Reductions and New types of Integrable models in two and more dimensions

Transcription

1 0-0 XIV International conference Geometry Integrabiliy and Quantization Varna, 8-13 June 2012, Bulgaria Reductions and New types of Integrable models in two and more dimensions V. S. Gerdjikov Institute for Nuclear Research and Nuclear Energy Sofia, Bulgaria

2 0-1 Based on: V. S. Gerdjikov. Z 2 -reductions of spinor models in two dimensions. Submitted to Physics of Atomic Nuclei 2012) V. S. Gerdjikov. Riemann-Hilbert Problems with canonical normalization and families of commuting operators. Submitted to Pliska Stud. Math. Bulgar. arxiv: v1 [nlin.si]. PLAN Reductions of spinor models RHP with canonical normalization Reductions of polynomial bundles New N-wave equations k = 2) in 2 and more dimensions Conclusions and open questions

3 0-2 1 Integrable 2-dimensional spinor models The integrability of the 2-dimensional versions of the Nambu Jona-Lasinio Vaks Larkin NJLVL), Gross Neveu model GN) and Zakharov and Mikhailov see Zakharov and Mikhailov 1981). NJLVL models are related to sun) algebras, Gross Neveu models to spn) and Zakharov Mikhailov ZM) models to son). Lax representations of these models: Ψ ξ = Uξ, η, λ)ψξ, η, λ), Ψ η = Uξ, η, λ)ψξ, η, λ), Uξ, η, λ) = U 1ξ, η) λ a, V ξ, η, λ) = V 1ξ, η) λ + a, where η = t + x, ξ = t x and a is a real number. We also impose the Z 2 -reduction: U x, t, λ) = Ux, t, λ ), V x, t, λ) = V x, t, λ ).

4 0-3 The compatibility condition of the above linear problems reads: which is equivalent to U η V ξ + [U, V ] = 0, U 1,η + 1 2a [U 1, V 1 ξ, η)] = 0, V 1,ξ 1 2a [V 1, U 1 ξ, η)] = 0. i) Nambu-Jona-Lasinio-Vaks-Larkin models. Here we choose g sun). Then ψξ, η) and ϕξ, η) are elements of the group SUN) and by definition ˆψξ, η) = ψ ξ, η), ˆϕξ, η) = ϕ ξ, η). Next we choose J = diag 1, 0,..., 0) and as a result only the first columns ϕ 1), ψ 1) and the first rows ˆϕ 1), ˆψ 1) enter into the systems. If we introduce the notations: ϕ α ξ, η) = ϕ 1) α,1, ψ αξ, η) = ψ 1) α,1,

5 0-4 then the explicit form of the system is: ϕ α η = i 2a ψ α ψ α ξ = i 2a ϕ α N ψβϕ β, β=1 N ϕ βψ β. β=1 The functional of the action is: N ) A NJLVL = dx dt i ϕ ϕ α α η + ψ α ψ α ξ α=1 1 2a N 2 ψαϕ α ). ii) Gross-Nevew models. Here we choose g sp2n, R); then ψξ, η) and ϕξ, η) are elements of the group G SP 2N, R). Following [?] we use the standard definition of symplectic group elements: α=1 ˆψξ, η) = Jψ T ξ, η)ĵ, ˆϕξ, η) = Jϕ T ξ, η)ĵ, J = ).

6 0-5 Then the corresponding Lie algebraic elements acquire the following block-matrix structure: ) A B U 1 ξ, η) = C A T, where A, B, C are arbitrary real N N matrices. Next we choose ) 0 B0 J =, B = diag 1, 0,..., 0, 0). As a consequence again only the first columns ϕ 1), ψ 1) and the first rows ˆϕ 1), ˆψ1) enter into the systems. If we introduce the N-component complex vectors: ϕ α ξ, η) = 1 2 ϕ1) α,1 + iϕ1) N+α,1 ), ψ αξ, η) = 1 2 ψ1) α,1 + iψ1) N+α,1 )

7 0-6 then the explicit form of the system is: ϕ α η = i a ψ α ψ α ξ = i a ϕ α N ψ β ϕ β ψβϕ β ), β=1 N ϕ β ψβ ϕ βψ β ). β=1 The functional of the action is: N ) A GN = dx dt i ϕ ϕ α α η + ψ α ψ α ξ α=1 1 2a N ) 2 ψαϕ α ϕ αψ α ). iii) Zakharov Mikhailov models. Now we choose g son, R); then ψξ, η) and ϕξ, η) are elements of the group G SON, R). Following [?] we use the standard definition of orthogonal group elements: α=1 ˆψξ, η) = ψ T ξ, η), ˆϕξ, η) = ϕ T ξ, η).

