Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 824167 | 13 pages | https://doi.org/10.1155/2011/824167

Legendre Polynomials Spectral Approximation for the Infinite-Dimensional Hamiltonian Systems

Academic Editor: Paulo Batista GonƧalves
Received11 Oct 2010
Revised06 Apr 2011
Accepted30 Apr 2011
Published12 Jul 2011

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 š“={š»[š‘¢]ā‰”š»(š‘„,š‘¢(š‘›))|Hisainļ¬nitediļ¬€erentiablesmoothfunction}; here š‘¢(š‘›)=(š‘¢1š‘‡,š‘¢2š‘‡,ā€¦,š‘¢š‘›š‘‡)š‘‡, and š‘¢š‘– denotes the š‘–th derivative of š‘¢. To each š»[š‘¢]āˆˆš“, there exists a functional āˆ«ā„‹=š»[š‘¢]š‘‘š‘„, and the corresponding set of all functional is āˆ«ā„±={ā„‹=š»[š‘¢]š‘‘š‘„|š»[š‘¢]āˆˆš“}. š›æā„‹/š›æš‘¢ is the variational derivative of the functional ā„‹āˆˆā„±. With the aid of the differential operator š’Ÿ, we can define a binary operator on ā„±: {ī€œā„‹,š’¢}=š›æā„‹š’Ÿš›æš‘¢š›æš’¢š›æš‘¢,āˆ€ā„‹,š’¢āˆˆā„±.(2.1)

If this binary operator satisfies the following conditions: (i){,} is antisymmetric, {ā„‹,š’¢}=āˆ’{š’¢,ā„‹},(2.2)(ii){,} is bilinear, {š›¼ā„‹+š›½š’¢,š’¦}=š›¼{ā„‹,š’¦}+š›½{š’¢,š’¦},āˆ€š›¼,š›½āˆˆš‘…,(2.3)(iii){,} satisfies the Jacobi identity, {{ā„‹,š’¢},š’¦}+{{š’¢,š’¦},ā„‹}+{{š’¦,ā„‹},š’¢}=0,(2.4) for all functionals ā„‹,š’¢,š’¦āˆˆā„±, then, it is called a Poisson bracket. In this case, š’Ÿ is called Hamiltonian operator.

For given a Hamiltonian functional ā„‹āˆˆā„± and a Hamiltonian operator š’Ÿ, Hamiltonian equation takes the following form: š‘¢š‘”=š’Ÿš›æā„‹.š›æš‘¢(2.5) This evolution equation is called an infinite-dimensional Hamiltonian system.

Consider the infinite-dimensional Hamiltonian system of the KdV equation: š‘¢š‘”+6š‘¢š‘¢š‘„+š‘¢š‘„š‘„š‘„=0.(2.6) It has the Hamiltonian structure š‘¢š‘”[š‘¢]=š’Ÿš›æā„‹,(2.7) where š’Ÿ=šœ•š‘„ is the Hamiltonian operator and ā„‹[š‘¢]=ī€œ1āˆ’1ī‚€12š‘¢2š‘„āˆ’š‘¢3ī‚š‘‘š‘„(2.8) is the Hamiltonian functional.

3. Semidiscrete Legendre Polynomials Spectral Approximation

Let šæš‘›(š‘„) be the š‘›th degree Legendre polynomial. The Legendre polynomials satisfy the three-term recurrence relation: šæ0(š‘„)=1,šæ1(š‘„)=š‘„,(š‘›+1)šæš‘›+1(š‘„)=(2š‘›+1)š‘„šæš‘›(š‘„)āˆ’š‘›šæš‘›āˆ’1(š‘„),š‘›ā‰„1(3.1) and the orthogonality relation: ī€œ1āˆ’1šæš‘˜(š‘„)šæš‘—1(š‘„)š‘‘š‘„=š›æš‘˜+(1/2)š‘˜š‘—,šæš‘›(Ā±1)=(Ā±1)š‘›.(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 {šæš‘›} be a sequence of orthogonal polynomials. As a general rule, for differential equations with š‘š boundary conditions, our task is to look for basis functions in the formšœ™š‘˜(š‘„)=šæš‘˜(š‘„)+š‘šī“š‘—=1š‘Žš‘—(š‘˜)šæš‘˜+š‘—(š‘„),(3.3) where š‘Žš‘—(š‘˜)(š‘—=1,2,ā€¦,š‘š) are chosen so that šœ™š‘˜(š‘„) satisfy the š‘š homogeneous boundary conditions.

Suppose that š‘ˆ={š»(š‘„)|š»(š‘„)isasmoothfunction,š‘„āˆˆ[āˆ’1,1]}, for the fixed homogeneous boundary conditions š»(āˆ’1)=š»(1)=0.(3.4) As š‘š=2, (3.3) has the form šœ™š‘˜(š‘„)=šæš‘˜(š‘„)+š‘Ž1šæš‘˜+1(š‘„)+š‘Ž2šæš‘˜+2(š‘„).(3.5) Using the basic properties of Legendre polynomials and the boundary value conditions, obviously šœ™š‘˜(āˆ’1)=0,šœ™š‘˜(1)=0.(3.6) We can verify readily that šœ™š‘˜(š‘„)=šæš‘˜(š‘„)āˆ’šæš‘˜+2(š‘„).(3.7) Easily, we obtain šœ™0(š‘„),šœ™1(š‘„),šœ™2(š‘„),ā€¦. The šæ2-inner product on š‘ˆ is defined by ī€œ(š‘,š‘ž)=1āˆ’1š‘ā‹…š‘žš‘‘š‘„,āˆ€š‘,š‘žāˆˆš‘ˆ.(3.8)

The basis functions šœ™š‘˜(š‘„)(š‘˜=1,2,ā€¦) can be orthogonalized standard on the šæ2- inner product. Thus, we can get the sequence of standard orthogonal basis functions šœ“š‘˜(š‘„).

After carefully calculation, the orthogonal basis isšœ“0=14ī‚€āˆš15āˆ’3š‘„2ī‚,šœ“1=14ī‚€āˆšāˆš105š‘„āˆ’105š‘„3ī‚,šœ“2=18ī‚€āˆšāˆ’3āˆš5+245š‘„2āˆšāˆ’215š‘„4ī‚,ā‹®(3.9)

Set ī€½šœ“šµ=span0,šœ“1,šœ“2,ā€¦,šœ“š‘ī€¾āŠ‚š‘ˆ,(3.10) and set š‘ƒ as an orthogonal projection. š‘ƒāˆ¶š‘ˆā†’šµ, š‘¢āŸ¶š‘¢=š‘ƒš‘¢=š‘Ž0šœ“0+š‘Ž1šœ“1+ā‹Æ+š‘Žš‘šœ“š‘,(3.11) where š‘Žš‘›=ī€œ1āˆ’1š‘¢(š‘„)šœ“š‘›(š‘„)š‘‘š‘„,š‘›=1,2,3,ā€¦.(3.12)

Denote īšµ={Ģ‚š‘¢=(š‘Ž0,š‘Ž1,ā€¦,š‘Žš‘)š‘‡āˆˆš‘…š‘+1}. The inner product of īšµ is usually denoted by Euclidean inner āŸØā‹…,ā‹…āŸ©, that is, āŸØĢ‚š‘,Ģ‚š‘žāŸ©=š‘Ž0ā‹…Ģƒš‘Ž0+š‘Ž1ā‹…Ģƒš‘Ž1+š‘Ž2ā‹…Ģƒš‘Ž2+ā‹Æ+š‘Žš‘ā‹…Ģƒš‘Žš‘ī,āˆ€Ģ‚š‘,Ģ‚š‘žāˆˆšµ,(3.13) where Ģ‚š‘ž=(Ģƒš‘Ž0,ī‚š‘Ž1,Ģƒš‘Ž2,ā€¦,Ģƒš‘Žš‘)š‘‡. Set īš¼āˆ¶šµā†’šµ, Ģ‚š‘¢āŸ¶š‘¢=š¼Ģ‚š‘¢=š‘Ž0šœ“0+š‘Ž1šœ“1+ā‹Æ+š‘Žš‘.(3.14)

