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

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 𝑁^𝑢 of 𝑁 defined by the formula (𝑑/𝑑𝜖)𝑁(𝑢+𝜖ℎ)|_𝜖=0=𝛿𝑁(𝑢,ℎ)=𝑁^𝑢ℎ.

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 𝐹_𝑁∶𝐷(𝐹_𝑁)=𝐷(𝑁)→𝑅 such that𝛿𝐹_𝑁[𝑢,ℎ]=Φ(𝑁(𝑢),ℎ),∀𝑢∈𝐷(𝑁),∀ℎ∈𝐷𝑁^𝑢.(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 𝐹_𝑁 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]: Φ𝑁^𝑢ℎ,𝑔=Φ𝑁^𝑢𝑔,ℎ,∀𝑢∈𝐷(𝑁),∀ℎ,𝑔∈𝐷𝑁^𝑢.(1.3) Under this condition, the potential 𝐹_𝑁 is given by𝐹_𝑁[𝑢]=¹⁰Φ𝑁𝑢₀+𝜆𝑢−𝑢₀,𝑢−𝑢₀𝑑𝜆+const,(1.4)

where 𝑢₀ is a fixed element of D(N).

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

2. Statement of the Problem

Let us consider the following differential-difference operator equation: 𝑁(𝑢)≡¹_𝜆₌₋₁𝑃_𝜆(𝑡)𝑢_𝑡(𝑡+𝜆𝜏)+𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=0,𝑢∈𝐷(𝑁),𝑡∈𝑡₀,𝑡₁⊂ℝ.(2.1)

Here, 𝑃_𝜆∶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∈𝑡₀−𝜏,𝑡₀,𝑢(𝑡)=𝜑₂(𝑡),𝑡∈𝑡₁,𝑡₁+𝜏,(2.2) where 𝜑_𝑖,(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₌₋₁𝑃_𝜆(𝑡)𝑢_𝑡(𝑡+𝜆𝜏)+𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=0,∀𝑡∈𝑡₀,𝑡₁⊂ℝ.(2.3)

Let us give the following bilinear form: Φ(⋅,⋅)≡^𝑡1_𝑡0⟨⋅,⋅⟩𝑑𝑡∶𝑉×𝑈⟶ℝ,(2.4) where the bilinear mapping Φ₁≡⟨⋅,⋅⟩ satisfies the following conditions: 𝐷_𝑡⟨𝑣₁,𝑣₂⟩=⟨𝑣₂,𝑣₁⟩,∀𝑣₁,𝑣₂∈𝑉₁,⟨𝑣,𝑔⟩=⟨𝐷_𝑡𝑣,𝑔⟩+⟨𝑣,𝐷_𝑡𝑔⟩,∀𝑣,𝑔∈𝑉.(2.5)

Our aim is to define the structure of operators 𝑃_𝜆(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 𝐷_𝑡=𝑑/𝑑𝑡 is the antisymmetric operator on 𝐷(𝑁^𝑢), that is, Φ𝐷_𝑡ℎ₁,ℎ₂=−Φ𝐷_𝑡ℎ₂,ℎ₁,∀ℎ₁,ℎ₂∈𝐷𝑁^𝑢.(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=−𝐷_𝑡 on the set 𝐷(𝑁^𝑢), 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(𝑁^𝑢): 𝑃_−𝜆+𝑃^∗𝜆|_{𝑡→𝑡−𝜆𝜏}=0,𝜕𝑃^∗𝜆𝜕𝑡||||_{𝑡→𝑡−𝜆𝜏}𝜆=−1,0,+1,𝑄∀𝑢∈^{𝑢(𝑡−𝜆𝜏)}𝐷−(𝑄𝑁),∀^{∗𝑢(𝑡+𝜆𝜏)}𝑡||∈𝑡_{𝑡→𝑡−𝜆𝜏0}=0,,𝑡₁.(3.1)

