Lattice geometry of the Hirota equation

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1 Lattice geometry of the Hirota equation arxiv:solv-int/ v1 8 Jul 1999 Adam Doliwa Instytut Fizyki Teoretycznej, Uniwersytet Warszawski ul. Hoża 69, Warszawa, Poland doliwa@fuw.edu.pl Abstract Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of the Laplace equation, (ii) the gauge invariant form of the Hirota equation for projective invariants of the Laplace sequence, (iii) the discrete Toda system for the rotation coefficients and (iv) the original form of the Hirota equation for the τ-function of the quadrilateral lattice. Keywords: Integrable discrete geometry; Hirota equation 1 Introduction The integrable discrete geometry deals with lattice submanifolds described by integrable equations. Among the integrable discrete (difference) equations an important role is played by the Hirota equation [12] which is the discrete analog of the two-dimensional Toda system [17]. Both the Toda system and the Hirota equation are important in the theory of integrable equations and in their applications. It turns out that the two-dimensional Toda system was studied in classical differential geometry and describes the so called Laplace sequences of two-dimensional conjugate nets [3]. The lattice geometric interpretation of the discrete Toda system was found in [6] and is based on the observation that the discrete analog of 1

2 the conjugate net on a surface, which is given by the two-dimensional lattice made of planar quadrilaterals [18], allows for definition of the corresponding Laplace sequence of discrete conjugate nets. Since then, the multidimensional quadrilateral lattice [10] (multidimensional lattice made of planar quadrilaterals the integrable discrete analog of a multidimensional conjugate net) became one of the central notions of the integrable discrete geometry. In particular, many classical results of the theory of conjugate nets and of their reductions have been generalized recently to the discrete level [1, 16, 2, 7, 11, 14, 5, 9]. The goal of the present article is to reinterpret and expand results obtained in[6] using new notions provided by the general theory of quadrilateral lattices. Different formulations of the Hirota equations are reviewed and, for each of them, the geometric interpretation of the corresponding functions is given. In Sections 2 and 3 we reformulate, using more convenient notation, results found in [6]. In Section 4 we present the discrete analog of the standard version of the Toda system as an equation governing Laplace transformations of the rotation coefficients. Then in Section 5, basing on the geometric interpretation of the τ function of the quadrilateral lattice given in [9], we show that the τ functions of the Laplace sequence of quadrilateral lattices solve the original Hirota s form of the discrete Toda system this last result fills out the missing point of paper [6]. 2 Laplace transformations of quadrilateral lattices Definition 2.1. Two dimensional quadrilateral lattice is a mapping of the two dimensional integer lattice into M dimensional projective space such that elementary quadrilaterals of the lattice are planar: x : Z 2 P M, T 1 T 2 x x,t 1 x,t 2 x. In the above definition T i, i = 1,2, denotes the shift operator along the i-th direction of the lattice. In the non-homogenous coordinates of the projective space a two dimensional quadrilateral lattice is represented by the mapping x : Z 2 R M 2

3 T j x T i T j x x T i x T j L ij (x) L (x) ij T j -1 x T i L (x) ij Figure 1: Laplace transformation of quadrilateral lattices satisfying the Laplace equation [6] 1 2 x = (T 1 A 12 ) 2 x+(t 2 A 21 ) 2 x, (2.1) which is equivalent to the planarity condition; here i = T i 1, i = 1,2, is the partial difference operator, and the functions A 12,A 21 : Z 2 R, define the position of the point T 1 T 2 x with respect to the points x, T 1 x and T 2 x. Planarity of elementary quadrilaterals implies that the tangent lines containing opposite sides of an elementary quadrilateral intersect. The Laplace transformation L ij, i j of the lattice x is defined [6, 11] as intersection of the line x,t i x with the line T 1 j x,t i T 1 j x (see Figure 1). Using elementary calculations one can show the following results [6, 11]. Proposition 2.1. In the non-homogenous representation the Laplace transformation L ij (x) of the quadrilateral lattice x is given by L ij (x) = x 1 A ji i x. (2.2) Proposition 2.2. The Laplace transformations of quadrilateral lattices are again quadrilateral lattices and the corresponding transformations of the coefficients of the Laplace equation (2.1) of the lattice L ij (x) read L ij (A ij ) = A ji (T i A ij +1) 1, (2.3) T j A ji ( ) L ij (A ji ) = Tj 1 Ti L ij (A ij ) L ij (A ij ) (A ji +1) 1. (2.4) 3

