Symmetry Reductions of Integrable Lattice Equations

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1 Isaac Newton Institute for Mathematical Sciences Discrete Integrable Systems Symmetry Reductions of Integrable Lattice Equations Pavlos Xenitidis University of Patras Greece March 11, 2009 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

2 Sketch of the presentation Analysis ➀ Class of lattice equations possessing the Klein symmetry ➁ Generalized symmetries Adler Bobenko Suris equations & Extended generalized symmetries ➊ Master symmetries Hierarchies of generalized symmetries ➋ Continuous Symmetry Reductions ➌ Relations to Soliton solutions Implementation The discrete potential Korteweg de Vries equation (H1) Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

3 General characteristics Q(u n,m, u n+1,m, u n,m+1, u n+1,m+1 ) = 0 (n, m + 1) (n + 1, m + 1) ❶ Autonomous lattice equations ❷ Affine linear w.r.t. the values of u ❸ Klein symmetry (n, m) (n + 1, m) An elementary quadrilateral on the lattice Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

4 Equation Q V (Viallet) a 1u n,mu n+1,mu n,m+1u n+1,m+1 +a 2(u n,mu n+1,mu n,m+1 +u n+1,mu n,m+1u n+1,m+1 +u n,m+1u n+1,m+1u n,m +u n+1,m+1u n,mu n+1,m) +a 3(u n,mu n+1,m+1 + u n+1,mu n,m+1) + a 4(u n,mu n+1,m + u n,m+1u n+1,m+1) +a 5(u n,mu n,m+1 + u n+1,mu n+1,m+1) + a 6(u n,m + u n+1,m + u n,m+1 + u n+1,m+1) + a 7 = 0 Viallet C. (2009) Integrable Lattice Maps: Q V, a Rational Version of Q 4 Glasgow Math. J. 51 Adler V.E., Bobenko A.I., Suris Y.B. (2003) Classification of integrable equations on quad graphs. The consistency approach Commun. Math. Phys. 233 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

5 Some important polynomials Q(u, x, y, z) = 0 y h 1 (y, z) z ❶ The horizontal polynomials h 1 h 1(u, x) = QQ,yz Q,yQ,z h 2 (u, y) h 2 (x, z) G(u, z) G(x, y) h 1(y, z) = QQ,ux Q,uQ,x ❷ The vertical polynomials h 2 h 2(u, y) = QQ,xz Q,xQ,z u h 1 (u, x) x The polynomials on the edges & the diagonals h 2(x, z) = QQ,uy Q,uQ,y ❸ The diagonal polynomials G G(u, z) = QQ,xy Q,xQ,y G(x, y) = QQ,uz Q,uQ,z Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

6 Symmetry analysis Three point generalized symmetries Every equation in this class admits a pair of symmetries with generators the vector fields h1(u n,m, v n [0] = R [0] u n+1,m) n un,m := 1 «u n+1,m u n 1,m 2 u n+1,mh 1(u n,m, u n+1,m) un,m and h2(u n,m, v m [0] = R [0] u n,m+1) m un,m := 1 «u n,m+1 u n,m 1 2 u n,m+1h 2(u n,m, u n,m+1) un,m. XP (2009) Integrability and Symmetries of Difference Equations: the Adler Bobenko Suris Case arxiv: Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

7 Extended symmetries Definition A symmetry acting on the lattice parameters, as well. The Adler Bobenko Suris equations Each one of them admits a pair of extended symmetries generated by the vector fields V n = n v [0] n r(α) α and respectively. V m = m v [0] m r(β) β, Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

8 Using the extended generalized symmetries ❶ Master symmetries Observation : Infinite hierarchies of generalized symmetries can be constructed by considering the commutators h i h i v n [k] = V n,v n [k 1], v m [k] = V m,v m [k 1], k = 1, 2,.... Rasin O., Hydon P. (2007) Symmetries of Integrable Difference Equations Stud. Appl. Math. 49 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

9 Using the extended generalized symmetries ❶ Master symmetries These infinite hierarchies can be constructed explicitly using a recursion operator v n [k] = R k R [0] n un,m, v m [k] = R k R [0] m un,m, k = 0, 1, 2,... where R = X l= l S n (l) R [0] n un+l,m + r(α) α r(β) β X l= l S m (l) R [0] m un,m+l XP (2009) Integrability and Symmetries of Difference Equations: the Adler Bobenko Suris Case arxiv: Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

10 Using the extended generalized symmetries ❷ Continuous symmetric reductions Lead to similarity solutions which are Invariant under the action of V n and V m Determined by an integrable system of partial differential equations Related to Painlevé transcendents and soliton solutions Tsoubelis D, XP (2009) Continuous symmetric reductions of the Adler Bobenko Suris equations to appear in J. Phys. A: Math. Theor. (arxiv: ) Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

11 Starting point 1 Assumption The solution u n,m depends continuously on the lattice parameters α, β. 2 Invariance under the extended symmetries The solution u n,m satisfies also the invariant surface conditions r(α) αu n,m = h1(u n,m, u n+1,m) n 1 «u n+1,m u n 1,m 2 u n+1,mh 1(u n,m, u n+1,m) r(β) β u n,m = h2(u n,m, u n,m+1) m 1 «u n,m+1 u n,m 1 2 u n,m+1h 2(u n,m, u n,m+1) Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

12 Next step 3 Construction of an equivalent system of PDEs The system of lattice equation + invariant surface conditions is overdetermined involving six different values of u. Eliminate three of them through compatibility procedures. A system of PDEs of hyperbolic type results. Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

13 Further properties of the resulting system of PDEs Integrability aspects 1. Lax pair 2. auto-bäcklund transformation Equivalent forms It can be written in four different ways as Σ(u n,m, u n+τ,m, u n,m+σ; τ n, σ m; α, β) where τ = ±1, σ = ±1. Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

