Conservation Laws of Discrete Korteweg-de Vries Equation

Rasin, Alexander G.; Hydon, Peter E.

doi:10.3842/sigma.2005.026

Cited by 13 publications

(26 citation statements)

References 3 publications

(4 reference statements)

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“…We use the method of calculating conservation laws described in [12,13], which works as follows. The three-point conservation laws can be determined directly by substituting (1) into F (k+1, l, u 1,0 , u 1,1 )−F (k, l, u 0,0 , u 0,1 )+G(k, l+1, u 0,1 , u 1,1 )−G(k, l, u 0,0 , u 1,0 ) = 0, (4) and solving the resulting functional equation.…”

Section: The Methodsmentioning

confidence: 99%

“…The first systematic technique for obtaining all conservation laws of a given type was introduced by Hydon [8], who found all three-point conservation laws of the discrete potential modified Korteweg-de Vries (dpmKdV) equation. This technique was improved in [12] and used to find all three-and five-point conservation laws of the discrete Korteweg-de Vries (dKdV) equation. Rasin & Hydon also classified all three-point conservation laws for a family of integrable difference equations due to Nijhoff, Quispel & Capel, showing that every equation from this class has at least two nontrivial conservation laws [13].…”

Section: Introductionmentioning

confidence: 99%

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Conservation laws for integrable difference equations

Rasin¹,

Hydon²

2007

J. Phys. A: Math. Theor.

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This paper deals with conservation laws for all integrable difference equations that belong to the famous Adler-Bobenko-Suris classification. All inequivalent three-point conservation laws are found, as are three five-point conservation laws for each equation. We also describe a method of generating conservation laws from known ones; this method can be used to generate higherorder conservation laws from those that are listed here. † O.Rasin@surrey.ac.uk ‡ P.Hydon@surrey.ac.uk

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Section: The Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Conservation laws for integrable difference equations

Rasin¹,

Hydon²

2007

J. Phys. A: Math. Theor.

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“…In this section, we find conservation laws for quad-graph equations by the direct method, which is explained fully in [10,16]. Three-point conservation laws have components …”

Section: Conservation Lawsmentioning

confidence: 99%

Integrability conditions for nonautonomous quad-graph equations

Sahadevan

Rasin

Hydon

2007

Journal of Mathematical Analysis and Applications

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This paper presents a systematic investigation of the integrability conditions for nonautonomous quadgraph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations.

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“…The Gardner CLaws for dpKdV. In [21], Rasin and Schiff used a discrete version of the Gardner transformation to construct an infinite number of CLaws for the dpKdV equation (22). (Rasin [20] has subsequently used the same approach to generate an infinite hierarchy of CLaws for all the equations in the ABS classification and one asymmetric equation.)…”

Section: Using Roots To Detect Equivalencementioning

confidence: 99%

Characteristics of Conservation Laws for Difference Equations

Grant

Hydon

2013

Found Comput Math

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Abstract. Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka-Volterra equation.

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Conservation Laws of Discrete Korteweg-de Vries Equation

Cited by 13 publications

References 3 publications

Conservation laws for integrable difference equations

Conservation laws for integrable difference equations

Integrability conditions for nonautonomous quad-graph equations

Characteristics of Conservation Laws for Difference Equations

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