4 Proposition 2.3. Under the assumption that the transformed lattices are non-degenerate, i.e. their quadrilaterals do not degenerate to segments or points, we have L ij L ji = L ji L ij = id. (2.5) In this way given two dimensional quadrilateral lattice x one can define a sequence of quadrilateral lattices x (l) = L l 12(x), l N, L 1 12 = L 21. In analogy to the Laplace sequence of conjugate nets, the above sequence can be called the Laplace sequence of quadrilateral lattices. Equations (2.3)-(2.4) can be then rewritten in the form 2 A (l) 21 A (l) 21 1 A (l) 12 A (l) 12 = = T 1 A (l) 12 A(l+1) 12 (T 1 A (l) 12 +1)(A(l+1) 12 +1), (2.6) T 2 A (l) 12 A(l 1) 21 (T 2 A (l) 21 +1)(A(l 1) 21 +1), (2.7) which is the discrete analog of the coupled Volterra system. 3 Projective invariants of the Laplace sequence The planarity of elementary quadrilaterals of the quadrilateral lattice and the construction of the Laplace sequence are essentially of the projective nature [6]. It would be therefore interesting to know the pure projectivegeometric version of the equation describing the Laplace sequence of quadrilateral lattices. The basic numeric invariant of projective transformations is the so called cross-ratio of four collinear points, which is given in the affine representation as notice the simple identity cr(a,b;c,d) = ( ) c a : c b ( ) d a ; d b cr(a,b;c,d) = cr(b,a;d,c). (3.1) 4

5 Define the function K ij as the cross-ratio of x, L ij (x), T i x and T j L ij (x). Elementary calculations show that and, therefore, T i x L ij (x) = 1+A ji A ji i x, T j L ij (x) x = T ia ij +1 i x, T j A ji ( 1 T j L ij (x) L ij (x) = 1+T ) ia ij i x, A ji T j A ji K ij = cr(x,l ij (x);t i x,t j L ij (x)) = A ji(t i A ij +1) T j A ji (1+T i A ij )(1+A ji ). Equations (2.3) and (2.4) allow to find the Laplace transforms of the projective invariants [6] ( ) L ij (K ij ) = Tj 1 Kij (T i T j K ij ) (T i K ij +1)(T j K ij +1) 1, (3.2) (T i K ij )(T j K ij ) T i K ji +1 L ij (K ji ) = K ij ; (3.3) notice that equation (3.3) is a simple consequence of Proposition 2.3 and property (3.1) of the cross-ratio. Equations (3.2) and (3.3) can be rewritten in terms of the function K = K 12 in the following form T 2 ( K (l+1) +1 K (l) +1 ) T 1 ( K (l 1) +1 K (l) +1 known as the gauge invariant form of the Hirota equation. ) = (T 1T 2 K (l) )K (l) (T 1 K (l) )(T 2 K (l) ), (3.4) 4 Rotation coefficients of the quadrilateral lattice As it was shown in [10] it is convenient to reformulate the Laplace equation (2.1) as a first order system. We introduce the suitably scaled tangent vectors X i, i = 1,2, i x = (T i H i )X i, (4.1) 5

6 T j x T j X i T i T j x (T j )X j X j T i X j x X i T i x Figure 2: Definition of the rotation coefficients in such a way that the j-th variation of X i is proportional to X j only (see Figure 2) j X i = (T j )X j, i j, (4.2) the coefficients in equation (4.2) are called the rotation coefficients. The scaling factors H i in equation (4.1), called the Lamé coefficients, satisfy the linear equations adjoint to (4.2); moreover i H j = (T i H i ), i j, A ij = jh i H i, i j. The Laplace transformation of the Lamé coefficients and the scaled tangent vectors was found in [11] and is presented in the following Proposition 4.1. The Lamé coefficients of the transformed lattice read L ij (H i ) = H j, L ij (H j ) = T 1 j ( j ( Hj )), 6

7 the tangent vectors of the new lattice are given by L ij (X i ) = i X i + i X i, L ij (X j ) = 1 X i. From above formulas follow transformation rules for the rotation coefficients. Proposition 4.2. The rotation coefficients transform according to ( L ij ( ) = T 1 j T i Q ) ij T i T j (1 (T i Q ji )(T j )), (4.3) T j L ij (Q ji ) = 1. (4.4) Equations (4.3) and (4.4) can be rewritten in terms of the function Q = Q 12 as ( 1 Q (l) T2 Q (l) 2 = T Q (l) 1 ) T 2Q (l+1). (4.5) Q (l 1) Q (l) 5 Geometry of the τ function To give the geometric meaning to the τ-function let us introduce [9] the vectors X i pointing in the negative directions i x = (T 1 i H i ) X i, or i x = H i (T i Xi ), where i = T 1 i 1 is the backward difference operator. The scaling factors H i (the backward Lamé coefficients) are chosen in such a way that the i variation of Xj is proportional to X i only (see Figure 3). We define the backward rotation coefficients as the corresponding proportionality factors i Xj = (T 1 i ) X i, or i Xj = (T i Xi ), i j. (5.1) The backward Lamé coefficients H i satisfy then the following system of linear equations j Hi = (T j Qij ) H j, i j, 7

