Soliton surfaces and generalized symmetries of integrable equations

Transcription

1 Soliton surfaces and generalized symmetries of integrable equations Sarah Post, joint with Michel Grundland Centre de Recherches Mathématques Université de Montreal Symmetries in Science, Bregenz August 2, / 34

2 Outline 1 CP N 1 sigma models and Generalized Weierstrass Formula for Immersion 2 Soliton Surfaces and the Fokas-Gel fand immersion Surfaces immersed in Lie groups and associated Lie algebras The three symmetries 3 Back to the example of CP N 1 sigma model 4 Conclusions and Future Perspectives New results in: A. M. Grundland and S. P. (2011). Soliton surfaces and generalized symmetries of integrable equations, with A. M. Grundland J.Phys. A Surfaces immersed in Lie algebras associated with elliptic integrals arxiv: A. M. Grundland and S. P. Analysis of CP N 1 sigma models via projective structures. arxiv: / 34

3 Soliton Surfaces We would like to consider surfaces whose structural equations are associated with integrable equations. Many classical surfaces, particularly in R 3, have been shown to be integrable: e.g. Minimal, constant mean curvature, linear Weingarten, Bonnet and Bianchi surfaces 1. Such surfaces can be defined by using the integrable equation written as a conservation law, using an associated Linear Spectral Problem (LSP) or continuous deformations of the LSP. Generalized Weierstrass immersion function for surfaces associated to CP N 1 sigma model is an example of using the integrable equations, written as a conservation law, to induce surface. The Sym-Tafel formula uses an LSP of an integrable equation to induce surfaces. The results of Fokas and Gel fand 2 and later with Finkel and Lui 3 extend the Sym-Tafel formula by making use of a gauge transformation of the wave function for the LSP and infinitesimal symmetries of the integrable equation. 1 A I Bobenko Functional Analysis and Its Appl., A. S. Fokas and I. M. Gelfand Comm. Math. Phys., 177: , A. S. Fokas, I. M. Gelfand, F. Finkel and Q. M. Liu Sel. Math / 34

4 Preliminaries of the model In studying Euclidean sigma models, one is interested in maps z : Ω R 2 { } C N with z z = 1 which are stationary points of the action functional S(z) = 1 (D µ z) D µ zdξ 1 dξ 2, (ξ 1, ξ 2 ) R 2 4 R 2 µ=1,2 given by solutions to the corresponding Euler-Lagrange (E-L) equations D µ D µ z + z((d µ z) D µ z) = 0 where D µ is given by µ=1,2 D µ z = µ z (z µ z)z, µ = ξ µ, µ = 1, 2. Such a model is explicitly U(1) invariant under rotations of z by a unitary phase. 4 / 34

5 We can extend the model to any non-normalized vector f by D µ f = µ f f µ f 1 ( D µ f ) f f, S(f D µ f ) = f 4 f f µ=1,2 and then for any two vectors f = kg with k : Ω C, S(f ) = S(kg) = S(g) S(f ) = S ( f f f R 2 ) dξ 1 dξ 2, ( ) f = S f f and so our model respects the equivalence relation f g f = kg and hence acts on complex projective space CP N 1 CP N 1 C N {0}/. 5 / 34

6 Projector Formalism It is natural to describe CP N 1 using the Projector Formalism. In terms of the projectors, the model is explicitly invariant under multiplication of f by scalar functions: f P f f f, P = P, tr(p) = 1, P 2 = P. f In terms of rank-1 Hermitian projectors, and complex variables ξ ± = ξ 1 ± iξ 2 ± ξ±. the Lagrangian density and action can be written as L(P) = tr( + P P) S(P) = tr( + P P)dξ + dξ. The E-L equations can be written in two equivalent forms [ + P, P] = 0, or + [ P, P] + [ + P, P] = 0. Also, we can associate with the model a topological charge density q(p) tr( + PP P) ( PP + P), Q(P) q(p)dξ + dξ Z Ω Ω 6 / 34