Proof. Taking into account formula (2.1), we get 𝑁^𝑢ℎ=¹_𝜆₌₋₁𝑃_𝜆ℎ_𝑡(𝑡+𝜆𝜏)+¹_𝜆₌₋₁𝑄^{𝑢(𝑡+𝜆𝜏)}ℎ(𝑡+𝜆𝜏).(3.2) The criterion of potentiality takes the following form: ^𝑡1_𝑡0¹_𝜆=−1𝑃^𝜆=ℎ_𝑡(𝑡+𝜆𝜏)+𝑄^{𝑢(𝑡+𝜆𝜏)}ℎ(𝑡+𝜆𝜏),𝑔(𝑡)𝑑𝑡_𝑡1𝑡0¹_𝜆=−1𝑃^𝜆𝑔_𝑡(𝑡+𝜆𝜏)+𝑄^{𝑢(𝑡+𝜆𝜏)}𝑔(𝑡+𝜆𝜏),ℎ(𝑡)𝑑𝑡,(3.3) or ^𝑡1_𝑡0¹_𝜆=−1𝑃^𝜆−ℎ_𝑡(𝑡+𝜆𝜏)+𝑄^{𝑢(𝑡+𝜆𝜏)}ℎ(𝑡+𝜆𝜏),𝑔(𝑡)₁_𝜆₌₋₁𝑃^𝜆𝑔_𝑡(𝑡+𝜆𝜏)+𝑄^{𝑢(𝑡+𝜆𝜏)}𝑔(𝑡+𝜆𝜏),ℎ(𝑡)𝑑𝑡=0,∀𝑢∈𝐷(𝑁),∀𝑔,ℎ∈𝐷𝑁^𝑢.(3.4) Bearing into account the condition 𝐷^∗𝑡=−𝐷_𝑡 on the set 𝐷(𝑁^𝑢), from (3.4), we get ^𝑡1_𝑡0¹_𝜆=−1𝑃^𝜆𝐷_𝑡+𝑄^{𝑢(𝑡+𝜆𝜏)}ℎ(𝑡+𝜆𝜏),𝑔(𝑡)−¹_𝜆₌₋₁−𝐷^𝑡𝑃^∗𝜆ℎ(𝑡)+𝑄^{∗𝑢(𝑡+𝜆𝜏)}ℎ(𝑡),𝑔(𝑡+𝜆𝜏)𝑑𝑡(3=.50), or ^𝑡1_𝑡0¹_𝜆=−1𝑃^−𝜆−𝐷_𝑡+𝑄^{𝑢(𝑡−𝜆𝜏)}ℎ(𝑡−𝜆𝜏),𝑔(𝑡)₁_𝜆₌₋₁−𝜕𝑃^∗𝜆𝜕𝑡−𝑃^∗𝜆𝐷_𝑡+𝑄^𝑢∗||||_{𝑡→𝑡−𝜆𝜏}ℎ(𝑡−𝜆𝜏),𝑔(𝑡)𝑑𝑡=0.(3.6) Thus, condition (3.4) can be reduced to the following form: ^𝑡1_𝑡0¹_𝜆=−1𝑃^−𝜆𝐷_𝑡+𝑄^{𝑢(𝑡−𝜆𝜏)}ℎ(𝑡−𝜆𝜏)−−𝜕𝑃^∗𝜆𝜕𝑡−𝑃^∗𝜆𝐷_𝑡+𝑄^𝑢∗||||_{𝑡→𝑡−𝜆𝜏}∀𝑢∈𝐷ℎ(𝑁(𝑡)−,𝜆𝜏∀)𝑔,,𝑔ℎ(𝑡∈)𝐷𝑑𝑡𝑁=_𝑢0,.(3.7) This equality is fulfilled identically if and only if ¹_𝜆=−1𝑃^−𝜆+𝑃^∗𝜆|_{𝑡→𝑡−𝜆𝜏}𝐷_𝑡+𝜕𝑃^∗𝜆𝜕𝑡||||_{𝑡→𝑡−𝜆𝜏}+𝑄^{𝑢(𝑡−𝜆𝜏)}−𝑄^{∗𝑢(𝑡+𝜆𝜏)}|_{𝑡→𝑡−𝜆𝜏}⋅ℎ(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𝑢_𝑡(𝑡−𝜆𝜏)+grad_Φ𝐵[𝑢]−¹_𝜆₌₋₁𝜕𝑅_𝜆∀𝑢∈𝐷𝑁₁𝜕𝑡𝑢(𝑡+𝜆𝜏)=0,,𝑡∈𝑡₀,𝑡₁⊂ℝ.(3.9)
The operators 𝑅_𝜆 and 𝐵 depend on 𝑃_𝜆(𝑡) and 𝑄(𝑡,𝑢(𝑡+𝜆𝜏)).

