Pentagram Maps and Integrable Hierarchies Boris Khesin (joint with Fedor Soloviev and Anton Izosimov) Geometry, Dynamics and Mechanics Seminar May 19, 2020 Boris Khesin Pentagram Maps and Integrable Hierarchies 1 / 35
Table of contents 1 Boussinesq and higher KdV equations 2 Pentagram map in 2D 3 Pentagram maps in any dimension 4 Duality 5 Numerical integrability and non-integrability Boris Khesin Pentagram Maps and Integrable Hierarchies 2 / 35
Geometry of the Boussinesq equation Let G : R → RP 2 be a nondegenerate curve, i.e. G ′ and G ′′ are not collinear ∀ x ∈ R . Reminder on envelopes: Define evolution of G ( x ) in time. Given ǫ > 0 take the envelope L ǫ of chords [ G ( x − ǫ ) , G ( x + ǫ )]. Expand the envelope L ǫ in ǫ : L ǫ ( x ) = G ( x ) + ǫ 2 B G ( x ) + O ( ǫ 4 ) Boris Khesin Pentagram Maps and Integrable Hierarchies 3 / 35
In the expansion L ǫ ( x ) = G ( x ) + ǫ 2 B G ( x ) + O ( ǫ 4 ) view ǫ 2 as the time variable. Theorem (Ovsienko-Schwartz-Tabachnikov 2010) The evolution equation ∂ t G ( x , t ) = B G ( x , t ) is equivalent to the Boussinesq equation u tt + 2( u 2 ) xx + u xxxx = 0 . Remark. The Boussinesq equation is the (2 , 3)-equation of the Korteweg-de Vries hierarchy. Boris Khesin Pentagram Maps and Integrable Hierarchies 4 / 35
Joseph Boussinesq and shallow water the Boussinesq shallow water approximation (1872) u tt + 2( u 2 ) xx + u xxxx = 0 he introduced the KdV equation (as a footnote, 1877) u t + ( u 2 ) x + u xxx = 0 it was rediscovered by D.Korteweg and G.de Vries (1895) Boris Khesin Pentagram Maps and Integrable Hierarchies 5 / 35
Reminder on the KdV hierarchy Let R = ∂ m + u m − 2 ( x ) ∂ m − 2 + u m − 3 ( x ) ∂ m − 3 + ... + u 1 ( x ) ∂ + u 0 ( x ) , where ∂ j := d j / dx j . Define its m th root Q = R 1 / m as a formal pseudo-differential operator Q = ∂ + a 1 ( x ) ∂ − 1 + a 2 ( x ) ∂ − 2 + ..., such that Q m = R . (Use the Leibniz rule ∂ f = f ∂ + f ′ .) Define its fractional power R k / m = ∂ k + ... for any k = 1 , 2 , ... and take its purely differential part Q k := ( R k / m ) + . Example for k = 1 one has Q 1 = ∂ , for k = 2 one has Q 2 = ∂ 2 + (2 / m ) u m − 2 ( x ). Boris Khesin Pentagram Maps and Integrable Hierarchies 6 / 35
The ( k , m ) -KdV equation is d dt R = [ Q k , R ] . Given order m , these evolution equations on R = ∂ m + ... + u 0 ( x ) commute for different k and form integrable hierarchies. Example the Korteweg-de Vries equation u t + uu x + u xxx = 0 is the “(3,2)-KdV equation”. It is the 3 rd evolution equation on Hill’s operator R = ∂ 2 + u ( x ) of order m = 2. the Boussinesq equation u tt + 2( u 2 ) xx + u xxxx = 0 is the “(2,3)-KdV equation”. It is the 2 nd evolution equation on operator R = ∂ 3 + u ( x ) ∂ + v ( x ) of order m = 3, after exclusion of v . Boris Khesin Pentagram Maps and Integrable Hierarchies 7 / 35
In higher dimensions... Let G : R → RP d be a nondegenerate curve, i.e. ( G ′ , G ′′ , ..., G ( d ) ) are linearly independent ∀ x ∈ R . Given ǫ > 0 and reals κ 1 < κ 2 < ... < κ d such that � j κ j = 0 define hyperplanes P ǫ ( x ) = [ G ( x + κ 1 ǫ ) , ..., G ( x + κ d ǫ )]. Example In RP 3 above κ j = − 1 , 0 , 1 Let L ǫ ( x ) be the envelope curve for the family of hyperplanes P ǫ ( x ) for a fixed ǫ . Boris Khesin Pentagram Maps and Integrable Hierarchies 8 / 35
Reminder on envelopes of plane families The envelope condition means that for each x the point L ǫ ( x ) and the derivative vectors ǫ ( x ) , ..., L ( d − 1) L ′ ( x ) belong to the ǫ plane P ǫ ( x ). Boris Khesin Pentagram Maps and Integrable Hierarchies 9 / 35
Expand the envelope in ǫ : L ǫ ( x ) = G ( x ) + ǫ 2 B G ( x ) + O ( ǫ 3 ) Theorem (K.-Soloviev 2016) The evolution equation ∂ t G ( x , t ) = B G ( x , t ) is equivalent to the (2 , d + 1) -KdV equation for any choice of κ 1 , ..., κ d . In particular, it is an integrable infinite-dimensional system. Remark. If � j κ j � = 0 , then the expansion is L ǫ ( x ) = G ( x ) + ǫ G ′ ( x ) + O ( ǫ 2 ) and the evolution equation ∂ t G ( x , t ) = G ′ ( x , t ) is equivalent to the d (1 , d + 1)-KdV equation dt R = [ ∂, R ]. Open question. Describe the geometry of higher ( k , d + 1)-KdV equations for all k ≥ 3. Boris Khesin Pentagram Maps and Integrable Hierarchies 10 / 35