8 0-7 Now we choose J = E 1,N E N,1, where the N N matrices E kp are defined by E kp ) nm = δ kn δ pm. As a consequence now the first and the last columns ϕ 1), ϕ N), ψ 1), ψ N) and the first and the last rows ˆϕ 1), ˆϕ N), ˆψ 1), ˆψ N) enter into the systems. If we introduce the N-component complex vectors: ϕ α ξ, η) = 1 2 ϕ1) α,1 + iϕn) α,n ), ψ αξ, η) = 1 2 ψ1) α,1 + iψn) α,n ) then the explicit form of the system becomes: i ψ α ξ = i a i ϕ α η = i a N ϕ αϕ β ψ β ϕ α ϕ β)ψ β ), β=1 N ψαψ β ϕ β ψ α ψβ)ϕ β, β=1

9 0-8 The functional of the action is: N A ZM = dx dt i 1 2a N α,β=1 α=1 ) ϕ ϕ α α η + ψ α ψ α ξ ϕ αϕ β ϕ βϕ α )ψαψ β ψβψ α ).

10 0-9 2 Z 2 -Reductions of the spinor models Apply the idea of the reduction group Mikhailov 1980) and obtain new types of spinor models generalizing the previous ones. Start with the Lax representation: Ψ ξ = U R ξ, η, λ)ψξ, η, λ), U R ξ, η, λ) = U 1ξ, η) λ a + CU 1ξ, η)c 1 ϵλ 1 a Ψ η = V R ξ, η, λ)ψξ, η, λ),, V R ξ, η, λ) = V 1ξ, η) λ + a where ϵ = ±1, a 1 is a real number and C is an involutive automorphism of g. It satisfy also: + CV 1ξ, η)c 1 ϵλ 1 + a U R ξ, η, λ) = CU R ξ, η, ϵλ 1 )C 1, V R ξ, η, λ) = CV R ξ, η, ϵλ 1 )C 1, The new Lax representation is:, which is equivalent to U R η V R ξ + [U R, V R ] = 0, U 1,η + [U 1, V R ξ, η, a)] = 0, V 1,ξ + [V 1, U R ξ, η, a)] = 0.

11 0-10 In the same way as above we get: i) Z 2 -NJLVL models. Here G SUN) and the system takes the form: i ϕ η + 1 2a ψ ψ ϕ) + 1 ϵa 1 + a C ψ ψ Ĉ ϕ)ξ, η) = 0, i ψ ξ + 1 2a ϕ ϕ ψ) + 1 ϵa 1 + a C ϕ ϕ Ĉ ψ)ξ, η) = 0. where ψ = ψ α,1,..., ψ α,n ) T and ϕ = ϕ α,1,..., ϕ α,n ) T. For the automorphism C of the SUN) group we may have a) C N = diag ϵ 1, ϵ 2,..., ϵ N ), ϵ j = ±1, b) C N = C N 1 ). where C N 1 belongs to the Weyl group of SUN 1) and is such that CN 1 2 = 11. These two special choices of C are such that lim ξ ± U R ξ, η) = lim ξ ± CU R ξ, η)ĉ.

12 0-11 ii) Z 2 -GN models. are given by: a) C = Here G SP 2N, R). Two typical choices of C C1 0 0 C 1 ), b) C = 0 C2 C 2 0 ), where C 2 1 = C 2 2 = 11. ϕ η = i ψ a ψ, ϕ) ϕ, ψ) ) ψ ξ = i a ϕ ψ, ϕ) ϕ, ψ) ) The corresponding action can be written as follows: A Z2,GNa = dx dt + 2i a + ϵa 1 C 1ψ ψ ) C 1ϕ) ϕ C 1ψ), 2i a + ϵa 1 C 1ϕ ψ ) C 1ϕ) ϕ C 1ψ). i ϕ ϕ η + ψ ψ ) 1 ψ ξ 2a, ϕ) ϕ, ψ) ) 2 1 ϵa 1 ψ + a ) ) 2 C 1ϕ) ϕ C 1ψ).

