By Henrik Aratyn, Johan Van de Leur (auth.), Ludwig Faddeev, Pierre Van Moerbeke, Franklin Lambert (eds.)

The CKP hierarchy and the WDVV prepotential; H. Aratyn, J. van de Leur.- Quantum invariance teams of particle algebras; M. Arik.- Algebraic Hirota maps; C. Athorne.- Boundary states in Susy Sine-Gordon version; Z. Bajnok et al.- Geometry of discrete integrability. The consistency technique; A.I. Bobenko.- Homoclinic orbits and dressing procedure; E.V. Doktorov, V.M. Rothos.- Riemann-Hilbert challenge and algebraic curves; V. Enolskii, T. Grava.- Analytic and algebraic facets of Toda box theories and their genuine Hamiltonian types; V.S. Gerdjikov.- Bilinear avatars of the discrete Painlevé II equation; B. Grammaticos et al.- Orthogonal polynomials pleasant Q-difference equations; L. Haine.- Discretization of coupled soliton equations; R. Hirota.- An adelic W-algebra and rank one bispectral operators; E. Horozov.- Toroidal Lie algebra and bilinear identification of the self-dual Yang-Mills hierarchy; S. Kakei.- From soliton equations to their 0 curvature formula; F. Lambert, J. Springael.- Covariant sorts of Lax one-field operators: from Abelian to non-commutative; S. Leble.- at the Dirichlet boundary challenge and Hirota equations; A. Marshakov, A. Zabrodin.- Functional-difference deformations of Darboux-Pöshl-Teller potentials; V.B. Matveev.- Maxwell equations for quantum space-time; R.M. Mir-Kasmov.- A solvable version of interacting photons; J. Naudts.- Discretization of a Sine-Gordon variety equation; Y. Ohta.- Hierarchy of quantum explicitly solvable and integrable versions; A.K. Pogrebkov.- A two-parameter elliptic extension of the lattice KDV procedure; S.E. Puttock, F.W. Nijhoff.- traveling waves in a per-turbed discrete Sine-Gordon equation; V.M. Rothos, M. Feckan.- Quantum VS classical Calogero-Moser structures; R. Sasaki.- Geometrical dynamics of an integrable piecewise-linear mapping; D. Takahashi, M. Iwao.- unfastened bosons and dispersionless restrict of Hirota tau-function; L.A. Takhtajan.- Similarity rate reductions of Hirota bilinear equations and Painlevé equations; K.M. Tamizhmani et al.- On primary cycle of periodic Box-Ball platforms; T. Tokihiro.- Combinatorics and integrable geometry; P. van Moerbeke.- On rate reductions of a few KDV-type platforms and their hyperlink to the quartic Hénon-Heiles Hamiltonian; C. Verhoeven et al.- at the bilinear different types of Painlevé’s 4th equation; R. Willox, J. Hietarinta.

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**Extra resources for Bilinear Integrable Systems: From Classical to Quantum, Continuous to Discrete**

**Example text**

Thus we obtain the decompositions: ⊗ (0,0,2) ⊗ (0,0,3) ⊗ (0,0,1) (0,0,1) (0,0,2) (0,0,3) = = = ⊕ (0,0,4) ⊕ (0,0,6) ⊕ (0,0,2) (0,1,0) ⊕ (0,1,4) ⊕ (0,1,2) (0,2,0) (0,2,2) ⊕ Diagrammatically, ⊗ ⊗ ⊗ ⊕ = ⊕ = ⊕ ⊕ = ⊕ ⊕ (0,3,0) Algebraic Hirota Maps 25 and for this choice of Hirota map, D34 , the effect is seen to be to add a single box to each of the top two rows because the tableau associated with δ = (0, 1, 0) is 6 THE CLASSICAL HIROTA DERIVATIVE We will consider only the cases of sl2 (C) and sl3 (C) in this section since these are the cases of direct relevance to the classical Hirota derivative and will confine ourselves to some remarks concerning the general case.

The mapping L(z 2 , z 1 ; α, λ) is associated to the oriented edge (z 1 , z 2 ). Going from ψ1 to ψ3 in two different ways and using the arbitrariness Geometry of Discrete Integrability 49 of ψ1 we get L(z 3 , z 2 ; β, λ)L(z 2 , z 1 ; α, λ) = L(z 3 , z 4 ; α, λ)L(z 4 , z 1 ; β, λ). (5) Using the matrix representation of M¨obius transformations az + b = L[z], cz + d where L= a c b , d and normalizing the matrices (for example by the condition det L = 1) we arrive at the zero curvature representation (5).

65–70. 3. Fulton, W. and Harris, J. (1991) Representation theory. A first course, Graduate Texts in Mathematics, Vol. 129, Readings in Mathematics, Springer-Verlag, New York. 4. Hilbert, D. (1993) Theory of Algebraic Invariants, CUP. 5. Hirota, R. (1982) Bilinearization of soliton equations, J. Phys. Soc. Japan 51, pp. 323–331. Algebraic Hirota Maps 33 6. Athorne, C. (2001) Hirota derivatives and representation theory in Integrable systems: linear and nonlinear dynamics (Islay, 1999), Glasg. Math.