Relation of curves and differential operators How to associate a curve G ⊂ RP d to a differential operator R = ∂ d +1 + u d − 1 ( x ) ∂ d − 1 + ... + u 0 ( x ) ? Consider the linear differential equation R ψ = 0. Take any fundamental system of solutions Ψ( x ) := ( ψ 1 ( x ) , ψ 2 ( x ) , ..., ψ d +1 ( x )) . Regard it as a map Ψ : R → R d +1 . Pass to the corresponding homogeneous coordinates: G ( x ) := ( ψ 1 ( x ) : ψ 2 ( x ) : ... : ψ d +1 ( x )) ∈ RP d , i.e. G : R → RP d . Boris Khesin Pentagram Maps and Integrable Hierarchies 11 / 35
We associated a curve G ⊂ RP d to a linear differential operator R . Moreover, Wronskian Ψ( x ) � = 0 ∀ x ⇐ ⇒ G ( x ) is nondegenerate ∀ x , i.e. ( G ′ , G ′′ , ..., G ( d ) ) are linearly independent for all x . Solution set Ψ is defined modulo SL d +1 transformations. The curve G ⊂ RP d is defined modulo PSL d +1 (i.e. projective) transformations. For differential operator R with periodic coefficients, solutions Ψ of R Ψ = 0 (and hence, the curve G ) are quasiperiodic: there is a monodromy M ∈ SL d +1 such that Ψ( x + 2 π ) = M Ψ( x ) for all x ∈ R . Boris Khesin Pentagram Maps and Integrable Hierarchies 12 / 35
Defining the pentagram map (R.Schwartz 1992) The pentagram map takes a (convex) n -gon P ⊂ RP 2 into a new polygon T ( P ) spanned by the “shortest” diagonals of P (modulo projective equivalence): T = id for n = 5 T 2 = id for n = 6 T is quasiperiodic for n ≥ 7 Hidden integrability? Boris Khesin Pentagram Maps and Integrable Hierarchies 13 / 35
Or, rather, a Pentagon map? Boris Khesin Pentagram Maps and Integrable Hierarchies 14 / 35
Properties of the pentagram map The integrability was proved for closed and twisted (i.e. with fixed monodromy) polygons in 2D (Ovsienko-Schwartz-Tabachnikov 2010, Soloviev 2012). There are first integrals, an invariant Poisson structure, and a Lax form. Pentagram map is related to cluster algebras, frieze patterns, etc. Its continuous limit is the Boussinesq equation. Extension to corrugated polygons, polygonal spirals, etc. How to generalize to higher dimensions? No generalizations to polyhedra: many choices of hyperplanes passing through neighbours of any given vertex. Diagonal chords of a space polygon may be skew and do not intersect in general. Boris Khesin Pentagram Maps and Integrable Hierarchies 15 / 35
Example and integrability of a pentagram map in 3D For a generic n -gon { v k } ⊂ RP 3 , for each k consider the two-dimensional “short-diagonal plane” P k := [ v k − 2 , v k , v k +2 ]. Then the space pentagram map T sh is the intersection point T sh v k := P k − 1 ∩ P k ∩ P k +1 . Theorem (K.-Soloviev 2012) The 3D short-diagonal pentagram map is a discrete integrable system (on a Zariski open subset of the complexified space of closed n-gons in 3D modulo projective transformations PSL 4 ). It has a Lax representation with a spectral parameter. Invariant tori have dimensions 3 ⌊ n / 2 ⌋ − 6 for odd n and 3( n / 2) − 9 for even n. Boris Khesin Pentagram Maps and Integrable Hierarchies 16 / 35
Twisted polygons Remark. There is a version for twisted space n -gons, where vertices are related by a fixed monodromy M ∈ PSL 4 : v k + n = Mv k for any k ∈ Z . The dimension of the space of closed n -gons modulo projective equivalence is 3 n − dim PSL 4 = 3 n − 15. This dimension for twisted n -gons is 3 n − 15 + 15 = 3 n . Boris Khesin Pentagram Maps and Integrable Hierarchies 17 / 35
Analogy and coordinates on the spaces of polygons A differential operator R = ∂ d +1 + ... + u 0 ( x ) defines a “solution curve” Ψ : R → R d +1 such that Ψ ( d +1) + u d − 1 ( x )Ψ ( d − 1) + ... + u 0 ( x )Ψ = 0 ∀ x ∈ R , which defines a nondegenerate curve G : R → RP d (mod projective equivalence). A difference operator defines a (twisted) “polygonal curve” V : Z → R d +1 such that V i + d +1 + a i , d V i + d + ... + a i , 1 V i +1 ± V i = 0 ∀ i ∈ Z , which defines a generic twisted polygon v : Z → RP d (mod projective equivalence). As n → ∞ a generic n -gon v i , i ∈ Z in RP d “becomes” a nondegenerate curve G ( x ) , x ∈ R in RP d . Boris Khesin Pentagram Maps and Integrable Hierarchies 18 / 35
Pentagram maps in any dimension For a generic n -gon { v k } ⊂ RP d and any fixed ( d − 1)-tuple I = ( i 1 , ..., i d − 1 ) of “jumps” i ℓ ∈ N define an I -diagonal hyperplane P I k by P I k := [ v k , v k + i 1 , v k + i 1 + i 2 , ..., v k + i 1 + ... + i d − 1 ] . Example The diagonal hyperplane P I k for the jump tuple I = (3 , 1 , 2) in RP 4 . Pentagram map T I in RP d is T I v k := P I k ∩ P I k +1 ∩ ... ∩ P I k + d − 1 . Boris Khesin Pentagram Maps and Integrable Hierarchies 19 / 35
Recommend
More recommend