Proof. If 𝐷^∗𝑡=−𝐷_𝑡 on the set 𝐷(𝑁^1𝑢) 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:𝐹_𝑁[𝑢]=^𝑡1_𝑡0¹_𝜆=−1𝑅_𝜆(𝑡)𝑢(𝑡+𝜆𝜏),𝑢_𝑡(𝑡)+𝐵[𝑢]𝑑𝑡+𝐹_𝑁𝑢₀.(3.10) It is easy to check that 𝛿𝐹_𝑁[𝑢,ℎ]=^𝑡1_𝑡0¹_𝜆=−1𝑅_𝜆ℎ(𝑡+𝜆𝜏),𝑢_𝑡(𝑡)+¹_𝜆₌₋₁=𝑅_𝜆𝑢(𝑡+𝜆𝜏),ℎ_𝑡(𝑡)+⟨grad_Φ𝐵[𝑢],ℎ(𝑡)⟩𝑑𝑡_𝑡1𝑡0¹_𝜆=−1𝑅^∗𝜆|_{𝑡→𝑡−𝜆𝜏}𝑢_𝑡(𝑡−𝜆𝜏),ℎ(𝑡)−¹_𝜆₌₋₁+⟨grad_Φ𝐷_𝑡𝑅_𝜆𝑢(𝑡+𝜆𝜏),ℎ(𝑡)=𝐵[𝑢],ℎ(𝑡)⟩𝑑𝑡_𝑡1𝑡0¹_𝜆=−1𝑅^∗𝜆|_{𝑡→𝑡−𝜆𝜏}𝑢_𝑡(𝑡−𝜆𝜏),ℎ(𝑡)−¹_𝜆₌₋₁𝜕𝑅_𝜆−𝜕𝑡𝑢(𝑡+𝜆𝜏),ℎ(𝑡)₁_𝜆₌₋₁=𝑅_−𝜆𝑢_𝑡(𝑡−𝜆𝜏),ℎ(𝑡)+⟨grad_Φ𝐵[𝑢],ℎ⟩𝑑𝑡_𝑡1𝑡0¹_𝜆=−1+grad_Φ𝑅^∗𝜆|_{𝑡→𝑡−𝜆𝜏}−𝑅_−𝜆𝑢_𝑡(𝑡−𝜆𝜏)𝐵[𝑢]−¹_𝜆₌₋₁𝜕𝑅_𝜆=𝜕𝑡𝑢(𝑡+𝜆𝜏),ℎ(𝑡)𝑑𝑡_𝑡1𝑡0⟨𝑁(𝑢),ℎ⟩𝑑𝑡,∀𝑢∈𝐷(𝑁),∀ℎ∈𝐷𝑁^𝑢.(3.11) Then functional (3.10) is a potential of evolutionary operator (2.1).
If 𝐷^∗𝑡=−𝐷_𝑡 on 𝐷(𝑁^𝑢), then𝑄(𝑡,𝑢(𝑡𝑃+𝜆_−𝜆𝜏()𝑡))==𝑅−^∗𝜆|_{𝑡→𝑡−𝜆𝜏1}−𝑅_−𝜆,_𝜆₌₋₁𝜕𝑅_𝜆𝜕𝑡𝑢(𝑡+𝜆𝜏)+grad^Φ𝐵[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₌₋₁𝑎^𝜆(𝑥,𝑡)𝜕𝑢𝜕𝑡(𝑥,𝑡+𝜆𝜏)−𝑏^𝜆𝑖𝑗(𝑥,𝑡)𝜕²𝑢𝜕𝑥_𝑖𝜕𝑥_𝑗(𝑥,𝑡+𝜆𝜏)=0,(4.1) where (𝑥,𝑡)∈𝑄=Ω×(𝑡₀,𝑡₁),𝑡₁−𝑡₀>2𝜏,𝑖,𝑗=1,𝑛,𝑎_𝜆(𝑥,𝑡)∈𝐶^0,1𝑥,𝑡(𝑄),𝑏^𝜆𝑖𝑗(𝑥,𝑡)∈𝐶^2,0𝑥,𝑡(𝑄).
Ω is a bounded domain in ℝ^𝑛 with piecewise smooth boundary 𝜕Ω.
The domain of definition 𝐷(𝑁₁) is given by the equality: 𝐷𝑁₁=𝑢𝑢(𝑥∈,𝑡𝑈)==𝜑𝐶^2,1𝑥,𝑡₂Ω×𝑡₀−𝜏,𝑡₁+𝜏∶𝑢(𝑥,𝑡)=𝜑₁(𝑥,𝑡),(𝑥,𝑡)∈𝐸₁=Ω×𝑡₀−𝜏,𝑡₀,(𝑥,𝑡),(𝑥,𝑡)∈𝐸₂=Ω×𝑡₁,𝑡₁+𝜏,𝜕^𝜇𝑢𝜕𝑛^𝜇𝑥||||_𝜕Γ𝜏=𝜓_𝜇(𝑥,𝑡),𝜇=0,1,(4.2) where Γ_𝜏=𝜕Ω×[𝑡₀−𝜏,𝑡₁+𝜏], 𝜑_𝑖∈𝐶(𝐸_𝑖),(𝑖=0,1),𝜓_𝜇∈𝐶(Γ_𝜏),(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 𝑁^1𝑢ℎ=¹_𝜆₌₋₁𝑎^𝜆(𝑥,𝑡)ℎ_𝑡(𝑥,𝑡+𝜆𝜏)−𝑏^𝜆𝑖𝑗(𝑥,𝑡)ℎ_{𝑥𝑖𝑥𝑗}(𝑥,𝑡+𝜆𝜏).(4.3)