Denote īš‘ƒ=š¼āˆ’1īšµāˆ˜š‘ƒāˆ¶š‘ˆā†’, īī€·š‘Žš‘¢āŸ¶š‘ƒš‘¢=0,š‘Ž1,š‘Ž2,ā€¦,š‘Žš‘ī€øš‘‡.(3.15)

Hamiltonian equation š‘¢š‘”=š’Ÿš›æā„‹š›æš‘¢(3.16) has the special Poisson structure; so we can exploit it to design numerical approximations. We can discretize Hamiltonian operator š’Ÿ and Hamiltonian functionals; then a numerical bracket can be defined.

The discretization of the Hamiltonian operator š’Ÿ is īīīīīīīš’Ÿ=š‘ƒāˆ˜š’Ÿāˆ˜š¼āˆ¶šµāŸ¶šµ,š’Ÿ(Ģ‚š‘)Ģ‚š‘ž=Ģ‚š‘āˆ˜š’Ÿ(š¼Ģ‚š‘)ā‹…š¼Ģ‚š‘ž,āˆ€Ģ‚š‘,Ģ‚š‘žāˆˆšµ.(3.17)

The discretization of a functional š»(š‘„) in š‘ˆ is īī€œš»(Ģ‚š‘¢)=1āˆ’1īšµš»(š¼Ģ‚š‘¢)š‘‘š‘„,āˆ€Ģ‚š‘¢āˆˆ.(3.18)

Let š‘ˆ be the set of discrete functionals; then we can define a bracket on š‘ˆ, ī‚†īīšŗī‚‡īš»īš’Ÿī‚€āˆ‡īšŗī‚š»,=āˆ‡š‘‡,āˆ‡īī‚µšœ•īš»š»=šœ•Ģ‚š‘¢1,šœ•īš»šœ•Ģ‚š‘¢2šœ•īš»,ā€¦,šœ•Ģ‚š‘¢š‘ī‚¶,(3.19) which is an approximation of bracket {š»,šŗ}.

Now we define the semidiscrete approximative equation in īšµ of the infinite-dimensional Hamiltonian system š‘¢š‘”=š’Ÿš›æā„‹š›æš‘¢ asš‘‘Ģ‚š‘¢=īš‘‘š‘”š’Ÿ(āˆ‡š»(Ģ‚š‘¢))š‘‡.(3.20)

If īš’Ÿ is still a Hamiltonian operator, then (3.20) is exactly a finite-dimensional Hamiltonian system. The function īš»(š‘ƒš‘¢) 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: š‘¢š‘”+6š‘¢š‘¢š‘„+š‘¢š‘„š‘„š‘„[],[].=0,š‘„āˆˆ(āˆ’1,1),š‘”āˆˆ0,š‘‡š‘¢(āˆ’1,š‘”)=š‘¢(1,š‘”)=0,š‘”āˆˆ0,š‘‡(4.1)

The KdV equation can be written as Hamiltonian form: š‘¢š‘”=šœ•š‘„š›æā„‹1š›æš‘¢,(4.2) where the Hamiltonian operator is š’Ÿ1=šœ•š‘„ and the the Hamiltonian functional is ā„‹1=āˆ«1āˆ’1((1/2)š‘¢2š‘„āˆ’š‘¢3)š‘‘š‘„.

