Legendre Polynomials Spectral Approximation for the Infinite-Dimensional Hamiltonian Systems

Abstract

This paper considers a Legendre polynomials spectral approximation for the infinite-dimensional Hamiltonian systems. As a consequence, the Legendre polynomials spectral semidiscrete system is a Hamiltonian system for the Hamiltonian system whose Hamiltonian operator is a constant differential operator.

1. Introduction

The numerical method for infinite-dimensional Hamiltonian Systems has been widely developed. One of the great challenges in the numerical analysis of PDEs is the development of robust stable numerical algorithms for Hamiltonian PDEs. For the numerical analysis, we always look for those discretizations which can preserve as much as possible some intrinsic properties of Hamiltonian equations. In fact, for Hamiltonian systems, the most important is its Hamiltonian structure. From this point of view, some semidiscrete numerical methods which are based on spectral methods have been developed. Spectral methods have proved to be particularly useful in infinite-dimensional Hamiltonian. Wang [1] discussed the semidiscrete Fourier spectral approximation of infinite-dimensional Hamiltonian systems, Hamiltonian of infinite-dimensional Hamiltonian systems, and Hamiltonian structure. Shen [2] studied the dual-Petrov-Galerbin method for third and higher odd-order equations. Ma and Sun [3] deliberated the third-order equations by using an interesting Legendre-Petrov-Galerbin method. So we consider that the Legendre polynomials basis is very important to analysis of the discretization of Hamiltonian systems.

In this paper, we consider a Legendre polynomials spectral approximation for the KdV equation and the wave equation. As a consequence, we show that the Legendre polynomials spectral semidiscrete system is also a Hamiltonian system for the Hamiltonian system whose Hamiltonian operator is a constant differential operator.

The paper is organized as follows. In Section 2, we give a brief description of infinite-dimensional Hamiltonian equations. In Section 3, we introduce semidiscrete Legendre polynomials spectral approximation. In the last two sections, we consider the Legendre polynomials spectral approximation for the boundary value problem of the KdV equation and the wave equation. Moreover, we give the conclusion about the Hamiltonian structure.

2. A Brief Description of Infinite-Dimensional Hamiltonian Equations

First, we get familiar with some basic knowledge about the infinite-dimensional Hamilton system.

Let the set 𝐴={𝐻[𝑢]≡𝐻(𝑥,𝑢^(𝑛))|Hisainfinitedifferentiablesmoothfunction}; here 𝑢^(𝑛)=(𝑢^1𝑇,𝑢^2𝑇,…,𝑢^𝑛𝑇)^𝑇, and 𝑢^𝑖 denotes the 𝑖th derivative of 𝑢. To each 𝐻[𝑢]∈𝐴, there exists a functional ℋ=∫𝐻[𝑢]𝑑𝑥, and the corresponding set of all functional is ℱ={ℋ=∫𝐻[𝑢]𝑑𝑥|𝐻[𝑢]∈𝐴}. 𝛿ℋ/𝛿𝑢 is the variational derivative of the functional ℋ∈ℱ. With the aid of the differential operator 𝒟, we can define a binary operator on ℱ: {ℋ,𝒢}=𝛿ℋ𝛿𝑢𝒟𝛿𝛿𝒢𝑢,∀ℋ,𝒢∈ℱ.(2.1)

If this binary operator satisfies the following conditions: (i){,} is antisymmetric, {ℋ,𝒢}=−{𝒢,ℋ},(2.2)(ii){,} is bilinear, {𝛼ℋ+𝛽𝒢,𝒦}=𝛼{ℋ,𝒦}+𝛽{𝒢,𝒦},∀𝛼,𝛽∈𝑅,(2.3)(iii){,} satisfies the Jacobi identity, {{ℋ,𝒢},𝒦}+{{𝒢,𝒦},ℋ}+{{𝒦,ℋ},𝒢}=0,(2.4) for all functionals ℋ,𝒢,𝒦∈ℱ, then, it is called a Poisson bracket. In this case, 𝒟 is called Hamiltonian operator.

For given a Hamiltonian functional ℋ∈ℱ and a Hamiltonian operator 𝒟, Hamiltonian equation takes the following form: 𝑢_𝑡=𝒟𝛿ℋ𝛿𝑢.(2.5) This evolution equation is called an infinite-dimensional Hamiltonian system.