Necessary and sufficient conditions of potentiality take the form: Φ𝑁≡^1𝑢ℎ−,𝑔_𝑡1𝑡0−Φ𝑁^1𝑢𝑔,ℎ_Ω⎧⎪⎨⎪⎩¹_𝜆₌₋₁𝐷_𝑡𝑎_𝜆(𝑥,𝑡)𝑔(𝑥,𝑡)|||||_{𝑡→𝑡−𝜆𝜏}+¹_𝜆₌₋₁−𝐷_{𝑥𝑖𝑥𝑗}𝑏^𝜆𝑖𝑗(𝑥,𝑡)𝑔(𝑥,𝑡)|||||_{𝑡→𝑡−𝜆𝜏}ℎ(𝑥,𝑡)⎫⎪⎬⎪⎭𝑑𝑥𝑑𝑡_𝑡1𝑡0_Ω1𝜆=−1𝑎^𝜆⋅ℎ(𝑥,𝑡)(𝑑𝑥𝑥,𝑡𝑑)𝑡𝑔_𝑡=(𝑥0,,𝑡+∀𝜆𝑢𝜏∈)−𝐷𝑏^𝜆𝑖𝑗𝑁(𝑥₁,𝑡)𝑔_{𝑥𝑖𝑥𝑗}(𝑥,𝑡+𝜆𝜏),∀𝑔,ℎ∈𝐷𝑁^1𝑢.(4.4)
From that, we come to the following: −¹_𝜆₌₋₁𝐷^𝑡𝑎_𝜆𝑔(𝑥,𝑡)−𝐷_{𝑥𝑖𝑥𝑗}𝑏^𝜆𝑖𝑗𝑔(𝑥,𝑡)|||_{𝑡→𝑡−𝜆𝜏}=¹_𝜆₌₋₁𝑎^𝜆𝑔_𝑡(𝑥,𝑡+𝜆𝜏)−𝑏^𝜆𝑖𝑗𝑔∀𝑔_{𝑥𝑖𝑥𝑗}∈(𝑥𝐷,𝑡+𝑁𝜆_1𝑢𝜏),.(4.5)
That is true if and only if −𝑎_−𝜆(𝑥)=𝑎_𝜆(𝑥),𝜆=1,−1,𝑖,𝑗𝑎=₀(𝑥)=0,𝑏^𝜆𝑖𝑗(𝑡)||_{𝑡→𝑡−𝜆𝜏}=𝑏^{𝑖𝑗−𝜆}(𝑡),𝜆=−1,0,1,1,𝑛,∀(𝑥,𝑡)∈𝑄.(4.6)
Under the fulfilment of that conditions, the corresponding functional is given by 𝐹_𝑁1[𝑢]=12^𝑡1_𝑡0_Ω𝑎¹⋅𝑢_𝑥𝑗(𝑥)𝑢_𝑡(𝑥,𝑡−𝜏)𝑢(𝑥,𝑡)−𝑎₁(𝑥)𝑢_𝑡(𝑥,𝑡+𝜏)𝑢(𝑥,𝑡)+𝑏^1𝑖𝑗(𝑡−𝜏)𝑢_𝑥𝑖(𝑥,𝑡−𝜏)(𝑥,𝑡)+𝑏^0𝑖𝑗(𝑡)𝑢_𝑥𝑖(𝑥,𝑡)𝑢_𝑥𝑗(𝑥,𝑡)+𝑏^𝑖𝑗−1(𝑡+𝜏)𝑢_𝑥𝑖(𝑥,𝑡+𝜏)𝑢_𝑥𝑗(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)=+𝑢_𝑡1(𝑡,𝑥)−2𝑢(𝑡,𝑥+2𝜏)𝑢_𝑥(𝑡,𝑥+2𝜏)−2𝑢_𝑥(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)−2𝑢(𝑡,𝑥)𝑢_𝑥(𝑡,𝑥−2𝜏)2𝑢^𝑥𝑥𝑥(𝑡,𝑥+2𝜏)+12𝑢^𝑥𝑥𝑥(𝑡,𝑥−2𝜏)=0,(𝑡,𝑥)∈𝑄_𝑇=(0,𝑇)×(−∞,+∞).(4.8) Let us note that this equation is a Korteweg-de Vries’ equation 𝜏=0.