By above analysis and the chosen orthogonal basis, for š‘=2,īš’Ÿ1=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽ02āˆš720āˆ’āˆš720āˆš212āˆš0āˆ’2120āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ ,īš»1ī€œ(Ģ‚š‘¢)=1āˆ’1ī‚ƒ12ī€·š‘Ž0šœ“0+š‘Ž1šœ“1+š‘Ž2šœ“2ī€ø2š‘„āˆ’ī€·š‘Ž0šœ“0+š‘Ž1šœ“1+š‘Ž2šœ“2ī€ø3ī‚„=5š‘‘š‘„4š‘Ž20+214š‘Ž21+āˆš32š‘Ž0š‘Ž2+514š‘Ž22+3āˆš15š‘Ž1430āˆ’3āˆš15š‘Ž1420š‘Ž2+āˆš155š‘Ž2221š‘Ž2+āˆš4235š‘Ž200232+āˆš152š‘Ž0š‘Ž21+āˆš695š‘Ž1540š‘Ž22.(4.3) Then āˆ‡īš»š‘‡1=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽ52š‘Ž0+āˆš32š‘Ž2+9āˆš15š‘Ž1420āˆ’3āˆš57š‘Ž0š‘Ž2+āˆš152š‘Ž21+āˆš5š‘Ž15422212š‘Ž1+āˆš155š‘Ž111š‘Ž2+āˆš157š‘Ž0š‘Ž1āˆš32š‘Ž0+512š‘Ž2āˆ’3āˆš5š‘Ž1420+āˆš155a2221+āˆš12695š‘Ž200222+āˆš695š‘Ž770š‘Ž2āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ .(4.4)

This equations can be written as Hamiltonian form in another way, that is, š‘¢š‘”=ī€·šœ•š‘„š‘„š‘„+4š‘¢šœ•š‘„+2š‘¢š‘„š¼ī€øš›æā„‹2š›æš‘¢,(4.5) where the Hamiltonian operator is š’Ÿ2=šœ•š‘„š‘„š‘„+4š‘¢šœ•š‘„+2š‘¢š‘„š¼ abd the Hamiltonian functional is ā„‹2=āˆ«1āˆ’1(āˆ’(1/2)š‘¢2)š‘‘š‘„.

In the same theory, for š‘=2, we can get īš’Ÿ2=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽī‚™āˆ’2157š‘Ž1āˆ’āˆš1372ī‚™+4157š‘Ž0ī‚™āˆ’657š‘Ž2ī‚™1057š‘Ž1āˆ’āˆšī‚™7āˆ’2157š‘Ž0ī‚™āˆ’457š‘Ž2ī‚™0ā„›āˆ’457š‘Ž1āˆ’5āˆš212ī‚™āˆ’657š‘Ž0āˆ’16ī‚™11157š‘Ž28ī‚™11157š‘Ž1āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ īš»2ī€œ(Ģ‚š‘¢)=1āˆ’1ī‚ƒāˆ’12ī€·š‘Ž0šœ“0+š‘Ž1šœ“1+š‘Ž2šœ“2ī€ø2ī‚„1š‘‘š‘„=āˆ’2ī€·š‘Ž20+š‘Ž21+š‘Ž22ī€ø(4.6) where āˆšā„›denotesāˆ’20ī‚™21+1057š‘Ž0+8ī‚™10157š‘Ž2.

Then āˆ‡īš»š‘‡2=ī€·āˆ’š‘Ž0,āˆ’š‘Ž1,āˆ’š‘Ž2ī€ø.(4.7) The corresponding semidiscrete approximation is š‘‘Ģ‚š‘¢=īš‘‘š‘”š’Ÿ(āˆ‡š»(Ģ‚š‘¢))š‘‡ī€·š‘Ž,Ģ‚š‘¢=0,š‘Ž1,š‘Ž2ī€øš‘‡.(4.8)

It is easy to verify that īš’Ÿ1 is Hamiltonian operator; so the approximating system can be written as š‘‘Ģ‚š‘¢=īš’Ÿš‘‘š‘”1ī€·āˆ‡š»1ī€ø(Ģ‚š‘¢)š‘‡,(4.9) adifferent Hamiltonian form, and it can be verified that īš’Ÿ2 is not a Hamiltonian operator. As īš’Ÿ1 is a constant antisymmetric matrix, š‘‘Ģ‚š‘¢=īš‘‘š‘”š‘ƒāˆ˜šœ•š‘„āˆ˜š¼ā‹…š›æš»1š›æš‘¢(š¼Ģ‚š‘¢)(4.10) is a finite-dimensional Hamiltonian system. So the approximating system can preserve the Poisson structure given by Hamiltonian operator īš’Ÿ1.