Consider the infinite-dimensional Hamiltonian system of the KdV equation: 𝑢_𝑡+6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0.(2.6) It has the Hamiltonian structure 𝑢_𝑡=𝒟𝛿ℋ[𝑢],(2.7) where 𝒟=𝜕_𝑥 is the Hamiltonian operator and ℋ[𝑢]=¹⁻¹12𝑢^2𝑥−𝑢³𝑑𝑥(2.8) is the Hamiltonian functional.

3. Semidiscrete Legendre Polynomials Spectral Approximation

Let 𝐿_𝑛(𝑥) be the 𝑛th degree Legendre polynomial. The Legendre polynomials satisfy the three-term recurrence relation: (𝑛+1)𝐿_𝑛+1𝐿₀(𝑥)=1,𝐿₁(𝑥)=𝑥,(𝑥)=(2𝑛+1)𝑥𝐿_𝑛(𝑥)−𝑛𝐿_𝑛−1(𝑥),𝑛≥1(3.1) and the orthogonality relation: ¹⁻¹𝐿_𝑘(𝑥)𝐿_𝑗(𝑥)𝑑𝑥=1𝑘+(1/2)𝛿^𝑘𝑗𝐿_𝑛,(±1)=(±1)^𝑛.(3.2)

As suggested in [4], the choice of compact combinations of orthogonal polynomials as basis functions to minimize the bandwidth and the conditions number of the coefficient matrix is very important. Let {𝐿_𝑛} be a sequence of orthogonal polynomials. As a general rule, for differential equations with 𝑚 boundary conditions, our task is to look for basis functions in the form𝜙_𝑘(𝑥)=𝐿_𝑘(𝑥)+^𝑚_𝑗=1𝑎^𝑗(𝑘)𝐿_𝑘+𝑗(𝑥),(3.3) where 𝑎^𝑗(𝑘)(𝑗=1,2,…,𝑚) are chosen so that 𝜙_𝑘(𝑥) satisfy the 𝑚 homogeneous boundary conditions.

Suppose that 𝑈={𝐻(𝑥)|𝐻(𝑥)isasmoothfunction,𝑥∈[−1,1]}, for the fixed homogeneous boundary conditions 𝐻(−1)=𝐻(1)=0.(3.4) As 𝑚=2, (3.3) has the form 𝜙_𝑘(𝑥)=𝐿_𝑘(𝑥)+𝑎₁𝐿_𝑘+1(𝑥)+𝑎₂𝐿_𝑘+2(𝑥).(3.5) Using the basic properties of Legendre polynomials and the boundary value conditions, obviously 𝜙_𝑘(−1)=0,𝜙_𝑘(1)=0.(3.6) We can verify readily that 𝜙_𝑘(𝑥)=𝐿_𝑘(𝑥)−𝐿_𝑘+2(𝑥).(3.7) Easily, we obtain 𝜙₀(𝑥),𝜙₁(𝑥),𝜙₂(𝑥),…. The 𝐿²-inner product on 𝑈 is defined by (𝑝,𝑞)=¹⁻¹𝑝⋅𝑞𝑑𝑥,∀𝑝,𝑞∈𝑈.(3.8)

The basis functions 𝜙_𝑘(𝑥)(𝑘=1,2,…) can be orthogonalized standard on the 𝐿²- inner product. Thus, we can get the sequence of standard orthogonal basis functions 𝜓_𝑘(𝑥).

After carefully calculation, the orthogonal basis is𝜓₀=14√𝜓₁15−3𝑥²,=14√𝜓₂105𝑥−√105𝑥³,=18−3√⋮5+24√5𝑥²−21√5𝑥⁴,(3.9)

Set 𝐵=span𝜓₀,𝜓₁,𝜓₂,…,𝜓_𝑁⊂𝑈,(3.10) and set 𝑃 as an orthogonal projection. 𝑃∶𝑈→𝐵, 𝑢⟶𝑢=𝑃𝑢=𝑎₀𝜓₀+𝑎₁𝜓₁+⋯+𝑎_𝑁𝜓_𝑁,(3.11) where 𝑎_𝑛=¹⁻¹𝑢(𝑥)𝜓_𝑛(𝑥)𝑑𝑥,𝑛=1,2,3,….(3.12)