7 Generalized Weierstrass formula for immersion The generalized Weierstrass formula for immersion (GWFI) introduced by Konopelchenko 4 provides an efficient tool for constructing a surface immersed in su(n) for any solution of the CP N 1 sigma model. Theorem If P is a rank-1 Hermitian projector is a solution to the Euler-Lagrange equations then there exists a surface F whose immersion is defined by F = i [ + P, P]dξ + + [ P, P]dξ, F = F su(n). The differential df is exact if and only if P satisfies the E-L eqs. Introduce the following inner product on su(n), proportional to the Killing form, (A, B) = 1 tr(a, B), 2 A, B su(n) 4 Konopelchenko, B. Stud. Appl. Math., 1996, 96, / 34

8 Geometric Characteristics and Physical Quantities It had been previously shown that the surfaces are conformally parameterized 5, in fact we can be more specific and show: Observation: A. M. Grundland and SP For a finite action solution to the E-L eqs, P, the conformal factor associated with the surface F defined by the GWFI is proportional to the Lagrangian density and area of the surface is equal to the action, S(P). In particular, the surface will have finite area. g ++ = 1 2 tr([ +P, P][ + P, P]) = tr( + PP + P) = D P, D + P = 0 = g g + = 1 2 tr([ +P, P][ P, P]) = 1 2 (tr( +PP P) + tr( PP + P)) = 1 2 L(P), A(F) = R 2 L(P)dξ + dξ = S(P) 5 Grundland, A.M. & Yurdusen, I. J. Phys. A: Math. Theor., 2009, 42, / 34

9 We can write other geometric characteristics in terms of the physical quantities of the model L(P) and q(p) : The Gaussian curvature The norm of the mean curvature The Willmore functional The Euler-Poincare character K (F) = 2 + (ln(l(p))). L(P) (H(F ), H(F )) = 4 L(P) 2 ( L(P) 2 + 3q(P) 2). W (F ) = 1 2 S(P) (F) = 1 π R2 q(p) 2 L(P) dξ +dξ, R 2 + ln(l(p))dξ + dξ. 9 / 34

10 Raising and lowering operators In 1980, Din and Zakrzewski 6 showed that for finite action solutions of the CP N 1 model, differentiable on the Riemann sphere C = C { }, there exists some holomorphic vector z 0 related to the original solution via a raising operator. In analogy with the raising and lowering operators defined for the vector case, there exist raising and lowering operators for projectors given by 7 Π ± X ±XX X tr( ± XX X), Π ±X 0 whenever ± XX X = 0. Theorem If P is a rank-1 Hermitian projector solution to the CP N 1 sigma model, then Π ± P will either be 0 or a rank-1 Hermitian operator. Further, for any rank-1 Hermitian projector solution of the E-L equations in C N+2 (C { }) with finite action, there exists some k and rank-1 Hermitian projector P 0 which is holomorphic Π P 0 = 0 and for which P = Π k +P 0 6 Din, A.M. & Zakrzewski, W.J. l Nucl. Phys. B, 1980, 174, Goldstein, P.P. & Grundland, A.M., 2009, arxiv: / 34

11 Generalized Weierstrass Formula for Projectors Theorem (Grundland & Yurdusen 8 ) The surface defined by the GWFI has an associated integer k and holomorphic projector P 0 so that the following holds k 1 F = F k i P k + 2 P j 1 + 2k N I N, P l = Π l +P 0. j=0 F k is equivalent to surfaces generated by the Sym-Tafel formula (ST) based on the linear spectral problem (LSP) associated with the integrable system. 8 Grundland, A.M. & Yurdusen, I. J. Phys. A: Math. Theor., 2009, 42, / 34