Theorem 4.1. The equation īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=1(āˆ‡š»1(Ģ‚š‘¢))š‘‡ is the discretization of the KdV equation š‘¢š‘”+6š‘¢š‘¢š‘„+š‘¢š‘„š‘„š‘„=0; then īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=1(āˆ‡š»1(Ģ‚š‘¢))š‘‡ has the property of energy conservation law.

Proof. īīīīīš’Ÿ=š‘ƒāˆ˜š’Ÿāˆ˜š¼āˆ¶šµāŸ¶šµ,š¼āˆ¶šµāŸ¶šµ,Ģ‚š‘¢āŸ¶š‘¢=š¼Ģ‚š‘¢=š‘Ž0šœ“0+š‘Ž1šœ“1+ā‹Æ+š‘Žš‘šœ“š‘,š‘ƒāˆ¶š‘ˆāŸ¶šµ,š‘¢āŸ¶š‘¢=š‘ƒš‘¢=š‘Ž0šœ“0+š‘Ž1šœ“1+ā‹Æ+š‘Žš‘šœ“š‘,īš‘ƒ=š¼āˆ’1īīī€·š‘Žāˆ˜š‘ƒāˆ¶š‘ˆāŸ¶šµ,š‘¢āŸ¶š‘ƒš‘¢=0,š‘Ž1,š‘Ž2,ā€¦,š‘Žš‘ī€øš‘‡,īāŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽī€·š’Ÿ=š’Ÿšœ“1,šœ“1ī€øī€·šœ“1,šœ“1ī€øī€·š’Ÿšœ“2,šœ“1ī€øī€·šœ“1,šœ“1ī€øā‹Æī€·š’Ÿšœ“š‘,šœ“1ī€øī€·šœ“1,šœ“1ī€øī€·š’Ÿšœ“1,šœ“2ī€øī€·šœ“2,šœ“2ī€øī€·š’Ÿšœ“2,šœ“2ī€øī€·šœ“2,šœ“2ī€øā‹Æī€·š’Ÿšœ“š‘,šœ“2ī€øī€·šœ“2,šœ“2ī€øī€·ā‹®ā‹®ā‹±ā‹®š’Ÿšœ“1,šœ“š‘ī€øī€·šœ“š‘,šœ“š‘ī€øī€·š’Ÿšœ“2,šœ“š‘ī€øī€·šœ“š‘,šœ“š‘ī€øā€¦ī€·š’Ÿšœ“š‘,šœ“š‘ī€øī€·šœ“š‘,šœ“š‘ī€øāŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ .(4.11) ā€‰ šœ“0,šœ“1,šœ“2,ā€¦,šœ“š‘ are a sequence of standard orthogonal basis, ī€·šœ“š‘–,šœ“š‘–ī€øī€·=1,š‘–=1,2,ā€¦,š‘,š’Ÿšœ“š‘–,šœ“š‘—ī€øī€·=āˆ’š’Ÿšœ“š‘—,šœ“š‘–ī€øī€·,(š‘–ā‰ š‘—),š’Ÿšœ“š‘–,šœ“š‘–ī€ø=ī€·šœ•š‘„šœ“š‘–,šœ“š‘–ī€ø=0,š‘–=1,2,ā€¦,š‘.(4.12) ā€‰ īš’Ÿ1 is a constant antisymmetric matrix.
According to {īīīš»īīš»,šŗ}=āˆ‡š’Ÿ(āˆ‡šŗ)š‘‡, then {īš»1,īš»1īš»}=āˆ‡1īīš»š’Ÿ(āˆ‡1)š‘‡=0.
The function īš»1(Ģ‚š‘¢) is a conservation law of energy, that is, š‘‘Ģ‚š‘¢=īš’Ÿš‘‘š‘”1(āˆ‡š»1(Ģ‚š‘¢))š‘‡ 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: šœ•2š‘¢šœ•š‘”2=šœ•2š‘¢šœ•š‘„2[],[].,š‘„āˆˆ(āˆ’1,1),š‘”āˆˆ0,š‘‡š‘¢(āˆ’1,š‘”)=š‘¢(1,š‘”)=0,š‘”āˆˆ0,š‘‡(5.1)