13 0-12 The second Z 2 -reduced GN-system is: ϕ η = i ψ a ψ, ϕ) ϕ, ψ) ) ψ ξ = i a ϕ ψ, ϕ) ϕ, ψ) ) These equations can be obtained from the action: A Z2,GNb = + + 2i a + ϵa 1 C 2ψ ψ ) T C 2ϕ) + ψ C 2ϕ ), 2i a + ϵa 1 C 2 ϕ ϕ T C 2 ψ) + ϕ C 2 ψ ) dx dt i ϕ ϕ η + ψ ψ ) 1 ψ ξ 2a, ϕ) ϕ, ψ) ) 2 1 ϵa 1 ϕ + a C 2ψ ) + ϕ ) ) 2 T C 2ψ). iii) Z 2 -ZM models. Here G SON, R). Again we used N-component ).

14 0-13 vectors to cast the Z 2 -reduced ZM systems in the form: ψ ξ = i a ϕ η = i a ϕ ϕ T, ψ) ϕ ϕ, ψ)) ψ ψ T, ϕ) ψ ψ, ϕ)) + + 2i a + ϵa 1 C ϕ ϕ T Cψ) ϕ ϕ Cψ) ), 2i a + ϵa 1 C ψ ψ T Ĉ ϕ) ψ ψ Ĉ ϕ) ), where the involutive automorphism C can be chosen as one of the type: a) C = diag ϵ 1, ϵ 2,..., ϵ 2, ϵ 1 ), ϵ j = ±1, b) C 1 0 = 0 C 3 ), with C 2 3 = 11. For these choices of C we have lim ξ ± U R ξ, η) = lim ξ ± CU R ξ, η)ĉ.

15 0-14 The action for the reduced ZM models is provided by: A Z2,ZM = + 1 a 2 + ϵa 1 + a dx dt i ϕ ϕ η + ψ ψ ) ξ ψ, ϕ ) ϕ T, ψ) ϕ, ψ) ψ, ϕ) ) ψ C ϕ ) ϕ T C ψ) ϕ C ψ) ψ C ϕ)) ). 3 RHP with canonical normalization ξ + x, t, λ) = ξ x, t, λ)g x, t, λ), λ k R, lim λ ξ+ x, t, λ) = 11, ξ ± x, t, λ) G Consider particular type of dependence G x, t, λ): i G x s λ k [J s, G x, t, λ)] = 0, i G t λk [K, G x, t, λ)] = 0.

16 0-15 where J s h g. The canonical normalization of the RHP: ξ ± x, t, λ) = exp Q x, t, λ), Q x, t, λ) = Q k x, t)λ k. k=1 where all Q k x, t) g. However, J s x, t, λ) = ξ ± x, t, λ)j s ˆξ± x, t, λ), K x, t, λ) = ξ ± x, t, λ)k ˆξ ± x, t, λ), belong to the algebra g for any J and K from g. If in addition K also belongs to the Cartan subalgebra h, then Zakharov-Shabat theorem [J s x, t, λ), K x, t, λ)] = 0. Theorem 1. Let ξ ± x, t, λ) be solutions to the RHP whose sewing function depends on the auxiliary variables x and t as above. Then ξ ± x, t, λ)

17 0-16 are fundamental solutions of the following set of differential operators: L s ξ ± i ξ± x s + U s x, t, λ)ξ ± x, t, λ) λ k [J s, ξ ± x, t, λ)] = 0, Mξ ± i ξ± t Proof. Introduce the functions: and prove that + V x, t, λ)ξ ± x, t, λ) λ k [K, ξ ± x, t, λ)] = 0. g ± s x, t, λ) = i ξ± x s ˆξ± x, t, λ) + λ k ξ ± x, t, λ)j s ˆξ± x, t, λ), g ± x, t, λ) = i ξ± t ˆξ ± x, t, λ) + λ k ξ ± x, t, λ)k ˆξ ± x, t, λ), g + s x, t, λ) = g s x, t, λ), g + x, t, λ) = g x, t, λ), which means that these functions are analytic functions of λ in the whole complex λ-plane. Next we find that: lim λ g+ s x, t, λ) = λ k J s, lim λ g+ x, t, λ) = λ k K.