Denote 𝐵={𝑢̂=(𝑎₀,𝑎₁,…,𝑎_𝑁)^𝑇∈𝑅^𝑁+1}. The inner product of 𝐵 is usually denoted by Euclidean inner ⟨⋅,⋅⟩, that is, ⟨𝑝,̂𝑞̂⟩=𝑎₀⋅𝑎₀̃+𝑎₁⋅𝑎₁̃+𝑎₂⋅𝑎₂̃+⋯+𝑎_𝑁⋅𝑎_𝑁̃,∀𝑝,̂𝑞̂∈𝐵,(3.13) where 𝑞̂=(𝑎₀̃,𝑎₁,𝑎₂̃,…,𝑎_𝑁̃)^𝑇. Set 𝐼∶𝐵→𝐵, 𝑢̂⟶𝑢=𝐼𝑢̂=𝑎₀𝜓₀+𝑎₁𝜓₁+⋯+𝑎_𝑁.(3.14)

Denote 𝑃=𝐼⁻¹∘𝑃∶𝑈→𝐵, 𝑢⟶𝑃𝑢=𝑎₀,𝑎₁,𝑎₂,…,𝑎_𝑁^𝑇.(3.15)

Hamiltonian equation 𝑢_𝑡=𝒟𝛿ℋ𝛿𝑢(3.16) has the special Poisson structure; so we can exploit it to design numerical approximations. We can discretize Hamiltonian operator 𝒟 and Hamiltonian functionals; then a numerical bracket can be defined.

The discretization of the Hamiltonian operator 𝒟 is 𝒟(𝑝)̂𝑞𝒟̂==𝑝𝑃̂∘𝒟∘𝒟(𝐼∘𝑝𝐼)̂⋅∶𝐼𝐵𝑞̂,⟶∀𝑝𝐵,̂𝑞,̂∈𝐵.(3.17)

The discretization of a functional 𝐻(𝑥) in 𝑈 is 𝐻(𝑢)̂=¹⁻¹𝐻(𝐼𝑢)̂𝑑𝑥,∀𝑢̂∈𝐵.(3.18)

Let 𝑈 be the set of discrete functionals; then we can define a bracket on 𝑈, ∇𝐻𝐻=,𝐺𝜕𝐻=∇𝐻𝒟∇𝐺^𝑇,𝜕𝑢₁̂,𝜕𝐻𝜕𝑢₂̂,…,𝜕𝐻𝜕𝑢_𝑁̂,(3.19) which is an approximation of bracket {𝐻,𝐺}.

Now we define the semidiscrete approximative equation in 𝐵 of the infinite-dimensional Hamiltonian system 𝑢_𝑡=𝒟𝛿ℋ𝛿𝑢 as𝑑𝑢̂𝑑𝑡=𝒟(∇𝐻(𝑢)̂)^𝑇.(3.20)

If 𝒟 is still a Hamiltonian operator, then (3.20) is exactly a finite-dimensional Hamiltonian system. The function 𝐻(𝑃𝑢) is a conservation law if and only if (3.20) always preserves conservation law.

4. Legendre Polynomials Spectral Approximation for the Boundary Value Problem of the KdV Equation

We consider the KdV equation with the fixed boundary conditions discussed above: 𝑢_𝑡+6𝑢𝑢𝑢(_𝑥−+1,𝑢𝑡)_𝑥𝑥𝑥==𝑢0(1,,𝑡𝑥)=∈0(,−1𝑡,∈1)[,0,𝑡𝑇∈].[0,𝑇],(4.1)

The KdV equation can be written as Hamiltonian form: 𝑢_𝑡=𝜕_𝑥𝛿ℋ₁𝛿𝑢,(4.2) where the Hamiltonian operator is 𝒟₁=𝜕_𝑥 and the the Hamiltonian functional is ℋ₁=∫¹⁻¹((1/2)𝑢^2𝑥−𝑢³)𝑑𝑥.