It can be rewritten as two forms of the first-order equations: šœ•š‘¢=šœ•š‘”šœ•š‘£,šœ•š‘„šœ•š‘£=šœ•š‘”šœ•š‘¢.šœ•š‘„(5.2)

This equation can be written as Hamiltonian form: šœ•š‘¢šœ•š‘”=š’Ÿ1š›æš»1,š›æš‘¢īƒ©š‘¢š‘£īƒŖš‘¢=.(5.3)

The Hamiltonian operator is š’Ÿ1=ī‚€0šœ•š‘„šœ•š‘„0ī‚, and the corresponding Hamiltonian functional is š»1āˆ«=1/21āˆ’1(š‘¢2+š‘£2)š‘‘š‘„.

There is another way to write the equation into Hamiltonian form, that is, šœ•š‘¢šœ•š‘”=š’Ÿ2š›æš»1,š›æš‘¢īƒ©š‘¢š‘£īƒŖš‘¢=.(5.4)

The corresponding Hamiltonian operator is š’Ÿ2=ī€·01āˆ’10ī€ø, and the Hamiltonian functional is š»2āˆ«=1/21āˆ’1(š‘¢2š‘„+š‘£2)š‘‘š‘„.

In this case, the element in š‘ˆ is denoted by š‘¢=(š‘¢1,š‘¢2)š‘‡. The inner product is denoted by āˆ‘(š‘¢,š‘£)=2š‘–=1(š‘¢š‘–,š‘£š‘–),ā€‰š‘¢,š‘£āˆˆš‘ˆ, where (š‘¢š‘–,š‘£š‘–āˆ«)=1āˆ’1š‘¢š‘–š‘£š‘–š‘‘š‘„.

Take the orthogonal basis: īƒ©šœ“00īƒŖ,īƒ©0šœ“0īƒŖ,īƒ©šœ“10īƒŖ,īƒ©0šœ“1īƒŖ,īƒ©šœ“20īƒŖ,īƒ©0šœ“2īƒŖ,ā€¦.(5.5)

Set šœ“šµ=spanīƒÆīƒ©00īƒŖ,īƒ©0šœ“0īƒŖ,īƒ©šœ“10īƒŖ,īƒ©0šœ“1īƒŖīƒ©šœ“,ā€¦,š‘0īƒŖ,īƒ©0šœ“š‘īƒŖīƒ°āŠ‚š‘ˆ.(5.6) That is, šµ is 2š‘+1-dimensional subspace of š‘ˆ.

The orthogonal projection is š‘ƒāˆ¶š‘ˆā†’šµ, š‘¢āŸ¶š‘¢=š‘ƒš‘¢=š‘Ž0īƒ©šœ“00īƒŖ+Ģƒš‘Ž0īƒ©0šœ“0īƒŖ+ā‹Æ+š‘Žš‘īƒ©šœ“š‘0īƒŖ+Ģƒš‘Žš‘īƒ©0šœ“š‘īƒŖ.(5.7)

Denote īšµ={Ģ‚š‘¢=(š‘Ž0,Ģƒš‘Ž0,š‘Ž1,Ģƒš‘Ž1,ā€¦,š‘Žš‘,Ģƒš‘Žš‘)š‘‡āˆˆš‘…2š‘+2}. The inner product of īšµ is usually denoted by Euclidean innerā€‰āŸØā‹…,ā‹…āŸ©, that is,ī«ī¬Ģ‚š‘,īš‘ž=š‘Ž0š‘0+Ģƒš‘Ž0Ģƒš‘0+ā‹Æ+š‘Žš‘š‘š‘+Ģƒš‘Žš‘Ģƒš‘š‘,(5.8) where Ģ‚š‘=(š‘Ž0,Ģƒš‘Ž0,ā€¦,š‘Žš‘,Ģƒš‘Žš‘)š‘‡ and Ģ‚š‘ž=(š‘0,Ģƒš‘0,ā€¦,š‘š‘,Ģƒš‘š‘)š‘‡.