18 0-17 and make use of Liouville theorem to get g + s x, t, λ) = g s x, t, λ) = λ k J s g + x, t, λ) = g x, t, λ) = λ k K k U s;l x, t)λ k l, l=1 k V l x, t)λ k l. l=1 We shall see below that the coefficients U s;l x, t) and V l x, t) can be expressed in terms of the asymptotic coefficients Q s. Lemma 1. The set of operators L s and M commute, i.e. the following set of equations hold: i U s x j i U j x s + [U s x, t, λ) λ k J s, U j x, t, λ) λ k J j ] = 0, i U s t i V x s + [U s x, t, λ) λ k J s, V x, t, λ) λ k K] = 0.

19 0-18 where U s x, t, λ) = k U s;l x, t)λ k l, V x, t, λ) = l=1 k V l x, t)λ k l. l=0 Proof. The set of the operators L s and M have a common FAS, i.e. they all must commute. 4 Jets of order k Consider the jets of order k of J x, λ) and Kx, λ): ) J s x, t, λ) λ k ξ ± x, t, λ)j ˆξ± l x, t, λ) = + λk J s U s x, t, λ), K x, t, λ) λ k ξ ± x, t, λ)k ˆξ ) ± x, t, λ) = + λk K V x, t, λ).

20 0-19 Express U s x) g in terms of Q s x): J s x, t, λ) = J s + k=1 1 k! ad k QJ s, K x, t, λ) = K + k=1 1 k! ad k QK, and therefore for U s;l we get: U s;1 x, t) = ad Q1 J s, U s;2 x, t) = ad Q2 J s 1 2 ad 2 Q 1 J s U s;3 x, t) = ad Q3 J s 1 2 ad Q 2 ad Q1 + ad Q1 ad Q2 ) J s 1 6 ad 3 Q 1 J s. U s;k x, t) = ad Qk J s s+p+r=k s+p=k ad Qs ad Qp J s ad Qs ad Qp ad Qr J s 1 k! ad k Q 1 J s, and similar expressions for V l x, t) with J s replaced by K.

21 Reductions of polynomial bundles a) Aξ +, x, t, ϵλ )Â = ˆξ x, t, λ), AQ x, t, ϵλ )Â = Qx, t, λ), b) Bξ +, x, t, ϵλ ) ˆB = ξ x, t, λ), BQ x, t, ϵλ ) ˆB = Qx, t, λ), c) Cξ +,T x, t, λ)ĉ = ˆξ x, t, λ), CQ x, t, λ)ĉ = Qx, t, λ), where ϵ 2 = 1 and A, B and C are elements of the group G such that A 2 = B 2 = C 2 = 11. As for the Z N -reductions we may have: Dξ ± x, t, ωλ) ˆD = ξ ± x, t, λ), DQx, t, ωλ) ˆD = Qx, t, λ), where ω N = 1 and D N = 11.

22 On N-wave equations k = 1) in 2 and more dimensions Lax representation involves two Lax operators linear in λ: Lξ ± i ξ± x + [J, Qx, t)]ξ± x, t, λ) λ[j, ξ ± x, t, λ)] = 0, Mξ ± i ξ± t + [K, Qx, t)]ξ ± x, t, λ) λ[k, ξ ± x, t, λ)] = 0. The corresponding equations take the form: [ i J, Q ] [ i K, Q ] [[J, Q], [K, Qx, t)]] = 0 t x Qx, t) = 0 u 1 u 3 J = diag a v 1 0 u 2 1, a 2, a 3 ),, v 3 v 2 0 K = diag b 1, b 2, b 3 ),

