Leading order integrability conditions for differential-difference equations

Hickman, Mark S.

doi:10.2991/jnmp.2008.15.1.6

Cited by 2 publications

(6 citation statements)

References 17 publications

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“…These terms yield the constraints on the undetermined coefficients or unknown functions in the density ρ. As shown in [55], the result of this split is that ρ is a density if and only if…”

Section: Leading Order Analysismentioning

confidence: 98%

“…It is quite effective for certain classes of DDEs, including the Kac-van Moerbeke and Toda lattices, but far less effective for more complicated lattices, such as the Bogoyavlenskii and the Gardner lattices. The latter examples are treated with a new method based on a leading order analysis proposed by Hickman [55].…”

Section: Part Ii: Nonlinear Differential-difference Equationsmentioning

confidence: 99%

“…It is easy to show [55] that if ρ is a density then D k ρ is also a density. Hence, using an appropriate "up-shift" all negative shifts in a density can been removed.…”

Section: Nonlinear Ddes and Conservation Lawsmentioning

confidence: 99%

“…, u q ) with ∂ 2 ρ ∂u ∂uq = 0. In [55], Hickman derived necessary conditions on this leading term (which, in the system case, is a matrix). First all terms in the candidate density ρ that contribute directly to the flux are removed.…”

Section: Leading Order Analysismentioning

confidence: 99%

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Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations

Hereman,

Adams,

Eklund

et al. 2008

Preprint

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We present direct methods, algorithms, and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).Our method for PDEs is based on calculus, linear algebra, and variational calculus. First, we compute the dilation symmetries of the given nonlinear system. Next, we build a candidate density as a linear combination with undetermined coefficients of terms that are scaling invariant. The variational derivative (Euler operator) is used to derive a linear system for the undetermined coefficients. This system is then analyzed and solved. Finally, we compute the flux with the homotopy operator.The method is applied to nonlinear PDEs in (1+1) dimensions with polynomial nonlinearities which include the Korteweg-de Vries (KdV), Boussinesq, and Drinfel'd-Sokolov-Wilson equations. An adaptation of the method is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville equations. For equations in laboratory coordinates, the coefficients of the candidate density are undetermined functions which must satisfy a mixed linear system of algebraic and ordinary differential equations.For the computation of conservation laws of nonlinear DDEs we use a splitting of the identity operator. This method is more efficient that an approach based on the discrete Euler and homotopy operators. We apply the method of undetermined coefficients to the Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. To overcome the shortcomings of the undetermined coefficient technique, we designed a new method that first calculates the leading order term and then the required terms of lower order. That method, which is no longer restricted to polynomial conservation laws, is applied to discretizations of the KdV and modified KdV equations, and a combination thereof. Additional examples include lattices due to Bogoyavlenskii, Belov-Chaltikian, and Blaszak-Marciniak.The undetermined coefficient methods for PDEs and DDEs have been implemented in Mathematica. The code TransPDEDensityFlux.m computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code DDEDensityFlux.m does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library discrete computes densities and fluxes of nonlinear DDEs.The software can be used to answer integrability questions and to gain insight in the physical and mathematical properties of nonlinear models. When applied to nonlinear systems with parameters, the software computes the conditions on the parameters for conservation laws to exist. The existence of a hierarchy of conservation laws is a predictor for complete integrability of the system and its solvability with the Inverse Scattering Transform.

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Section: Leading Order Analysismentioning

confidence: 98%

Section: Part Ii: Nonlinear Differential-difference Equationsmentioning

confidence: 99%

“…It is easy to show [55] that if ρ is a density then D k ρ is also a density. Hence, using an appropriate "up-shift" all negative shifts in a density can been removed.…”

Section: Nonlinear Ddes and Conservation Lawsmentioning

confidence: 99%

Section: Leading Order Analysismentioning

confidence: 99%

See 2 more Smart Citations

Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations

Hereman,

Adams,

Eklund

et al. 2008

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It was shown in [17] that if ρ is a density then D k ρ is also a density. Hence, using an appropriate "up-shift" all negative shifts in a density can be removed.…”

Section: Conservation Lawmentioning

confidence: 99%

Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential–Difference Equations

Göktaş

Hereman

2011

Dynamical Systems and Methods

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Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in Mathematica. The codes InvariantsSymmetries.m and DDERecursionOperator.m can aid researchers interested in properties of nonlinear DDEs.

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Leading order integrability conditions for differential-difference equations

Cited by 2 publications

References 17 publications

Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations

Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations

Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential–Difference Equations

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