12 Surfaces immersed in Lie groups and associated Lie algebras Consider an immersion function for a 2D surface in a Lie group: Φ : (x 1, x 2 ) R 2 G The tangent vectors of such a function can be written as 1 Φ = u 1 Φ, 2 Φ = u 2 Φ, with compatibility conditions equivalent to a matrix system of NPDE s, 2 u 1 1 u 2 + [u 1, u 2 ] = 0 where Φ(u, v) takes its values in the Lie group G and u 1 (x), u 2 (x) g, the associated Lie algebra. Note that the existence of u 1 and u 2 satisfying (1), implies the existence of the function Φ. 12 / 34

13 Consider now an infinitesimal deformation of the system of NPDE s (1) u α u α + ɛr α. The R α (x) g must satisfy The following theorem holds: Theorem 1: Fokas-Gel fand 2 R 1 1 R 2 + [R 1, u 2 ] + [u 1, R 2 ] = 0. Assume that u α g are matrix functions which satisfy the system of NPDE s, = 0, Φ[u] G is a solution of (1). Then, the following statements are equivalent: 1 There exist g-valued functions R 1 and R 2 which satisfy (2). 2 There exists an immersion function F(x) in the Lie algebra g with tangent vectors α F = Φ 1 R α Φ, α = 1, 2 3 There exists a matrix function Ψ such that u1 u 2 u1 u 2 Φ Φ + ɛ R 1 R 2 Ψ gives a generalized infinitesimal symmetry of the system of NPDE s = 0 together with the equation for the tangents of Φ.

14 Furthermore, if statement 2 holds then Ψ = ΦF is an admissible infinitesimal deformation of Φ and if statement 3 holds, then F = Φ 1 Ψ has tangent vectors which coincide with 2. We call immersion functions defined this way Fokas-Gel fand immersion function (FGIF).

15 Generalized vector fields and Jet space Instead of continuing the analysis with infinitesimal symmetries and Fréchet derivatives, we will use the apparatus of generalized vector fields and their prolongation structure. We consider functions on the extended jet space: f [θ n ] = f (x i, θj n ) composed of the independent variables, the dependent variables θ n and their derivatives denoted by: θj n n θ n =, J = (j 1,..., j n ), j k = 1, 2 xj1... xjn The derivatives of a function f [θ] are obtained via the total derivative operator D J,i = D i D J, D i x i + θj,i n θ n J, i = 1, / 34

16 Consider a (generalized) vector field with evolutionary representative v = ξ i [θ] x i + φ n [θ] θ n. v Q = Q n [θ] θ n, Q n[θ] = (φ n [θ] ξ i [θ]θi n ). It is associated with an infinitesimal deformation of independent variables x i and dependent variables θ n as (x i ) = x i + ɛξ i [θ], (θ n ) = θ n + ɛq n [θ], (1) The action of the infinitesimal deformation on functions of the jet space is via th prolongation of v Q : pr v Q = D J Qα[θ] j Symmetry Criterion: P J Olver 1993 The infinitesimal deformation given by (1) is a symmetry of some nondegenerate system of PDE s [θ] = 0 if and only if θ n J pr v Q ( [θ]) = 0 whenever [θ] = 0.

17 Back to the surfaces and integrability Suppose that it is possible to reparametrize the matrices u α (x) by a finite set of unknown functions θ n (x 1, x 2 ) and a spectral parameter λ so that = 0 is independent of the spectral parameter, i.e. M k [θ]: ([θ], λ) [u α ([θ], λ)] = 0 M k [θ] = 0. In this notation, the Φ and F are functions on the jet space. The tangent vectors of Φ give a linear spectral problem (LSP) D α Φ([θ], λ) = u α ([θ], λ)φ([θ], λ), α = 1, 2, and the tangent vectors of F([θ], λ) will be given by where R α ([θ], λ) satisfy D α F = Φ 1 R α Φ D 2 R 1 D 1 R 2 + [R 1, u 2 ] + [u 1, R 2 ] = / 34

18 In Fokas et al, the authors demonstrated three possible forms of R α ([θ], λ) : In particular, they proved the existence of a g-valued immersion function F with tangent vectors D 1 F = Φ 1 Q 1 Φ, D 2 F = Φ 1 Q 2 Φ R 1 = a(λ) λ u1 + 1 S + [S, u 1 ] + pr v Q u 1, R 2 = a(λ) λ u2 + 2 S + [S, u 2 ] + pr v Q u 2, Next, we discuss the necessary and sufficient conditions so that the g-valued immersion function can be integrated as : ( F = Φ 1 a(λ) Φ ) λ + SΦ + pr v Q Φ g. Let s consider each linearly independent term separately.

