐵[𝑢]=,ℎ(𝑡)⟩𝑑𝑡𝑡1𝑡01𝜆=−1𝑅∗𝜆|𝑡→𝑡−𝜆𝜏𝑢𝑡−(𝑡−𝜆𝜏),ℎ(𝑡)1𝜆=−1𝐷𝑡𝑅𝜆𝑢(𝑡+𝜆𝜏),ℎ(𝑡)+⟨gradΦ𝐵[𝑢]=,ℎ(𝑡)⟩𝑑𝑡𝑡1𝑡01𝜆=−1𝑅∗𝜆|𝑡→𝑡−𝜆𝜏𝑢𝑡−(𝑡−𝜆𝜏),ℎ(𝑡)1𝜆=−1𝜕𝑅𝜆−𝜕𝑡𝑢(𝑡+𝜆𝜏),ℎ(𝑡)1𝜆=−1𝑅−𝜆𝑢𝑡(𝑡−𝜆𝜏),ℎ(𝑡)+⟨gradΦ𝐵[𝑢]=,â„ŽâŸ©ğ‘‘ğ‘¡ğ‘¡1𝑡01𝜆=−1𝑅∗𝜆|𝑡→𝑡−𝜆𝜏−𝑅−𝜆𝑢𝑡(+𝑡−𝜆𝜏)gradΦ𝐵[𝑢]−1𝜆=−1𝜕𝑅𝜆=𝜕𝑡𝑢(𝑡+𝜆𝜏),ℎ(𝑡)𝑑𝑡𝑡1𝑡0𝑁⟨𝑁(𝑢),â„ŽâŸ©ğ‘‘ğ‘¡,∀𝑢∈𝐷(𝑁),âˆ€â„Žâˆˆğ·î…žğ‘¢î€¸.(3.11) Then functional (3.10) is a potential of evolutionary operator (2.1).
If 𝐷∗𝑡=−𝐷𝑡 on 𝐷(ğ‘î…žğ‘¢), then𝑃−𝜆(𝑡)=𝑅∗𝜆|𝑡→𝑡−𝜆𝜏−𝑅−𝜆,𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=−1𝜆=−1𝜕𝑅𝜆𝜕𝑡𝑢(𝑡+𝜆𝜏)+gradΦ𝐵[𝑢].(3.12)
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.