14 The discrete potential KdV equation and its symmetries The discrete potential KdV equation (u n,m u n+1,m+1 )(u n+1,m u n,m+1 ) α + β = 0 Extended Generalized symmetries n V n = un,m α u n+1,m u n 1,m V m = m u n,m+1 u n,m 1 un,m β Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

15 The corresponding continuous system u n+1,m β u n,m+1 α 2 u n,m α β = = = un+1,m un,m+1 α β un+1,m un,m+1 1 α β α β m (u n+1,m u n,m+1) un,m β n + (u n+1,m u n,m+1) un,m α 2 (u n+1,m u n,m+1) un,m u n,m α β ««+ n un,m β «m un,m α Nijhoff F., Hone A., Joshi N.(2000) On a Schwarzian PDE associated with the KdV Hierarchy Phys. Lett. A 267 Tongas A., Tsoubelis D., XP (2001) A family of Integrable nonlinear equations of Hyperbolic type J. Math. Phys. 42 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

16 Finding solutions: Symmetry reduction to Painlevé V Invariant form of the similarity solution u n,m = T n,m (α β) exp [ ( 1) n+m µ(α + β) ] Relation to Painlevé V transcendents The function T n,m is determined by the quadrature d dy ln Tn,m(y) = ( 1)n+m µ 1 + Gn,m(y) 1 G, y := α β, n,m(y) where G n,m is a solution of the Painlevé V equation with parameters A = n2 2, B = m2, Γ = λ ( 1) n+m µ, = 2µ 2 2 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

17 Finding solutions: Symmetry reduction to Painlevé V Invariant form of the similarity solution u n,m = T n,m (α β) exp [ ( 1) n+m µ(α + β) ] Relation to asymmetric, alternate discrete Painlevé II The above solution also satisfies the similarity constraint n u n+1,m u n 1,m + m u n,m+1 u n,m µ( 1) n+m u n,m = 0. Such solutions are determined by the asymmetric, alternate discrete Painlevé II equation. Tongas A., Tsoubelis D., XP (2007) Affine linear and D 4 symmetric lattice equations: Symmetry Analysis and Reductions J. Phys. A: Math. Theor. 40 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

18 Finding solutions: Symmetry reduction to Painlevé VI Invariant form of the similarity solution ( ) α u n,m = S n,m (αβ) (1+2( 1)n+m µ)/4 β Relation to Painlevé VI transcendents The function S n,m is determined by the quadrature d dy ln Sn,m(y) = 1 + 2( 1)n+m µ y + H n,m(y) 4y y H, y := α n,m(y) β, where H n,m is a solution of the Painlevé VI equation with parameters A = n2 2, B = m2, Γ = λ, = `1 + 2 ( 1) n+m µ 2 4 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

19 Finding solutions: Symmetry reduction to Painlevé VI Invariant form of the similarity solution ( ) α u n,m = S n,m (αβ) (1+2( 1)n+m µ)/4 β Relation to discrete generalized Painlevé equation The above solution also satisfies the similarity constraint n α u n+1,m u n 1,m + m β u n,m+1 u n,m µ( 1)n+m «u n,m = 0. Such solutions were shown to be related to discrete generalized Painlevé equation. Nijhoff F., Ramani A., Grammaticos B., Ohta Y. (2001) On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation Stud. Appl. Math. 106 Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

20 Finding solutions: Soliton solutions B(u, ũ;λ) : auto-bäcklund transformation of the continuous system ũ n,m α ũ n,m β = = un+1,m ũn,m α λ un,m+1 ũn,m β λ «n (u n+1,m ũ n,m) un,m α «m (u n,m+1 ũ n,m) un,m β (u n,m ũ n+1,m) (u n+1,m ũ n,m) = α λ (u n,m ũ n,m+1) (u n,m+1 ũ n,m) = β λ Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

21 Finding solutions: Soliton solutions Superposition principle ũ u λ 1 λ 2 û u n,m û n,m (ũ n,m û n,m) = λ 1 λ 2 u n+1,m û n+1,m (ũ n+1,m û n+1,m ) = λ 1 λ 2 u n,m+1 û n,m+1 (ũ n,m+1 û n,m+1 ) = λ 1 λ 2 λ 2 λ 1 û Bianchi commuting diagram Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

22 Finding solutions: Soliton solutions Seed solution : fixed point of the auto-bäcklund transformation B(u (0), u (0) ; λ) = u (0) n,m = n λ α + m λ β One soliton solution u n,m (1) from B(u (0), u (1) ;λ µ 2 1) u n,m (1) = u n,m (0) ρ µ 1, ρ1 := ρ λ α µ1 λ α + µ1 «n λ β µ1 λ β + µ1 «m c 1 Two solitons solution u (2) n,m from the superposition principle of B u (2) n,m = u (0) n,m (µ 1 + µ 2) 1 + ρ 1 + ρ 2 + ρ 1ρ 2 1 µ 1+µ 2 µ 1 µ 2 (ρ 1 ρ 2) ρ 1ρ 2 Nijhoff F., Atkinson J., Hietarinta J. (2009) Soliton Solutions for the ABS Lattice Equations: I Cauchy Matrix Approach arxiv: Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17

23 Summary Conclusions This symmetry approach led to 1 infinite hierarchies of generalized symmetries 2 integrable systems of PDEs and soliton solutions 3 reductions to continuous Painlevé equations Perspectives Study the reductions of 1 the other equations on the Adler Bobenko Suris lists 2 lattice equations defined on a black white lattice 3 systems of lattice equations Pavlos Xenitidis (Un. Patras) Symmetry Reductions March 11, / 17