19 Sym-Tafel Formula The first symmetry to be considered is a conformal symmetry of λ: λ = λ ɛa(λ), This induces an infinitesimal symmetry of the system of NDPE s and its LSP as u1 u 2 u1 u 2 + ɛa(λ) λu 1 λ u 2. Φ Φ Ψ = λ Φ This symmetry can be used to form a surface with tangent vectors D 1 F ST = Φ 1 a λ u 1 Φ D 2 F ST = Φ 1 a λ u 2 Φ The surfaces can be given, modulo a constant of integration as F ST = aφ 1 Ψ = aφ 1 λ Φ g. This is the Sym-Tafel formula for immersion. 19 / 34

20 Gauge transformation of the wave function Consider the symmetry of the LSP and system of NPDE s generated by a gauge transformation, S(x i ) g of the wave function u 1 u 1 1 S + [S, u 1 ] u 2 Φ u 2 Φ + ɛ 2 S + [S, u 2 ]. Ψ = SΦ This symmetry induces a surface F S as with tangent vectors F S = Φ 1 Ψ = Φ 1 SΦ g D α F S = Φ 1 ( α S + [S, u α ]) Φ, α = 1, / 34

21 For the final term, consider a generalized vector field v Q associated with a (generalized) symmetry of ([θ], λ) = 0 i.e. pr v Q (D 2 u 1 ) pr v Q (D 1 u 2 ) + [(pr v Q u 1 ), u 2 ] + [u 1, (pr v Q u 2 )] = 0, whenever ([θ], λ) D 1 u 2 D 2 u 2 + [u 1, u 2 ] = 0. Similarly, there exists a surface F with tangent vectors if and only if the compatibility conditions D α Φ = Φ 1 pr v Q (u α )Φ, α = 1, 2 (2) D 2 (pr v Q u 1 ) D 1 (pr v Q u 2 ) + [(pr v Q u 1 ), u 2 ] + [u 1, (pr v Q u 2 )] = 0 hold. But, the two equations are equivalent since Lemma 5.12: Olver 1993 A generalized vector field in evolutionary form commutes with the total derivative, i.e. [D α, pr v Q ] = 0. Thus, the generalized vector field v Q is a symmetry of = 0 if and only if there exists a surfaces with tangent vectors given by (2).

22 Proposition 2 Suppose that ([θ], λ) = 0 is an integrable equation and Φ([θ], λ) is a solution to its LSP. Then the following are equivalent 1 There exists a vector field v Q which is a generalized symmetry of ([θ], λ) = 0. 2 There exists an immersion function F ([θ], λ) in the Lie algebra g with tangent vectors D α F = Φ 1 pr v Q u α Φ, α = 1, 2. 3 There exists a matrix function Ψ such that the infinitesimal deformation u 1 u 1 pr v Q u 1 u 2 Φ u 2 Φ + ɛ pr v Q u 2 Ψ gives a generalized infinitesimal symmetry of ([θ], λ) = 0 together with its LSP. Furthermore, if statement 2 holds then Ψ = ΦF is an admissible infinitesimal deformation of Φ and if statement 3 holds, then F = Φ 1 Ψ has tangent vectors which coincide with 2.

23 We next consider the claim that F given by F([θ], λ) = Φ 1 pr v Q Φ, D α F = Φ 1 pr v Q u α Φ? Proposition 3 The proposed immersion function F = Φ 1 pr v Q Φ has the given tangent vectors if and only if v Q is a symmetry of the integrable equation, ([θ], λ) = 0, and the LSP in the sense that pr v Q (D α Φ u α Φ) = 0 whenever D α Φ u α Φ = 0.