By above analysis and the chosen orthogonal basis, for 𝑁=2,𝒟₁=⎛⎜⎜⎜⎜⎜⎜⎜⎝0²√7−√20720√210−√221𝐻₁20⎞⎟⎟⎟⎟⎟⎟⎟⎠,(𝑢)̂=¹⁻¹12𝑎⁰=5𝜓₀+𝑎₁𝜓₁+𝑎₂𝜓₂^2𝑥−𝑎₀𝜓₀+𝑎₁𝜓₁+𝑎₂𝜓₂³𝑑𝑥4𝑎²⁰+214𝑎²¹+√32𝑎⁰𝑎₂+514𝑎²²+3√1514𝑎³⁰−3√1514𝑎²⁰𝑎₂+15√522𝑎²¹𝑎₂+423√52002𝑎³²+√152𝑎⁰𝑎²¹+69√5154𝑎⁰𝑎²².(4.3) Then ∇𝐻^𝑇1=⎛⎜⎜⎜⎜⎜⎜⎜⎝52𝑎⁰+√32𝑎²+9√1514𝑎²⁰−3√57𝑎⁰𝑎₂+√152𝑎²¹+√5154𝑎²²212𝑎¹+15√511𝑎¹𝑎₂+√157𝑎⁰𝑎₁√32𝑎⁰+512𝑎²−3√514𝑎²⁰+15√522a²¹+1269√52002𝑎²²+69√577𝑎⁰𝑎₂⎞⎟⎟⎟⎟⎟⎟⎟⎠.(4.4)

This equations can be written as Hamiltonian form in another way, that is, 𝑢_𝑡=𝜕_𝑥𝑥𝑥+4𝑢𝜕_𝑥+2𝑢_𝑥𝐼𝛿ℋ²𝛿𝑢,(4.5) where the Hamiltonian operator is 𝒟₂=𝜕_𝑥𝑥𝑥+4𝑢𝜕_𝑥+2𝑢_𝑥𝐼 abd the Hamiltonian functional is ℋ₂=∫¹⁻¹(−(1/2)𝑢²)𝑑𝑥.

In the same theory, for 𝑁=2, we can get 𝒟₂=⎛⎜⎜⎜⎜⎜⎜⎜⎝−2157𝑎¹−13√72+4157𝑎⁰−657𝑎²1057𝑎¹−√7−2157𝑎⁰−457𝑎²−40ℛ57𝑎¹−5√212−657𝑎⁰−1611157𝑎²811157𝑎¹𝐻₂⎞⎟⎟⎟⎟⎟⎟⎟⎠(𝑢)̂=¹⁻¹−12𝑎⁰𝜓₀+𝑎₁𝜓₁+𝑎₂𝜓₂²𝑑𝑥=−12𝑎²⁰+𝑎²¹+𝑎²²(4.6) where ℛdenotes−20√21+1057𝑎⁰+810157𝑎².

Then ∇𝐻^𝑇2=−𝑎₀,−𝑎₁,−𝑎₂.(4.7) The corresponding semidiscrete approximation is 𝑑𝑢̂𝑑𝑡=𝒟(∇𝐻(𝑢)̂)^𝑇,𝑢̂=𝑎₀,𝑎₁,𝑎₂^𝑇.(4.8)

It is easy to verify that 𝒟₁ is Hamiltonian operator; so the approximating system can be written as 𝑑𝑢̂𝑑𝑡=𝒟¹∇𝐻₁(𝑢)̂^𝑇,(4.9) adifferent Hamiltonian form, and it can be verified that 𝒟₂ is not a Hamiltonian operator. As 𝒟₁ is a constant antisymmetric matrix, 𝑑𝑢̂𝑑𝑡=𝑃∘𝜕^𝑥∘𝐼⋅𝛿𝐻¹𝛿𝑢(𝐼𝑢)̂(4.10) is a finite-dimensional Hamiltonian system. So the approximating system can preserve the Poisson structure given by Hamiltonian operator 𝒟₁.

Theorem 4.1. The equation 𝑑𝑢/̂𝑑𝑡=𝒟₁(∇𝐻₁(𝑢)̂)^𝑇 is the discretization of the KdV equation 𝑢_𝑡+6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0; then 𝑑𝑢/̂𝑑𝑡=𝒟₁(∇𝐻₁(𝑢)̂)^𝑇 has the property of energy conservation law.