23 0-22 Then the 3-wave equations take the form: where u 1 t a 1 a 2 u 1 b 1 b 2 x + κϵ 1ϵ 2 u 2u 3 = 0, u 2 t a 2 a 3 u 2 b 2 b 3 x + κϵ 1u 1u 3 = 0, u 3 t a 1 a 3 u 3 b 1 b 3 x + κϵ 2u 1u 2 = 0, κ = a 1 b 2 b 3 ) a 2 b 1 b 3 ) + a 3 b 1 b 2 ). For 3-dimensional space-time we consider Q as above, but now let u j and v j be functions of x 1 = x, x 2 = y and t. Let also J 1 = J and J 2 = I = diag c 1, c 2.c 3 ). Now the corresponding solution of the RHP ξ ± x, y, t, λ) will be FAS not only of L and M above, but also of P ξ ± i ξ± y + [I, Qx, t)]ξ+ x, t, λ) λ[i, ξ + x, t, λ)] = 0,

24 0-23 and all these three operators will mutually commute, i.e. along with [L, M] = 0 we will have also [L, P ] = 0 and [P, M] = 0. As a result Qx, y, t) will satisfy two more 3-wave NLEE 2 u 1 t a 1 a 2 b 1 b 2 u 1 x a 1 a 2 c 1 c 2 u 1 y + κ 1 + κ 2 )ϵ 1 ϵ 2 u 2u 3 = 0, 2 u 2 t a 1 a 3 b 1 b 3 u 2 x a 1 a 3 c 1 c 3 u 2 y + κ 1 + κ 2 )ϵ 1 u 1u 3 = 0, 2 u 3 t a 2 a 3 b 2 b 3 u 3 x a 2 a 3 c 2 c 3 u 3 y + κ 1 + κ 2 )ϵ 2 u 1u 2 = 0. κ 1 = a 1 b 2 b 3 ) a 2 b 1 b 3 ) + a 3 b 1 b 2 ), κ 2 = a 1 c 2 c 3 ) a 2 c 1 c 3 ) + a 3 c 1 c 2 ). For N-wave equations related to Lie algebras g of higher rank r we can add up to r auxiliary variables: r Q r t ad 1 J s ad J ) Q i x s s=1 r s=1 ad 1 J s [[J, Q], [J s, Q x, t)]] = 0

25 0-24 where Q is an n n off-diagonal matrix depending on r + 1 variables. We remind that if J = diag a 1,..., a n ) then ad J Q) jk [J, Q]) jk = a j a k )Q jk, and similarly for the other J s. ad 1 J Q) jk = 1 a j a k Q jk, 7 New N-wave equations k = 2) in 2 and more dimensions Let g = sl3) and 0 u 1 u 3 Q 1 x, t) = v 1 0 u 2, Q 2 x, t) = v 3 v 2 0 q 11 w 1 w 3 z 1 q 22 w 2 z 3 z 2 q 33,

26 0-25 Fix up k = 2. Then the Lax pair becomes Lξ ± i ξ± x + Ux, t, λ)ξ± x, t, λ) λ 2 ]J, ξ ± x, t, λ)] = 0, Mξ ± i ξ± t + V x, t, λ)ξ ± x, t, λ) λ 2 ]K, ξ ± x, t, λ)] = 0, where U U 2 + λu 1 = [J, Q 2 x)] 1 ) 2 [[J, Q 1], Q 1 x)] + λ[j, Q 1 ], V V 2 + λv 1 = [K, Q 2 x)] 1 ) 2 [[K, Q 1], Q 1 x)] + λ[k, Q 1 ]. Impose a Z 2 -reduction of type a) with A = diag 1, ϵ, 1), ϵ 2 = 1. Thus Q 1 and Q 2 get reduced into: 0 u w 3 Q 1 = ϵu 1 0 u 2, Q 2 = 0 0 0, 0 ϵu 2 0 w3 0 0