24 Similarly, the infinitesimal deformation u 1 u 1 u 2 Φ u 2 Φ + ɛ pr v Q u 1 pr v Q u 2 pr v Q Φ gives a generalized infinitesimal symmetry of ([θ], λ) = 0 together with its LSP if and only if v Q is a symmetry of the LSP as well as the integrable equation. The relation between the Fréchet derivative and generalized vector fields is given in Olver pr v R Φ([θ], λ) = ( DΦ Dθ n )Q n lim ɛ 0 ɛ Φ(θn + ɛq n ). This provides the relation between our theorem and the results of Fokas, Gel fand et al. Namely, F = Φ 1 ( DΦ ) Dθ n )Q n D α F = Φ (( 1 Duα Dθ n )Q n Φ if and only if {Q n } is a symmetry of the LSP as well as its set of integrable equations.

25 Surfaces associated with CP N 1 sigma model Let us rewrite the rank-one Hermitian projector P in terms of an element of the Lie algebra su(n) and E = I/N θ i(p E) su(n), P P = P θ θ = i(2 N) N θ + 1 N N E The E-L equations can be written as the compatibility conditions of a linear spectral problem x 1 = ξ +, x 2 = ξ [θ] = D 2 u 1 D 1 u 2 + [u 1, u 2 ] = [θ 12, θ] = 0 u 1 = λ [θ 1, θ] u 2 = 2 1 λ [θ 2, θ] D α Φ = u α Φ, α = 1, 2 We can decompose θ in terms of a basis for su(n) θ = θ n e n, [θ] = c j kl θk 12θ l e j 25 / 34

26 The E-L equations for the CP N 1 sigma model are invariant under conformal symmetries associated with the vector field in evolutionary form ( ) v C f (x 1 )θ j 1 + g(x 2 )θ j 2 θ j. That is pr v C ([θ 12, θ]) = 0 whenever [θ 12, θ] = 0. Thus, there exist an su(n)-valued FGIF F with tangent vector D α F = Φ 1 pr v C u α Φ. The surface can be explicitly integrated and is given by F = Φ 1 (fu 1 + gu 2 )Φ. The following infinitesimal deformation is a symmetry of = 0 and its LSP: u 1 u 1 pr v C u 1 u 2 Φ u 2 Φ + ɛ pr v C u 2 Ψ = ΦF = (fu 1 + gu 2 )Φ

27 Integrated form of the FGIF Next, we consider the proposed integrated form of the FGFI. That is, let s consider the immersion function Question F = Φ pr v C Φ When is F a FGIF, ie when is D α F = Φ pr v C u α Φ? We know this is equivalent to the requirement that pr v C (D α Φ u α Φ) = 0, whenever D α Φ u α Φ = 0. In order to understand this restriction, we shall look at two examples where the wave function can be given explicitly for the CP N 1 sigma model: finite action solutions on Euclidean space traveling wave solutions on Minkowski space. 27 / 34

28 The CP N 1 model defined on 2d Euclidean space Let us consider finite action solutions of the E-L equations defined on the Euclidean plane. Recall that any such solution belongs to some family of projectors Λ = {P 0 = Π k P, Π k 1 P,..., Π P, P, Π + P,..., Π N k 1 + P} Again, we use θ su(n) with θ i(p E) su(n). The raising and lowering operators in this basis are Π + P = D 1PPD 2 P tr(d 1 PPD 2 P) = θ 1(E iθ)θ 2 tr(θ 1 (E iθ)θ 2 ) Π +(E ip), Π P = D 2PPD 2 P tr(d 2 PPD 2 P) = θ 2(E iθ)θ 1 tr(θ 2 (E iθ)θ 1 ) Π (E ip). The wave functions can be written as 4λ Φ = I + (1 λ) 2 Π j (E iθ) 2 (E iθ). 1 + λ j=1 28 / 34