Set īš¼āˆ¶šµā†’šµ, Ģ‚š‘¢āŸ¶š‘¢=š¼Ģ‚š‘¢=š‘Ž0īƒ©šœ“00īƒŖ+Ģƒš‘Ž0īƒ©0šœ“0īƒŖ+ā‹Æ+š‘Žš‘īƒ©šœ“š‘0īƒŖ+Ģƒš‘Žš‘īƒ©0šœ“š‘īƒŖ.(5.9)

Denote īš‘ƒ=š¼āˆ’1īšµāˆ˜š‘ƒāˆ¶š‘ˆā†’, īī€·š‘Žš‘¢āŸ¶š‘ƒš‘¢=0,Ģƒš‘Ž0,š‘Ž1,Ģƒš‘Ž1,š‘Ž2,Ģƒš‘Ž2,ā€¦,š‘Žš‘,Ģƒš‘Žš‘ī€øš‘‡.(5.10)

The discretization of the Hamiltonian operator š’Ÿ is īīīīīīīš’Ÿ=š‘ƒāˆ˜š’Ÿāˆ˜š¼āˆ¶šµāŸ¶šµ,š’Ÿ(Ģ‚š‘)Ģ‚š‘ž=Ģ‚š‘āˆ˜š’Ÿ(š¼Ģ‚š‘)ā‹…š¼Ģ‚š‘ž,āˆ€Ģ‚š‘,Ģ‚š‘žāˆˆšµ.(5.11)

The discretization of a functionals š»(š‘„) in š‘ˆ is īī€œš»(Ģ‚š‘¢)=1āˆ’1īšµš»(š¼Ģ‚š‘¢)š‘‘š‘„,āˆ€Ģ‚š‘¢āˆˆ.(5.12)

By the above analysis and the chosen orthogonal basis, for š‘=2, īš’Ÿ1=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽāˆš0072āˆš00000072āˆ’āˆš0072āˆš0002120āˆš0āˆ’72āˆš000212āˆš00āˆ’212āˆš000000āˆ’212āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ ,īš»0011(Ģ‚š‘¢)=2ī€œ1āˆ’1ī‚ƒī€·š‘Ž0šœ“0+š‘Ž1šœ“1+š‘Ž2šœ“2ī€ø2+ī€·Ģƒš‘Ž0šœ“0+Ģƒš‘Ž1šœ“1+Ģƒš‘Ž2šœ“2ī€ø2ī‚„=1š‘‘š‘„2ī€·š‘Ž20+Ģƒš‘Ž20+š‘Ž21+Ģƒš‘Ž21+š‘Ž22+Ģƒš‘Ž22ī€ø.(5.13)

Then āˆ‡īš»š‘‡1=ī€·š‘Ž0,Ģƒš‘Ž0,š‘Ž1,Ģƒš‘Ž1,š‘Ž2,Ģƒš‘Ž2ī€ø.(5.14)

The corresponding semidiscrete approximation is š‘‘š‘Ž0=āˆšš‘‘š‘”72š‘Ž1,š‘‘Ģƒš‘Ž0=āˆšš‘‘š‘”72Ģƒš‘Ž1,š‘‘š‘Ž1āˆšš‘‘š‘”=āˆ’72š‘Ž0+āˆš212š‘Ž2,š‘‘Ģƒš‘Ž1āˆšš‘‘š‘”=āˆ’72Ģƒš‘Ž0+āˆš212Ģƒš‘Ž2,š‘‘š‘Ž2āˆšš‘‘š‘”=āˆ’212š‘Ž1,š‘‘Ģƒš‘Ž2āˆšš‘‘š‘”=āˆ’212Ģƒš‘Ž1.(5.15)

