Formal diagonalisation of Lax-Darboux schemes

We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schrodinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model, principal chiral field model as well as with differential-difference integrable systems (including the Toda lattice and differential-difference Heisenberg chain) and integrable partial difference systems. We show that there exists a transformation which formally diagonalises all elements of the Lax-Darboux scheme simultaneously. It provides us with generating functions of local conservation laws for all integrable systems obtained. We discuss the relations between conservation laws for systems belonging to the Lax-Darboux scheme.Comment: 26 page

Darboux transformation with dihedral reduction group

We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix, we obtain local conservation laws of the system

Phenomenology of B -> pi pi, pi K Decays at O(alpha^2 beta_0) in QCD Factorization

We study O(alpha^2 beta_0) perturbative corrections to matrix elements entering two-body exclusive decays of the form B -> pi pi, pi K in the QCD factorization formalism, including chirally enhanced power corrections, and discuss the effect of these corrections on direct CP asymmetries, which receive their first contribution at O(alpha). We find that the O(alpha^2 beta_0) corrections are often as large as the O(alpha) corrections. We find large uncertainties due to renormalization scale dependence as well as poor knowledge of the non-perturbative parameters. We assess the effect of the perturbative corrections on the direct CP violation parameters of B -> pi^+ pi^-.Comment: 27 pages, 5 figures. Updated input parameters and added citations; expanded discussio

Dobivanje amplitude pionske raspodjele iz mjerenja CLEO i E791

Using QCD perturbation theory in NLO and light-cone QCD sum rules, we extract from the CLEO experimental results the data on the Fγ ∗γπ (Q2) transition form factor constraints, on the Gegenbauer coefficients a2 and a4, as well as on the inverse moment (x −1)π of the pion distribution amplitude. We show that both the asymptotic and the Chernyak–Zhitnitsky pion distribution amplitudes are excluded at the 3σ- and 4σ-level, respectively, while the data confirm the end-point suppressed shape of the pion DA that we previously obtained with QCD sum rules and nonlocal condensates. These findings are also supported by the data of the Fermilab E791 experiment on diffractive di-jet production.Primjenom teorije smetnje QCD u blizu-vodećem redu i zbrojnih pravila QCD, iz podataka mjerenja CLEO o prijelaznom faktoru oblika za Fγ ∗γπ (Q2) , izvodimo ograničenja na Gegenbauerove koeficijente a2 i a4, kao i na inverzni moment amplitude pionske raspodjele (x −1)π. Pokazujemo da su asimptotska i Chernyak– Zhitnitskyjeva amplituda pionske raspodjele isključene na razini 3σ odn. 4σ, dok podaci potvrđuju potisnuti oblik kraja pionske distribucijske amplitude izvedene zbrojnim pravilima QCD i nelokalnim kondenzatima. Naše zaključke podržavaju također mjerenja difraktivne tvorbe dvojnih mlazova E791 u Fermilabu

Dobivanje amplitude pionske raspodjele iz mjerenja CLEO i E791

Using QCD perturbation theory in NLO and light-cone QCD sum rules, we extract from the CLEO experimental results the data on the Fγ ∗γπ (Q2) transition form factor constraints, on the Gegenbauer coefficients a2 and a4, as well as on the inverse moment (x −1)π of the pion distribution amplitude. We show that both the asymptotic and the Chernyak–Zhitnitsky pion distribution amplitudes are excluded at the 3σ- and 4σ-level, respectively, while the data confirm the end-point suppressed shape of the pion DA that we previously obtained with QCD sum rules and nonlocal condensates. These findings are also supported by the data of the Fermilab E791 experiment on diffractive di-jet production.Primjenom teorije smetnje QCD u blizu-vodećem redu i zbrojnih pravila QCD, iz podataka mjerenja CLEO o prijelaznom faktoru oblika za Fγ ∗γπ (Q2) , izvodimo ograničenja na Gegenbauerove koeficijente a2 i a4, kao i na inverzni moment amplitude pionske raspodjele (x −1)π. Pokazujemo da su asimptotska i Chernyak– Zhitnitskyjeva amplituda pionske raspodjele isključene na razini 3σ odn. 4σ, dok podaci potvrđuju potisnuti oblik kraja pionske distribucijske amplitude izvedene zbrojnim pravilima QCD i nelokalnim kondenzatima. Naše zaključke podržavaju također mjerenja difraktivne tvorbe dvojnih mlazova E791 u Fermilabu

Symbolic representation and classification of integrable systems

This is a review paper of recent results in the perturbative symmetry approach in the symbolic representation

Cosymmetries and Nijenhuis recursion operators for difference equations

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