Proof. 𝑢̂⟶𝒟=𝑃𝐼∘∶𝒟𝐵∘⟶𝐼∶𝐵𝐵,⟶𝐵,𝑢⟶𝑢=𝐼𝑢̂=𝑃𝑎₀∶𝜓𝑈₀+⟶𝑎₁𝜓𝐵₁,+⋯+𝑎_𝑁𝜓_𝑁,𝑢=𝑃𝑃𝑢==𝐼𝑎₀₋₁𝜓₀+𝑎₁𝜓₁+⋯+𝑎_𝑁𝜓_𝑁,𝑢⟶𝑃𝑢=𝑎∘₀𝑃∶𝑈⟶𝐵,𝒟=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝒟𝜓₁,𝑎₁,𝑎₂,…,𝑎_𝑁^𝑇,,𝜓₁𝜓₁,𝜓₁𝒟𝜓²,𝜓₁𝜓₁,𝜓₁⋯𝒟𝜓^𝑁,𝜓₁𝒟𝜓₁𝜓₁,𝜓₁,𝜓₂𝜓₂,𝜓₂𝒟𝜓²,𝜓₂𝜓₂,𝜓₂⋯𝒟𝜓^𝑁,𝜓₂𝒟𝜓⋮₁⋮⋱𝜓₂⋮,𝜓₂,𝜓_𝑁𝜓_𝑁,𝜓_𝑁𝒟𝜓²,𝜓_𝑁𝜓_𝑁,𝜓_𝑁…𝒟𝜓^𝑁,𝜓_𝑁𝜓_𝑁,𝜓_𝑁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(4.11)   𝜓₀,𝜓₁,𝜓₂,…,𝜓_𝑁 are a sequence of standard orthogonal basis, 𝒟𝜓𝜓_𝑖𝑖,𝜓_𝑖=1,𝑖=1,2,…,𝑁,𝒟𝜓_𝑖,𝜓_𝑗=−𝒟𝜓_𝑗,𝜓_𝑖,(𝑖≠𝑗),,𝜓_𝑖=𝜕_𝑥𝜓_𝑖,𝜓_𝑖=0,𝑖=1,2,…,𝑁.(4.12)   𝒟₁ is a constant antisymmetric matrix.
According to {𝐻,𝐺}=∇𝐻𝒟(∇𝐺)^𝑇, then {𝐻₁,𝐻₁}=∇𝐻₁𝒟(∇𝐻₁)^𝑇=0.
The function 𝐻₁(𝑢)̂ is a conservation law of energy, that is, 𝑑𝑢̂𝑑𝑡=𝒟¹(∇𝐻₁(𝑢)̂)^𝑇 has the property of energy conservation law.

5. Legendre Polynomials Spectral Approximation for the Boundary Value Problem of the Wave Equation

Now we consider the wave equation with the fixed boundary conditions discussed above: 𝜕²𝑢𝜕𝑡²=𝜕²𝑢𝜕𝑥²𝑢(−1,𝑡)=,𝑢𝑥(1∈,𝑡)(−=1,0,1),𝑡∈𝑡∈[0[,0𝑇,𝑇].],(5.1)

It can be rewritten as two forms of the first-order equations: 𝜕𝑢𝜕𝜕𝑣𝑡=𝜕𝜕𝑥𝑣,𝜕𝑡=𝜕𝜕𝑥𝑢.(5.2)

This equation can be written as Hamiltonian form: 𝜕𝑢𝜕𝑡=𝒟¹𝛿𝐻₁𝛿𝑢,𝑢=𝑣𝑢.(5.3)

The Hamiltonian operator is 𝒟₁=^0𝜕𝑥_𝜕𝑥0, and the corresponding Hamiltonian functional is 𝐻₁=1/2∫¹⁻¹(𝑢²+𝑣²)𝑑𝑥.

There is another way to write the equation into Hamiltonian form, that is, 𝜕𝑢𝜕𝑡=𝒟²𝛿𝐻₁𝛿𝑢,𝑢=𝑣𝑢.(5.4)

The corresponding Hamiltonian operator is 𝒟₂=⁰¹⁻¹⁰, and the Hamiltonian functional is 𝐻₂=1/2∫¹⁻¹(𝑢^2𝑥+𝑣²)𝑑𝑥.