For the other form,we can also obtain īš’Ÿ2=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽāŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ ,īš»010000āˆ’10000000010000āˆ’10000000010000āˆ’1021(Ģ‚š‘¢)=2ī€œ1āˆ’1ī‚ƒī€·š‘Ž0šœ“0+š‘Ž1šœ“1+š‘Ž2šœ“2ī€ø2š‘„+ī€·Ģƒš‘Ž0šœ“0+Ģƒš‘Ž1šœ“1+Ģƒš‘Ž2šœ“2ī€ø2ī‚„=1š‘‘š‘„2ī‚€52š‘Ž20+Ģƒš‘Ž20+212š‘Ž21+Ģƒš‘Ž21+āˆš3š‘Ž0š‘Ž2+Ģƒš‘Ž22ī‚.(5.16)(5.17)

Then āˆ‡īš»2=īƒ©52š‘Ž0+āˆš32š‘Ž2,Ģƒš‘Ž0,212š‘Ž1,Ģƒš‘Ž1,512š‘Ž2+āˆš32š‘Ž0,ī‚š‘Ž2īƒŖ.(5.18)

The corresponding semidiscrete approximation is š‘‘š‘Ž0š‘‘š‘”=Ģƒš‘Ž0,š‘‘Ģƒš‘Ž05š‘‘š‘”=āˆ’2š‘Ž0āˆ’āˆš32š‘Ž2,š‘‘š‘Ž1=Ģƒaš‘‘š‘”1,š‘‘Ģƒš‘Ž1š‘‘š‘”=āˆ’212š‘Ž1,š‘‘š‘Ž2š‘‘š‘”=Ģƒš‘Ž2,š‘‘Ģƒš‘Ž2āˆšš‘‘š‘”=āˆ’32š‘Ž0āˆ’512š‘Ž2.(5.19)

Similar to the analysis of the KdV equation, for the situation of š‘=2, we can verify that īš’Ÿ1 and īš’Ÿ2 are all Hamiltonian operators; so the approximating system can be written as īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=1(āˆ‡š»1(Ģ‚š‘¢))š‘‡ and īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=2(āˆ‡š»2(Ģ‚š‘¢))š‘‡, two different Hamiltonian forms. As š·1 and īš’Ÿ2 both are constant antisymmetric matrix for š‘>2, š‘‘Ģ‚š‘¢=īīƒ©š‘‘š‘”š‘ƒāˆ˜0šœ•š‘„šœ•š‘„0īƒŖāˆ˜š¼ā‹…š›æš»1š›æš‘¢(š¼Ģ‚š‘¢),š‘‘Ģ‚š‘¢=īīƒ©īƒŖš‘‘š‘”š‘ƒāˆ˜01āˆ’10āˆ˜š¼ā‹…š›æš»2š›æš‘¢(š¼Ģ‚š‘¢).(5.20) are finite-dimensional Hamiltonian systems. The approximating systems can preserve the Poisson structure given by Hamiltonian operators īš’Ÿ1 and īš’Ÿ2.

Theorem 5.1. The equations īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=1(āˆ‡š»1(Ģ‚š‘¢))š‘‡ and īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=2(āˆ‡š»2(Ģ‚š‘¢))š‘‡ are the discretizations of the 1-dim wave equation šœ•š‘¢=šœ•š‘”šœ•š‘£,šœ•š‘„šœ•š‘£=šœ•š‘”šœ•š‘¢.šœ•š‘„(5.21) Then īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=1(āˆ‡š»1(Ģ‚š‘¢))š‘‡ and īš’Ÿš‘‘Ģ‚š‘¢/š‘‘š‘”=2(āˆ‡š»2(Ģ‚š‘¢))š‘‡ 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 ā€œ333 Projectā€ Foundation of Jiangsu Province.

References

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Copyright Ā© 2011 Zhongquan Lv et al. 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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