We denote 𝐷𝑁₂=𝑢∈𝑈∶𝐶^1,3𝑡,𝑥((0,𝑇)×(−∞,+∞))∶𝑢|_𝑡=0=𝑢₀(𝑥),𝑢|_𝑡=𝑇=𝑢₁(𝑥),lim_{|𝑥|→+∞}𝐷^𝑛𝑥𝑢=0𝑛=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^+∞−∞𝑣(𝑡,𝑥)⋅𝑔(𝑡,𝑥)𝑑𝑥𝑑𝑡,𝑣∈𝑉,𝑔∈𝑈.(4.10) Here, 𝑉={𝑣∈𝐶^0,3𝑡,𝑥((0,𝑇)×(−∞,+∞))∶lim_{|𝑥|→+∞}𝐷^𝑛𝑥𝑣=0,(𝑛=0,3)}.
We define the integrating operator 𝑀 as 𝑀𝑣(𝑡,𝑥)=∫^𝑥−∞𝑣(𝑡,𝑦)𝑑𝑦.
Then, the operator, 𝑁(𝑢)≡𝑀𝑁₂(𝑢)=+^𝑥−∞1𝑢_𝑡(𝑡,𝑦)𝑑𝑦−𝑢²(𝑡,𝑥+2𝜏)−2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)2𝑢^𝑥𝑥(𝑡,𝑥+2𝜏)+12𝑢^𝑥𝑥(𝑡,𝑥−2𝜏)𝑑𝑦(4.11)
is potential on set (4.9) relative to bilinear form (4.10). The corresponding functional 𝐹_𝑁2[𝑢] has the form: 𝐹=_𝑁2[1𝑢]2^𝑇0^+∞−∞𝑢(𝑡,𝑥)^𝑥−∞𝑢_𝑡(𝑡,𝑦)𝑑𝑦−𝑢_𝑥(𝑡,𝑥−𝜏)𝑢_𝑥(𝑡,𝑥+𝜏)−2𝑢²(𝑡,𝑥+𝜏)𝑢(𝑡,𝑥−𝜏)𝑑𝑦𝑑𝑥𝑑𝑡.(4.12)
Indeed, using (4.12), we find that 𝛿𝐹_𝑁2[𝑢,ℎ]=^𝑇0^+∞−∞^𝑥−∞𝑢_𝑡(𝑡,𝑦)𝑑𝑦+12𝑢^𝑥𝑥(𝑡,𝑥+2𝜏)+12𝑢^𝑥𝑥−𝑢₂(𝑡,𝑥−2𝜏)−2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)(𝑡,𝑥+2𝜏)𝑑𝑦ℎ(𝑡,𝑥)𝑑𝑥𝑑𝑡,∀𝑢∈𝐷(𝑁),∀ℎ∈𝐷𝑁^𝑢.(4.13)
From the condition 𝛿𝐹_𝑁[𝑢,ℎ]=0,𝑢∈𝐷(𝑁), for all ℎ∈𝐷(𝑁^𝑢), we obtain 𝑁(𝑢)≡𝑀𝑁₂(𝑢)≡^𝑥−∞𝑢_𝑡(𝑡,𝑦)𝑑𝑦−𝑢²(𝑡,𝑥+2𝜏)−2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥−2𝜏)+12𝑢^𝑥𝑥+1(𝑡,𝑥+2𝜏)2𝑢^𝑥𝑥(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)𝑢_𝑥(𝑡,𝑥−𝜏)𝑢_𝑥(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).