4. Examples

Example 4.1. Let us consider the evolutionary differential-difference equation with partial derivatives in the following form: 𝑁1(𝑢)≡1𝜆=−1î‚µğ‘Žğœ†(𝑥,𝑡)𝜕𝑢𝜕𝑡(𝑥,𝑡+𝜆𝜏)−𝑏𝜆𝑖𝑗𝜕(𝑥,𝑡)2𝑢𝜕𝑥𝑖𝜕𝑥𝑗(𝑥,𝑡+𝜆𝜏)=0,(4.1) where (𝑥,𝑡)∈𝑄=Ω×(𝑡0,𝑡1),𝑡1−𝑡0>2𝜏,𝑖,𝑗=1,𝑛,ğ‘Žğœ†(𝑥,𝑡)∈𝐶0,1𝑥,𝑡(𝑄),𝑏𝜆𝑖𝑗(𝑥,𝑡)∈𝐶2,0𝑥,𝑡(𝑄).
Ω is a bounded domain in ℝ𝑛 with piecewise smooth boundary 𝜕Ω.
The domain of definition 𝐷(𝑁1) is given by the equality: 𝐷𝑁1=𝑢∈𝑈=𝐶2,1𝑥,𝑡𝑡Ω×0−𝜏,𝑡1+𝜏∶𝑢(𝑥,𝑡)=𝜑1(𝑥,𝑡),(𝑥,𝑡)∈𝐸1=𝑡Ω×0−𝜏,𝑡0,𝑢(𝑥,𝑡)=𝜑2(𝑥,𝑡),(𝑥,𝑡)∈𝐸2=𝑡Ω×1,𝑡1,𝜕+𝜏𝜇𝑢𝜕𝑛𝜇𝑥||||𝜕Γ𝜏=𝜓𝜇,(𝑥,𝑡),𝜇=0,1(4.2) where Γ𝜏=𝜕Ω×[𝑡0−𝜏,𝑡1+𝜏], 𝜑𝑖∈𝐶(𝐸𝑖),(𝑖=0,1),𝜓𝜇∈𝐶(Γ𝜏),(𝜇=0,1) 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 ğ‘î…ž1ğ‘¢â„Ž=1𝜆=−1î‚€ğ‘Žğœ†(𝑥,𝑡)â„Žğ‘¡(𝑥,𝑡+𝜆𝜏)−𝑏𝜆𝑖𝑗(𝑥,𝑡)â„Žğ‘¥ğ‘–ğ‘¥ğ‘—î‚(𝑥,𝑡+𝜆𝜏).(4.3)
Necessary and sufficient conditions of potentiality take the form: Î¦î€·ğ‘î…ž1ğ‘¢î€¸î€·ğ‘â„Ž,ğ‘”âˆ’Î¦î…ž1𝑢𝑔,â„Žâ‰¡âˆ’ğ‘¡1𝑡0Ω⎧⎪⎨⎪⎩1𝜆=−1ğ·ğ‘¡î€·ğ‘Žğœ†îƒª|||||(𝑥,𝑡)𝑔(𝑥,𝑡)𝑡→𝑡−𝜆𝜏+1𝜆=−1𝐷𝑥𝑖𝑥𝑗𝑏𝜆𝑖𝑗(|||||𝑥,𝑡)𝑔(𝑥,𝑡)ğ‘¡â†’ğ‘¡âˆ’ğœ†ğœâŽ«âŽªâŽ¬âŽªâŽ­âˆ’î€œâ„Ž(𝑥,𝑡)𝑑𝑥𝑑𝑡𝑡1𝑡0Ω1𝜆=−1î‚†ğ‘Žğœ†(𝑥,𝑡)𝑔𝑡(𝑥,𝑡+𝜆𝜏)−𝑏𝜆𝑖𝑗(𝑥,𝑡)𝑔𝑥𝑖𝑥𝑗𝑁(𝑥,𝑡+𝜆𝜏)⋅ℎ(𝑥,𝑡)𝑑𝑥𝑑𝑡=0,∀𝑢∈𝐷1𝑁,∀𝑔,â„Žâˆˆğ·î…ž1𝑢.(4.4)
From that, we come to the following: −1𝜆=−1î‚€ğ·ğ‘¡î€·ğ‘Žğœ†î€¸ğ‘”(𝑥,𝑡)−𝐷𝑥𝑖𝑥𝑗𝑏𝜆𝑖𝑗|||𝑔(𝑥,𝑡)𝑡→𝑡−𝜆𝜏=1𝜆=−1î‚†ğ‘Žğœ†ğ‘”ğ‘¡(𝑥,𝑡+𝜆𝜏)−𝑏𝜆𝑖𝑗𝑔𝑥𝑖𝑥𝑗,𝑁(𝑥,𝑡+𝜆𝜏)âˆ€ğ‘”âˆˆğ·î…ž1𝑢.(4.5)
That is true if and only if âˆ’ğ‘Žâˆ’ğœ†(𝑥)=ğ‘Žğœ†(𝑥),𝜆=1,−1,ğ‘Ž0(𝑥)=0,𝑏𝜆𝑖𝑗||(𝑡)𝑡→𝑡−𝜆𝜏=𝑏𝑖𝑗−𝜆(𝑡),𝜆=−1,0,1,𝑖,𝑗=1,𝑛,∀(𝑥,𝑡)∈𝑄.(4.6)
Under the fulfilment of that conditions, the corresponding functional is given by 𝐹𝑁1[𝑢]=12𝑡1𝑡0î€œÎ©î‚†ğ‘Ž1(𝑥)𝑢𝑡(𝑥,𝑡−𝜏)𝑢(𝑥,𝑡)âˆ’ğ‘Ž1(𝑥)𝑢𝑡(𝑥,𝑡+𝜏)𝑢(𝑥,𝑡)+𝑏1𝑖𝑗(𝑡−𝜏)𝑢𝑥𝑖(𝑥,𝑡−𝜏)⋅𝑢𝑥𝑗(𝑥,𝑡)+𝑏0𝑖𝑗(𝑡)𝑢𝑥𝑖(𝑥,𝑡)𝑢𝑥𝑗(𝑥,𝑡)+𝑏𝑖𝑗−1(𝑡+𝜏)𝑢𝑥𝑖(𝑥,𝑡+𝜏)𝑢𝑥𝑗(𝑥,𝑡)𝑑𝑥𝑑𝑡+const.(4.7)
Let us consider an example when this criterion of potentiality fails.