8 T i -1 T i -1 ~ X j x X ~ i x ~ X j T -1-1 i T j x T -1 i Q ~ ~ ( ij )X i T -1 ~ j X i T j -1 x Figure 3: Definition of the backward data adjoint to system (5.1) Since the forward and backward rotation coefficients and describe the same lattice x but from different points of view, then one cannot expect that they are independent. Indeed, defining the functions ρ i : Z 2 R, i = 1,2, as the proportionality factors between X i and T i Xi (both vectors are proportional to i x): X i = ρ i (T i Xi ), T i H i = 1 ρ i Hi, i = 1,2, we have the following result [9] Proposition 5.1. The forward and backward rotation coefficients of the lattice x are related through the following formulas ρ j T j Qij = ρ i T i Q ji, and the factors ρ i are first potentials satisfying equations T j ρ i ρ i = 1 (T i Q ji )(T j ),i j. (5.2) The right hand side of equation (5.2) is symmetric with respect to the interchange of i and j, which implies the existence of a potential τ : Z 2 R, 8

9 such that ρ i = T iτ τ therefore equation (5.2) defines the second potential τ: ; (T i T j τ)τ (T i τ)(t j τ) = 1 (T iq ji )(T j ), i j. (5.3) The potential τ connecting the forward and backward data is the τ-function of the quadrilateral lattice [9]. Let us find the Laplace transformation of the τ-function. Formulas (4.3) and (4.4) imply that 1 (T i L ij (Q ji ))(T j L ij ( )) = T i T j T j T i τt i T j τ T i τt j τ, (5.4) which, due to equation (5.3), allows for identification L ij (τ) = τ. (5.5) It should be mentioned here that the above formula was strongly suggested by the identification of the Schlesinger transformation of the theory of the multicomponent Kadomtsev Petviashvili hierarchy [4, 13, 15] with the Laplace transformation of conjugate nets [8]. Corollary 5.2. The geometric meaning of τ ij as the Laplace transformation L ij (τ) of the τ-function applies for any dimension of the quadrilateral lattice. Finally, equation(5.3) rewritten in terms of the τ-function and its Laplace transformations take the following form τ (l) T 1 T 2 τ (l) = (T 1 τ (l) )(T 2 τ (l) ) (T 1 τ (l 1) )(T 2 τ (l+1) ), (5.6) which is the original Hirota s bilinear form of the discrete Toda system. 6 Conclusion The geometric interpretation of the Hirota equation (integrable discrete analog of the Toda system) is given by the Laplace sequence of quadrilateral 9

10 lattices, therefore various representations of the lattice give different versions of the equation. In the paper we presented four different forms of the Hirota equation: (i) the discrete coupled Volterra system (2.6)-(2.7) for the coefficients of the Laplace equations, (ii) the gauge invariant form of the Hirota equation (3.4) for projective invariants of the Laplace sequence, (iii) the discrete Toda system for the rotation coefficients (4.5), and (iv) the original form of the Hirota equation (5.6) for the τ-function of the quadrilateral lattice. Acknowledgements The author would like to thank the organizers of the SIDE III meeting for invitation and support. References [1] A. Bobenko and U. Pinkall, Discrete isothermic surfaces, J. reine angew. Math. 475 (1996), [2] J. Cieśliński, A. Doliwa, and P. M. Santini, The integrable discrete analogues of orthogonal coordinate systems are multidimensional circular lattices, Phys. Lett. A 235 (1997), [3] G. Darboux, Leçons sur la théorie générale des surfaces I IV, Gauthier Villars, Paris, [4] E. Date, M. Kashiwara, M. Jimbo, and T. Miwa, Transformation groups for soliton equations, Nonlinear integrable systems classical theory and quantum theory, Proc. of RIMS Symposium (M. Jimbo and T. Miwa, eds.), World Scientific, Singapore, 1983, pp [5] A. Doliwa, Quadratic reductions of quadrilateral lattices, J. Geom. Phys. 30 (1999) [6], Geometric discretization of the Toda system, Phys. Lett. A 234 (1997),

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