27 0-26 New type of integrable 3-wave equations: ia 1 a 2 ) u 1 t ib 1 b 2 ) u 1 x + ϵκu 2u 3 + ϵ κa 1 a 2 ) a 1 a 3 ) u 1 u 2 2 = 0, ia 2 a 3 ) u 2 t ib 2 b 3 ) u 2 x + ϵκu 1u 3 ϵ κa 2 a 3 ) a 1 a 3 ) u 1 2 u 2 = 0, ia 1 a 3 ) u 3 t ib 1 b 3 ) u 3 x iκ u 1 u 2 ) a 1 a 3 x a1 a 2 + ϵκ u a ) 2 a 3 u 2 2 u 1 u 2 + ϵκu 3 u 1 2 u 2 2 ) = 0, a 1 a 3 a 1 a 3 where: u 3 = w 3 + 2a 2 a 1 a 3 2a 1 a 3 ) u 1 u 2. The diagonal terms in the Lax representation are λ-independent.

28 0-27 Two of them read: ia 1 a 2 ) u 1 2 t ia 2 a 3 ) u 2 2 t ib 1 b 2 ) u 1 2 x ϵκu 1u 2 u 3 u 1u 2u 3 ) = 0, ib 2 b 3 ) u 2 2 x ϵκu 1u 2 u 3 u 1u 2u 3 ) = 0, These relations are satisfied identically as a consequence of the NLEE. Let the sewing function G of the RHP depends on 3 variables: t, x 1 = x and x 2 = y with J 1 = J and J 2 = I = diag c 1, c 2, c 3 ). For k = 2 we obtain: L, M and P ξ ± i ξ± y + W x, y, t, λ)ξ± x, y, t, λ) λ 2 ]I, ξ ± x, y, t, λ)] = 0, W W 2 + λw 1 = [I, Q 2 x, y, t)] 1 ) 2 [[I, Q 1], Q 1 x, y, t)] + λ[i, Q 1 x, y, t)],

29 0-28 commuting identically with respect to λ. It is obvious that [L, P ] = 0 if ia 1 a 2 ) u 1 t ic 1 c 2 ) u 1 y + ϵκ 2u 2u 3 + ϵ κ 2a 1 a 2 ) a 1 a 3 ) u 1 u 2 2 = 0, ia 2 a 3 ) u 2 t ic 2 c 3 ) u 2 y + ϵκ 2u 1u 3 ϵ κ 2a 2 a 3 ) a 1 a 3 ) u 1 2 u 2 = 0, ia 1 a 3 ) u 3 t ic 1 c 3 ) u 3 y iκ 2 u 1 u 2 ) a 1 a 3 y a1 a 2 + ϵκ 2 u a ) 2 a 3 u 2 2 u 1 u 2 + ϵκ 2 u 3 u 1 2 u 2 2 ) = 0, a 1 a 3 a 1 a 3

30 0-29 2i u 1 u t i v 1) )u 1 + ϵκ 1 + κ 2 ) 2 u 3 2i u 2 t i v 2) )u 2 + ϵκ 1 + κ 2 ) + u 1 u 2 2 ) = 0, a 1 a 2 a 1 a 3 ) u 1 u 3 u 2 u 1 2 ) = 0, a 1 a 3 a 1 a 3 ) 2i u 3 t i v 3) )u 3 i κ )u 1u 2 ) a 1 a 3 ) 2 + ϵκ 1 + κ 2 ) a 1 a 3 u 1 2 u 2 2 )u 3 + ϵκ 1 + κ 2 ) a 1 a 3 ) 2 a1 a 2 ) u a 2 a 3 ) u 2 2) u 1 u 2 = 0. Here = x, y ) T, the characteristic velocities v j), j = 1, 2, 3 and κ are two-component vectors given by: ) ) 1 b1 b v 1) = 2 1 b2 b, v a 1 a 2 c 1 c 2) = 3, 2 a 2 a 3 c 2 c 3 ) ) 1 b1 b v 3) = 3 κ1, κ =, a 1 a 3 c 1 c 3 κ 2 and κ 1 = κ.

31 Conclusions and open questions We constructed new integrable spinor models and new integrable 3- and N-wave equations. These new NLEE must be Hamiltonian. The Poisson brackets for polynomial bundles see Kulish, Reyman and Semenov-Tyan- Shanskii ); The method allows one also to apply Zakharov-Shabat dressing method for constructing their explicit N-soliton) solutions T. Valchev. This approach improves Gel fand-dickey approach.