In this case, the element in 𝑈 is denoted by 𝑢=(𝑢₁,𝑢₂)^𝑇. The inner product is denoted by (𝑢,𝑣)=∑^2𝑖=1(𝑢_𝑖,𝑣_𝑖), 𝑢,𝑣∈𝑈, where (𝑢_𝑖,𝑣_𝑖)=∫¹⁻¹𝑢_𝑖𝑣_𝑖𝑑𝑥.

Take the orthogonal basis: 𝜓0⁰,𝜓0₀,𝜓0¹,𝜓0₁,𝜓0²,𝜓0₂,….(5.5)

Set 𝐵=span𝜓⁰0,𝜓0₀,𝜓0¹,𝜓0₁,…,𝜓0^𝑁,𝜓0_𝑁⊂𝑈.(5.6) That is, 𝐵 is 2𝑁+1-dimensional subspace of 𝑈.

The orthogonal projection is 𝑃∶𝑈→𝐵, 𝑢⟶𝑢=𝑃𝑢=𝑎₀𝜓0⁰+𝑎⁰̃𝜓0₀+⋯+𝑎^𝑁𝜓0^𝑁+𝑎^𝑁̃𝜓0_𝑁.(5.7)

Denote 𝐵={𝑢̂=(𝑎₀,𝑎₀̃,𝑎₁,𝑎₁̃,…,𝑎_𝑁,𝑎_𝑁̃)^𝑇∈𝑅^2𝑁+2}. The inner product of 𝐵 is usually denoted by Euclidean inner ⟨⋅,⋅⟩, that is,𝑝,̂𝑞=𝑎₀𝑏₀+𝑎₀̃𝑏̃₀+⋯+𝑎_𝑁𝑏_𝑁+𝑎_𝑁̃𝑏̃_𝑁,(5.8) where 𝑝̂=(𝑎₀,𝑎₀̃,…,𝑎_𝑁,𝑎_𝑁̃)^𝑇 and 𝑞̂=(𝑏₀,𝑏₀̃,…,𝑏_𝑁,𝑏̃_𝑁)^𝑇.

Set 𝐼∶𝐵→𝐵, 𝑢̂⟶𝑢=𝐼𝑢̂=𝑎₀𝜓0⁰+𝑎⁰̃𝜓0₀+⋯+𝑎^𝑁𝜓0^𝑁+𝑎^𝑁̃𝜓0_𝑁.(5.9)

Denote 𝑃=𝐼⁻¹∘𝑃∶𝑈→𝐵, 𝑢⟶𝑃𝑢=𝑎₀,𝑎₀̃,𝑎₁,𝑎₁̃,𝑎₂,𝑎₂̃,…,𝑎_𝑁,𝑎_𝑁̃^𝑇.(5.10)

The discretization of the Hamiltonian operator 𝒟 is 𝒟(𝑝)̂𝑞𝒟̂==𝑝𝑃̂∘𝒟∘𝒟(𝐼∘𝑝𝐼)̂⋅∶𝐼𝐵𝑞̂,⟶∀𝑝𝐵,̂𝑞,̂∈𝐵.(5.11)

The discretization of a functionals 𝐻(𝑥) in 𝑈 is 𝐻(𝑢)̂=¹⁻¹𝐻(𝐼𝑢)̂𝑑𝑥,∀𝑢̂∈𝐵.(5.12)

By the above analysis and the chosen orthogonal basis, for 𝑁=2, 𝒟₁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00√70002√0007−√20072000√210−√2072000√2100−√2210002−√00021𝐻200₁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(𝑢)̂=12¹⁻¹𝑎⁰=1𝜓₀+𝑎₁𝜓₁+𝑎₂𝜓₂²+𝑎₀̃𝜓₀+𝑎₁̃𝜓₁+𝑎₂̃𝜓₂²𝑑𝑥2𝑎²⁰+𝑎²⁰̃+𝑎²¹+𝑎²¹̃+𝑎²²+𝑎²²̃.(5.13)

Then ∇𝐻^𝑇1=𝑎₀,𝑎₀̃,𝑎₁,𝑎₁̃,𝑎₂,𝑎₂̃.(5.14)