Example 4.2. Consider the equation: 𝑁2(𝑢)=𝑢𝑡(𝑡,𝑥)−2𝑢(𝑡,𝑥+2𝜏)𝑢𝑥(𝑡,𝑥+2𝜏)−2𝑢𝑥(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)−2𝑢(𝑡,𝑥)𝑢𝑥+1(𝑡,𝑥−2𝜏)2𝑢𝑥𝑥𝑥1(𝑡,𝑥+2𝜏)+2𝑢𝑥𝑥𝑥(𝑡,𝑥−2𝜏)=0,(𝑡,𝑥)∈𝑄𝑇=(0,𝑇)×(−∞,+∞).(4.8) Let us note that this equation is a Korteweg-de Vries’ equation 𝜏=0.
We denote 𝐷𝑁2=𝑢∈𝑈∶𝐶1,3𝑡,𝑥((0,𝑇)×(−∞,+∞))∶𝑢|𝑡=0=𝑢0(𝑥),𝑢|𝑡=𝑇=𝑢1(𝑥),lim|𝑥|→+âˆžğ·ğ‘›ğ‘¥î‚€ğ‘¢=0𝑛=.0,3(4.9) It is easy to be convinced that operator (4.8) is not potential on set (4.9) relative to the bilinear form: Φ(𝑣,𝑔)=𝑇0+âˆžâˆ’âˆžğ‘£(𝑡,𝑥)⋅𝑔(𝑡,𝑥)𝑑𝑥𝑑𝑡,𝑣∈𝑉,𝑔∈𝑈.(4.10) Here, 𝑉={𝑣∈𝐶0,3𝑡,𝑥((0,𝑇)×(−∞,+∞))∶lim|𝑥|→+âˆžğ·ğ‘›ğ‘¥ğ‘£=0,(𝑛=0,3)}.
We define the integrating operator 𝑀 as ∫𝑀𝑣(𝑡,𝑥)=ğ‘¥âˆ’âˆžğ‘£(𝑡,𝑦)𝑑𝑦.
Then, the operator, 𝑁(𝑢)≡𝑀𝑁2(𝑢)=ğ‘¥âˆ’âˆžğ‘¢ğ‘¡(𝑡,𝑦)𝑑𝑦−𝑢2+1(𝑡,𝑥+2𝜏)−2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)2𝑢𝑥𝑥1(𝑡,𝑥+2𝜏)+2𝑢𝑥𝑥(𝑡,𝑥−2𝜏)𝑑𝑦(4.11)
is potential on set (4.9) relative to bilinear form (4.10). The corresponding functional 𝐹𝑁2[𝑢] has the form: 𝐹𝑁2[𝑢]=12𝑇0+âˆžâˆ’âˆžî‚»ğ‘¢î€œ(𝑡,𝑥)ğ‘¥âˆ’âˆžğ‘¢ğ‘¡(𝑡,𝑦)𝑑𝑦−𝑢𝑥(𝑡,𝑥−𝜏)𝑢𝑥(𝑡,𝑥+𝜏)−2𝑢2(𝑡,𝑥+𝜏)𝑢(𝑡,𝑥−𝜏)𝑑𝑦𝑑𝑥𝑑𝑡.(4.12)
Indeed, using (4.12), we find that 𝛿𝐹𝑁2[]=𝑢,â„Žğ‘‡0+âˆžâˆ’âˆžî‚»î€œğ‘¥âˆ’âˆžğ‘¢ğ‘¡1(𝑡,𝑦)𝑑𝑦+2𝑢𝑥𝑥1(𝑡,𝑥+2𝜏)+2𝑢𝑥𝑥(𝑡,𝑥−2𝜏)−2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)−𝑢2𝑁(𝑡,𝑥+2𝜏)ğ‘‘ğ‘¦â„Ž(𝑡,𝑥)𝑑𝑥𝑑𝑡,∀𝑢∈𝐷(𝑁),âˆ€â„Žâˆˆğ·î…žğ‘¢î€¸.(4.13)
From the condition 𝛿𝐹𝑁[𝑢,ℎ]=0,𝑢∈𝐷(𝑁), for all â„Žâˆˆğ·(ğ‘î…žğ‘¢), we obtain 𝑁(𝑢)≡𝑀𝑁2(𝑢)â‰¡ğ‘¥âˆ’âˆžğ‘¢ğ‘¡(𝑡,𝑦)𝑑𝑦−𝑢21(𝑡,𝑥+2𝜏)−2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)+2𝑢𝑥𝑥+1(𝑡,𝑥+2𝜏)2𝑢𝑥𝑥(𝑡,𝑥−2𝜏)𝑑𝑦=0,𝑢∈𝐷(𝑁),(4.14) this equation is equivalent to (4.8).
Let us note that the formula ∫𝐼[𝑢]=+∞−∞(1/2)𝑢𝑥(𝑡,𝑥−𝜏)𝑢𝑥(𝑡,𝑥+𝜏)+𝑢2(𝑡,𝑥+𝜏)𝑢(𝑡,𝑥−𝜏)𝑑𝑥 defines the first integral 𝐼[𝑢]=const of (4.8).

Acknowledgment

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.


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