Integrable deformations of Lotka-Volterra systems from Poisson-Lie structures ´ Angel Ballesteros, Alfonso Blasco , Fabio Musso (angelb@ubu.es, ablasco@ubu.es, fmusso@ubu.es ) Physics Department, Universidad de Burgos (Spain) Abstract The quadratic Poisson structure underlying the integrability structure of 3D Lotka-Volterra (LV) equations is shown to be a particular Poisson-Lie structure on a three-dimensional group. By considering the most generic Poisson-Lie structure compatible with the coalgebra map Δ defined through the group multiplication, a two-parametric integrable deformation of the LV systems is explicitly found. Moreover, 3N-dimensional integrable systems containing deformed versions of the 3D Lotka-Volterra equations can be obtained by making use of the Poisson comultiplication map Δ. 1. Hamiltonian structure of the Lotka-Volterra equations Poisson algebra and Hamiltonian function Let us consider the quadratic Poisson algebra P : {X , Y } = αXY {X , Z } = β XZ {Y , Z } = γ YZ α, β = 0 whose Casimir function reads C = X −γ Y β Z −α . If we consider the Hamiltonian function (which generalizes [1]) H = a 1 X + a 2 Y + a 3 Z + b 1 log X + b 2 log Y + b 3 log Z we get the LV equations as the integrable dynamical system given by ˙ F = {F , H}. Namely, ˙ X = X [αa 2 Y + β a 3 Z +(αb 2 + β b 3 )] ˙ Y = Y [−αa 1 X + γ a 3 Z +(γ b 3 − αb 1 )] ˙ Z = Z [−β a 1 X − γ a 2 Y − (β b 1 + γ b 2 )] The LV Poisson algebra as a Poisson-Lie group Key observation [2]: The map Δ(X )= X ⊗ X → Δ (2) (X )= X 2 X 1 Δ(Y )= X γ β ⊗ Y + Y ⊗ 1 → Δ (2) (Y )= X γ β 2 Y 1 + Y 2 Δ(Z )= X − γ α ⊗ Z + Z ⊗ 1 → Δ (2) (Z )= X − γ α 2 Z 1 + Z 2 (where F 1 = I⊗ F , F 2 = F ⊗I ) is a Poisson map with respect to the LV Poisson algebra. Therefore, the LV equations are an integrable Hamiltonian system defined on this Poisson-Hopf algebra. Remark: The coproduct Δ is just the group law for the Lie group generated by the multiparametric Lie algebra g α,β,γ given by [x , y ]= γ β y [x , z ]= − γ α z [y , z ]= 0. Therefore, (P , Δ) is a multiparametric Poisson-Lie (PL) group. 2. The deformed LV Poisson algebra Theorem [2] The most generic Poisson structure quadratic in {X , X γ β , X − γ α , Y , Z , 1} and for which the comultiplication Δ is a Poisson map is given by the following (3+2)-parameter Poisson bracket P δ,ǫ : {X , Y } = αXY + δ X (1 − X γ β ) {X , Z } = β XZ + ǫX (1 − X − γ α ) {Y , Z } = γ YZ + γǫ β Y + γδ α Z + γδǫ βα  1 − X γ β − γ α  . Moreover, the Casimir function for P δ,ǫ is given by C δ,ǫ =  δ (1 − X γ β )+ αY  − β α  ǫ(X γ α − 1)+ β ZX γ α  . Corollary: (P δ,ǫ , Δ) is a 5-parametric PL structure on the Lie group G = exp(g α,β,γ ). A new integrable deformation of LV equations Now, from H = a 1 X + a 2 Y + a 3 Z + b 1 log X + b 2 log Y + b 3 log Z but considering that the dynamical variables generate P δ,ǫ , we get the following integrable (δ, ǫ)-deformation of the LV equations: ˙ X = X [αa 2 Y + β a 3 Z +(αb 2 + β b 3 )] +δ X  1 − X γ β   a 2 + b 2 Y  + ǫX  1 − X − γ α   a 3 + b 3 Z  ˙ Y = Y [−αa 1 X + γ a 3 Z +(−αb 1 + γ b 3 )] + δ  (X γ β − 1)(a 1 X + b 1 )+ γ α (a 3 Z + b 3 )  + ǫγ β  Y  a 3 + b 3 Z  + δ α  a 3 + b 3 Z   1 − X γ β − γ α   ˙ Z = Z [−β a 1 X − γ a 2 Y +(−γ b 2 − β b 1 )] − δγ α Z  a 2 + b 2 Y  +ǫ  (X − γ α − 1)(a 1 X + b 1 ) − γ β (a 2 Y + b 2 )  + δǫγ αβ  (X γ β − γ α − 1)  a 2 + b 2 Y  that include both polynomial and rational perturbation terms. 3. A higher dimensional integrable system containing a deformation of LV equations The existence of the coproduct map Δ allows the definition of a completely integrable 6D Hamiltonian given by [2] H (2) := Δ (2) (H)= a 1 (X 2 X 1 )+ a 2  X γ β 2 Y 1 + Y 2  + a 3  X − γ α 2 Z 1 + Z 2  + b 1 log (X 2 X 1 )+ b 2 log  X γ β 2 Y 1 + Y 2  + b 3 log  X − γ α 2 Z 1 + Z 2  . The Hamilton equations for this 6D system, when computed on P , read: ˙ X 2 = X 2 (αa 2 Y 2 + β a 3 Z 2 )+ X 2   αY 2   b 2 X γ β 2 Y 1 + Y 2   + β Z 2   b 3 X − γ α 2 Z 1 + Z 2     ˙ Y 2 = Y 2 (−αa 1 X 2 X 1 + γ a 3 Z 2 )+ γ Y 2   b 3 Z 2 X − γ α 2 Z 1 + Z 2   −αY 2   b 1 + γ β X γ β 2 Y 1   a 2 + b 2 X γ β 2 Y 1 + Y 2   − γ α X − γ α 2 Z 1   a 3 + b 3 X − γ α 2 Z 1 + Z 2     ˙ Z 2 = Z 2 (−β a 1 X 2 X 1 − γ a 2 Y 2 ) − γ Z 2   b 2 Y 2 X γ β 2 Y 1 + Y 2   −β Z 2   b 1 + γ β X γ β 2 Y 1   a 2 + b 2 X γ β 2 Y 1 + Y 2   − γ α X − γ α 2 Z 1   a 3 + b 3 X − γ α 2 Z 1 + Z 2     Under the constraint ˙ X 1 = 0 the equations for {X 2 , Y 2 , Z 2 } are just an integrable deformation of LV equations, and the same happens for the first block {X 1 , Y 1 , Z 1 }, when ˙ X 2 = 0: ˙ X 1 = X 1  α a 2 Y 1 X γ β 2 + β a 3 Z 1 X − γ α 2  + X 1   αY 1   b 2 X γ β 2 X γ β 2 Y 1 + Y 2   + β Z 1   b 3 X − γ α 2 X − γ α 2 Z 1 + Z 2     ˙ Y 1 = Y 1  −αa 1 X 1 X 2 + γ a 3 Z 1 X − γ α 2  + γ Y 1   b 3 Z 1 X − γ α 2 X − γ α 2 Z 1 + Z 2   − α b 1 Y 1 ˙ Z 1 = Z 1  −β a 1 X 1 X 2 − γ Y 1 X γ β 2  − γ Z 1   b 2 Y 1 X γ β 2 X γ β 2 Y 1 + Y 2   − β b 1 Z 1 The integrals of the motion are the Hamiltonian and the Casimir functions C 1 = X −γ 1 Y β 1 Z −α 1 C 2 = X −γ 2 Y β 2 Z −α 2 C (2) := Δ (2) (C )=(X 1 X 2 ) −γ  X γ β 2 Y 1 + Y 2  β  X − γ α 2 Z 1 + Z 2  −α A further deformation is got by using P δ,ǫ , and by making use of the N -th coproduct [3], a 3N-dimensional integrable LV system is constructed [2]. Bibliography [1] Y. Nutku, ‘Hamiltonian structure of the Lotka-Volterra equations’, Phys. Lett. A. 145, p.27 (1990). [2] A. Ballesteros, A. Blasco, F. Musso, ‘Integrable deformations of Lotka-Volterra systems’ (submitted, 2011), arXiv:1106.0805. [3] A. Ballesteros, O. Ragnisco, ‘A systematic construction of completely integrable Hamiltonians from coalgebras’, J. Phys. A: Math. Gen. 31, p. 3791 (1998).