29 Action of the conformal transformation on Π ± P,Φ Given a generalized vector field v C associated with a conformal symmetry transformation. The action of pr v C on Π k ±(E iθ) is given by pr v C (Π k ± (E iθ)) = fd 1 Π k ± (E iθ) + gd 2 Π k ± (E iθ). That is, the infinitesimal deformation (x 1 ) = x 1 + ɛf (x 1 ), (x 2 ) = x 2 + ɛg(x 2 ), induces the infinitesimal deformation Π k ± (E iθ) = Π k ± (E iθ) + Consequently the action of v C on Φ is given by Proof: Induction. θ = θ + ɛ(f θ 1 + gθ 2 ), ɛ ( fd 1 Π k ± (E iθ) + gd 2 Π k ± (E iθ) ). pr v C Φ = fd 1 Φ + gd 2 Φ. 29 / 34

30 The surface The su(n)-valued immersion functions F and F coincide Φ pr v C Φ = Φ (fu 1 + gu 2 )Φ, F = Φ pr v C Φ = F, and in particular they have the same tangent vectors D α F = Φ pr v C u α Φ = D α F, α = 1, 2 and so F is a FGIF generated by conformal symmetries of the CP N 1 sigma model defined on the Euclidean plane. Further, the infinitesimal deformation u1 u 2 Φ u1 u 2 Φ + ɛ pr v C u 1 pr v C u 2 pr v C Φ gives a generalized infinitesimal symmetry of the system of the E-L equations together with its associated LSP. 30 / 34

31 CP N 1 sigma model defined on Minkowski space It is possible to define the CP N 1 sigma model on Minkowski space by taking light-cone coordinates x 1 = ξ 1 + ξ 2, x 2 = ξ 1 ξ 2. In this case, we do not have raising and lowering operators nor is there a closed form for the wave function in terms of an arbitrary solution. If we consider traveling wave solutions, P(x 1 + cx 2 ), we can construct the solutions for Φ, however, they will not satisfy pr v C (D α Φ u α Φ) = 0 and so, we have shown by example that this condition is not automatically satisfied. 31 / 34

32 Conclusions We have: Given a formula for the systematic construction of surfaces immersed in Lie algebras in terms of conservation laws generalized vector fields and their prolongation structure. Written the necessary and sufficient conditions for the existence of such surfaces in terms of the invariance criterion for generalized symmetries. Given sufficient conditions that allow for the explicit integration of the immersion function and, in particular, proposed a form for the explicit integration of such surfaces. The necessary and sufficient conditions for the proposed form of the immersion function are given in terms of the generalized vector fields and their action on the LSP. Illustrated these theoretical considerations using the completely integrable CP N 1 sigma model to show that the necessary and sufficient conditions are new, non-trivial conditions. 32 / 34

33 Future Perspectives 1 The next immediate step is to construct surfaces in a Lie algebra g related to integrable equations and identify those surfaces which have an invariant geometric characterization. 2 For finite action solutions of the CP N 1 sigma model defined on the complex Euclidean plane, the surfaces can be characterized via L, q, and A. A natural question is to see how the deformations of a FGIF affect the geometric characteristics. Also, whether some of this data are superfluous, such as in the case of holomorphic and anti-holomorphic solutions where L and q are not independent. 3 Further, it is an open question to see how the surfaces obtained in this fashion manifest the structures of intergrable equations: e.g. Hamiltonian structures, hierarchy of conserved quantities and recursion operators for the generalized symmetries. 33 / 34

34 Thank you for your attention. F ST : λ = 1, g(λ) = 5 F u 1 : λ = 1, g(λ) = 5 Figure: Soliton surfaces for for θ = P(x, 0, 1) with x 1 [0.2, 3] and x 2 [ π/g(1), π/g(1)] for the negative discriminate cases and x [ 1, 1] y [ 0.5, 0.5] for the positive discriminant cases. The axes indicate the components of the immersion function in the basis {e 1, e 2, e 3 }