Darboux transformations and recursion operators for differential-difference equations

Abstract.We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.

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“…Summarising the computations above we get a dynamical system of the form 11) which is reduced to the well-known Toda lattice 12) where…”

Section: Laplace Cascade For the Linear Hyperbolic Type Equationsmentioning

confidence: 99%

On a method for constructing the Lax pairs for nonlinear integrable equations

Habibullin

Khakimova

Poptsova³

2015

J. Phys. A: Math. Theor.

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“…The theory of variational principle (least action) for DDEs was introduced in for the most general case, namely, for a Lagrangian defined on dependent variables

u \in {double-struckR}^{q}

and finitely many of their derivatives and shifts, with multidimensional differential and difference variables

x \in {double-struckR}^{p_{1}}

and

n \in {double-struckZ}^{p_{2}}

playing as independent variables. The derivations and applications (e.g., as an integrability criterion or for conducting reductions) of symmetries and conservation laws for DDEs have also been deeply investigated during the last few decades, see, for instance, for symmetry analysis, for derivation of conservation laws, and for integrability method.…”

Section: Noether's Theorem For Differential‐difference Equationsmentioning

confidence: 99%

“…In the next running example, the Volterra equation, we show how symmetries can be calculated from the linearized symmetry condition. Example Let us consider Lie point symmetries for the Volterra equation (e.g., )

u^{'} - u false(u_{1} - u_{- 1} false) = 0,

which is one of the most simple integrable Volterra type of equations of the form

u^{'} = f false(u_{- 1}, u, u_{1} false) .

Equation has also been called the KvM lattice (e.g., ). Here,

t \in R

is the differential independent variable, while

n \in Z

is the difference independent variable.…”

Section: Noether's Theorem For Differential‐difference Equationsmentioning

confidence: 99%

Symmetries, Conservation Laws, and Noether's Theorem for Differential‐Difference Equations

Peng

2017

Stud Appl Math

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This paper mainly contributes to the extension of Noether's theorem to differential‐difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self‐adjointness method is adapted to the computation of conservation laws for differential‐difference equations. Several differential‐difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.

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“…Namely, given a Lax representation for a partial differential equation, we can systematically construct Darboux transformations whose Bianchi permutability condition leads to an integrable difference equation, while the corresponding Bäcklund transformations are (often nonlocal) symmetries of these difference equations and are integrable differential-difference equations in their own right (see, for instance, [1][2][3][4]). …”

Section: Introductionmentioning

confidence: 99%

“…System (6) is known, and it was found by Adler [20] in his classification of isotropic integrable Volterra-type lattices on the sphere with generalised symmetries. In this paper, we equip this system with a Lax-Darboux representation and connect it to the vector sine-Gordon equation (1) and its Bäcklund chains (3) and (4). The Bianchi permutability condition for two Darboux transformations with distinct parameters μ = ±ν resulting in two shift operators S ν , S μ leads to the integrable discrete equation…”

Section: Introductionmentioning

confidence: 99%

Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere

2016

Self Cite

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We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations, we derive new vector Yang-Baxter map and integrable discrete vector sineGordon equation on a sphere.Mathematics Subject Classification. 35Q51, 37K10, 37K35, 39A14.

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Darboux transformations and recursion operators for differential-difference equations

Cited by 73 publications

References 83 publications

On a method for constructing the Lax pairs for nonlinear integrable equations

On a method for constructing the Lax pairs for nonlinear integrable equations

Symmetries, Conservation Laws, and Noether's Theorem for Differential‐Difference Equations

Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere

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