Pavlos Xenitidis

Publications (8)1.56 Total impact

• Article: Second order integrability conditions for difference equations. An integrable equation

  Alexandre V. Mikhailov, Pavlos Xenitidis
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  ABSTRACT: Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws is discussed. In the generic case, nonlocal conservation laws are also generated. A new integrable equation satisfying the second order integrability conditions is presented and its integrability is established by the construction of symmetries, conservation laws and a 3x3 Lax representation. Finally, the relation of the symmetries of this equation to a generalized Bogoyavlensky lattice and a new integrable lattice are derived.
  05/2013;
• Source

  Available from: ArXiv

  Article: Lattice Schwarzian Boussinesq equation and two-component systems

  Pavlos Xenitidis, Frank Nijhoff
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  ABSTRACT: Various new two-component systems related to the lattice Schwarzian Boussinesq equation are constructed in a systematic way from conservation laws. Their multidimensional consistency is demonstrated, Lax pairs, symmetries and conservation laws are derived and an auto-Backlund transformation is constructed. Finally, Yang-Baxter maps from these systems are constructed.
  02/2012;
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  Available from: ArXiv

  Article: Symmetries and conservation laws of lattice Boussinesq equations

  Pavlos Xenitidis, Frank Nijhoff
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  ABSTRACT: Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations leads to corresponding hierarchies of such equations for which conservation laws and Lax pairs are constructed. Finally, using the continuous symmetry reduction approach, an integrable, multidimensionally consistent system of partial differential equations is derived in relation with the lattice modified Boussinesq system.
  12/2011;
• Article: Symmetries and conservation laws of the ABS equations and corresponding differential–difference equations of Volterra type

  Pavlos Xenitidis
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  ABSTRACT: A sequence of canonical conservation laws for all the Adler–Bobenko–Suris (ABS) equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1–H3, Q1–Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless Virasoro-type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1–H3, Q1–Q3 as differential–difference equations of Yamilov's discretization of Krichever–Novikov equation, corresponding hierarchies of master symmetries along with isospectral and nonisospectral zero curvature representations are derived for all of them.
  Journal of Physics A Mathematical and Theoretical 10/2011; 44(43):435201. · 1.56 Impact Factor
• Source

  Available from: ArXiv

  Article: Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type

  Pavlos Xenitidis
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  ABSTRACT: A sequence of canonical conservation laws for all the Adler-Bobenko-Suris equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and non-local master symmetries in each lattice direction form centerless Virasoro type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-difference equations of Yamilov's discretization of Krichever-Novikov equation, corresponding hierarchies of isospectral and non-isospectral zero curvature representations are derived for all of them.
  05/2011;
• Source

  Available from: ArXiv

  Article: Cosymmetries and Nijenhuis recursion operators for difference equations

  Alexander V. Mikhailov, Jing Ping Wang, Pavlos Xenitidis
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  ABSTRACT: In this paper we discuss the concept of cosymmetries and co--recursion operators for difference equations and present a co--recursion operator for the Viallet equation. We also discover a new type of factorisation for the recursion operators of difference equations. This factorisation enables us to give an elegant proof that the recursion operator given in arXiv:1004.5346 is indeed a recursion operator for the Viallet equation. Moreover, we show that this operator is Nijenhuis and thus generates infinitely many commuting local symmetries. This recursion operator and its factorisation into Hamiltonian and symplectic operators can be applied to Yamilov's discretisation of the Krichever-Novikov equation.
  09/2010;
• Source

  Available from: ArXiv

  Article: On the Lagrangian formulation of multidimensionally consistent systems

  Pavlos Xenitidis, Frank Nijhoff, Sarah Lobb
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  ABSTRACT: Multidimensional consistency has emerged as a key integrability property for partial difference equations (P$\Delta$Es) defined on the "space-time" lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral P$\Delta$Es possessing this property, leading to the so-called ABS list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly nontrivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding P$\Delta$E the Lagrange forms are closed, i.e. they obey a {\it closure relation}. Here we extend those results to the continuous case: it is known that associated with the integrable P$\Delta$Es there exist systems of PDEs, in fact differential equations with regard to the parameters of the lattice as independent variables, which equally possess the property of multidimensional consistency. In this paper we establish a universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrange multi-form structure for the corresponding continuous PDEs, and we show that the Lagrange forms possess the closure property. Comment: 22 pages
  08/2010;
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  Available from: ArXiv

  Article: Recursion operators, conservation laws and integrability conditions for difference equations

  Alexander V. Mikhailov, Jing Ping Wang, Pavlos Xenitidis
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  ABSTRACT: In this paper we make an attempt to give a consistent background and definitions suitable for the theory of integrable difference equations. We adapt a concept of recursion operator to difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. Similar to the case of partial differential equations these canonical densities can serve as integrability conditions for difference equations. We have found the recursion operators for the Viallet and all ABS equations.
  04/2010;