The corresponding semidiscrete approximation is 𝑑𝑎₀𝑑𝑡=√72𝑎¹𝑑𝑎₀̃,𝑑𝑡=√72𝑎¹̃𝑑𝑎₁,𝑑𝑡=−√72𝑎⁰+√212𝑎²𝑑𝑎₁̃,𝑑𝑡=−√72𝑎⁰̃+√212𝑎²̃𝑑𝑎₂,𝑑𝑡=−√212𝑎¹𝑑𝑎₂̃,𝑑𝑡=−√212𝑎¹̃.(5.15)

For the other form,we can also obtain 𝐻𝒟₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−000001010000−000001010000−000001000010₂⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(𝑢)̂=12¹⁻¹𝑎⁰=1𝜓₀+𝑎₁𝜓₁+𝑎₂𝜓₂^2𝑥+𝑎₀̃𝜓₀+𝑎₁̃𝜓₁+𝑎₂̃𝜓₂²𝑑𝑥252𝑎²⁰+𝑎²⁰̃+212𝑎²¹+𝑎²¹̃+√3𝑎₀𝑎₂+𝑎²²̃.((55..1167))

Then ∇𝐻₂=52𝑎⁰+√32𝑎²,𝑎₀̃,212𝑎¹,𝑎₁̃,512𝑎²+√32𝑎⁰,𝑎₂.(5.18)

The corresponding semidiscrete approximation is 𝑑𝑎₀𝑑𝑡=𝑎⁰̃𝑑𝑎₀̃,𝑑𝑡=−52𝑎⁰−√32𝑎²𝑑𝑎₁,𝑑𝑡=a¹̃𝑑𝑎₁̃,𝑑𝑡=−221𝑎¹𝑑𝑎₂,𝑑𝑡=𝑎²̃𝑑𝑎₂̃,𝑑𝑡=−√32𝑎⁰−512𝑎².(5.19)

Similar to the analysis of the KdV equation, for the situation of 𝑁=2, we can verify that 𝒟₁ and 𝒟₂ are all Hamiltonian operators; so the approximating system can be written as 𝑑𝑢/̂𝑑𝑡=𝒟₁(∇𝐻₁(𝑢)̂)^𝑇 and 𝑑𝑢/̂𝑑𝑡=𝒟₂(∇𝐻₂(𝑢)̂)^𝑇, two different Hamiltonian forms. As 𝐷₁ and 𝒟₂ both are constant antisymmetric matrix for 𝑁>2, 𝑑𝑢̂𝑑𝑑𝑡𝑢̂=𝑃∘𝜕0_𝑥𝜕0^𝑥∘𝐼⋅𝛿𝛿𝐻𝑢¹(𝐼𝑢)̂,𝑑𝑡=𝑃∘−0101∘𝐼⋅𝛿𝛿𝐻𝑢²(𝐼𝑢)̂.(5.20) are finite-dimensional Hamiltonian systems. The approximating systems can preserve the Poisson structure given by Hamiltonian operators 𝒟₁ and 𝒟₂.

Theorem 5.1. The equations 𝑑𝑢/̂𝑑𝑡=𝒟₁(∇𝐻₁(𝑢)̂)^𝑇 and 𝑑𝑢/̂𝑑𝑡=𝒟₂(∇𝐻₂(𝑢)̂)^𝑇 are the discretizations of the 1-dim wave equation 𝜕𝑢𝜕𝜕𝑣𝑡=𝜕𝜕𝑥𝑣,𝜕𝑡=𝜕𝜕𝑥𝑢.(5.21) Then 𝑑𝑢/̂𝑑𝑡=𝒟₁(∇𝐻₁(𝑢)̂)^𝑇 and 𝑑𝑢/̂𝑑𝑡=𝒟₂(∇𝐻₂(𝑢)̂)^𝑇 both have the property of energy conservation law.

The proof of Theorem 5.1 is similar to that of Theorem 4.1.

Acknowledgments

The topic is proposed by the project team of Professor Yongzhong Song and Professor Yushun Wang. The aythors gratefully acknowlege their considerable help by means of suggestion, comments and criticism. The work is supported by the Foundation for the Authors of the National Excellent Doctoral Thesis Award of China under Grant no. 200720, the National Natural Science Foundation of China under Grant no. 10971102, and “333 Project” Foundation of Jiangsu Province.