Typeset with ps2.cls <ver.1.2> Full Paper Determinant Expressions for Discrete Integrable Maps Kiyoshi Sogo Department of Physics, School of Science, Kitasato University, Kanagawa 228-8555, Japan Explicit formulas for several discrete integrable maps with periodic boundary condition are obtained, which give the sequential time developments in a form of the quotient of successive determinants of tri-diagonal matrices. We can expect that such formulas make the corresponding numerical simulations simple and stable. The cases of discrete Lotka-Volterra and discrete KdV equations are demonstrated by using the common algorithm computing determinants of tri-diagonal matrices. KEYWORDS: discrete Lotka-Volterra equation, discrete KdV equation, discrete Toda lattice equation, discrete KP equation, tropical R model 1. Introduction The concept of solitons was discovered by Zabusky and Kruscal in the well known pioneering paper of 1965 1) with the aid of computer simulation of periodic Korteweg de Vries (KdV) equation. Although the algorithm employed has enough accuracy to convince their conclusions, when the computations are continued further, they diverge at certain time due to a numerical instability. One of the purposes considering a discretization which preserves the integrability of the equation is to avoid such inherent instability brought by insufficient discretization schemes. There have been already proposed many such integrable discretizations for several models including the discrete Lotka-Volterra, the discrete KdV, the discrete time Toda lattice and so on. The obtained difference equations, often called integrable maps, however are written in implicit forms and are not necessarily appropriate for numerical computations. The aim of the present paper is to rewrite such implicit equations into explicit formulas, for some models with periodic boundary condition, so that we can use them in numerical computations. The obtained formulas have the common property possessing the form of the quotient of two determinants which are tri-diagonal matrices. Because there are some standard algorithms to compute determinant of tri-diagonal matix, one of which is shown in the appendix, it is very simple to write simulation programs for these integrable maps by using such formulas. In the next section we derive the explicit formulas for discrete Lotka-Volterra and discrete KdV equations. These two models are in common with having the first order time derivative E-mail address: sogo@sci.kitasato-u.ac.p 1/11
in the continuous time limit. Such property leads to discrete equations with respect to a set of variables of one kind. On the other hand in 3 we discuss the discrete Toda lattice, the discrete KP equation and tropical R model, which are discrete maps of two sets of variables, which will be called here the second order discrete maps. The last example, the tropical R model, may be not familiar to physicists, since it is a rather mathematical model not clarified enough about its physical relevances. Here it is taken up only because it still has similar explicit formula using determinants of tri-diagonal matrices. The last section is devoted to summary and concluding remarks. 2. First Order Discrete Integrable Maps 2.1 Discrete Lotka-Volterra Equation The discrete Lotka-Volterra equation is given by 2) u n+1 u n = δ ( ) u n u n 1 u n+1 u n+1 +1, (1) where δ is the time interval and the periodic boundary condition u n +N = un for any integer and n is assumed. Equation (1) can be changed into a dimensionless form by setting x = δ u n and X = δ u n+1 as X x = x x 1 X X +1 or (1 + X +1 )X = (1 + x 1 )x. (2) We use hereafter the upper characters for the time step n+1 variables and the lower characters for the time step n variables because of the notational simplicity. For the case of N = 3 and N = 4, explicit solutions for (2) are given by Saito et al. 3), which are read for N = 3 and for N = 4 X 0 = x 0 1 + x 2 + x 2 x 1 1 + x 1 + x 1 x 0, X 1 = x 1 1 + x 0 + x 0 x 2 1 + x 2 + x 2 x 1, X 2 = x 2 1 + x 1 + x 1 x 0 1 + x 0 + x 0 x 2, X 0 = x 0 1 + x 3 + x 2 (1 + x 1 + x 3 ) 1 + x 2 + x 1 (1 + x 0 + x 2 ), X 1 = x 1 1 + x 0 + x 3 (1 + x 0 + x 2 ) 1 + x 3 + x 2 (1 + x 1 + x 3 ), X 2 = x 2 1 + x 1 + x 0 (1 + x 1 + x 3 ) 1 + x 0 + x 3 (1 + x 0 + x 2 ), X 3 = x 3 1 + x 2 + x 1 (1 + x 0 + x 2 ) 1 + x 1 + x 0 (1 + x 1 + x 3 ). Although they do not infer anymore, these results strongly suggest that for arbitrary N we can expect the solution having the form where the periodicity N+ = is implied. X = x +1, ( = 0, 1,, N 1) (3) Suppose that this is the case, then by substituting (3) into (2) we have N linear simultaneous equations for s (1 + x 2 ) 1 + + x +1 = 0, ( = 0, 1,, N 1) (4) where the periodicity x N+ = x is as well implied. Because the determinant of the coefficient 2/11
matrix is zero, we can conclude that 0,, N 1 are linearly dependent, and can be determined except an irrelevant common factor. In fact after some calculations we can derive the formula 1 x 1 (1 + x 0 ) 1 x 2 0 =, (5) (1 + x N 4 ) 1 x N 2 (1 + x N 3 ) 1 which is a determinant of (N 1) (N 1) tri-diagonal matrix. Other s are obtained by shifting the indices one by one cyclically. For example we have 1 x 2 (1 + x 1 ) 1 x 3 1 =. (6) (1 + x N 3 ) 1 x N 1 (1 + x N 2 ) 1 From this expression it is easy to confirm the above results of N = 3 and N = 4. Although the implicit difference equations (1) are not adequate for numerical computations, our explicit formulas (3) with (5) are easily applied for simulations by using standard algorithms to compute determinants of tri-diagonal matrices. For the sake of completeness we give a brief account of one of such algorithms in the appendix. Let us show an example of such numerical calculations. Figure 1 is a snapshot of simulations N = 300, δ = 0.01, where four solitons will be recognized clearly. Figure shows a situation when the highest soliton overtakes the other two, and chases the last one. Since we used as an initial condition the four soliton solution for continuous Lotka-Volterra equation of infinite system ( < < + ), we have small ripples around, because it is not anymore the exact soliton solution for discrete time Lotka-Volterra equation with periodic boundary. Since our difference equations preserve the integrability, our computations are guaranteed to be numerically stable, and that is in fact achieved. We have checked the accuracy of calculations by comparing at each step the conserved quantity N 1 =0 X = N 1 =0 x, which is the first one of series of conserved quantities. 3/11
Fig. 1. Discrete Lotka-Volterra equation. 2.2 Discrete KdV Equation Let us turn to the discrete KdV equation. While Hirota 4) gave firstly a discrete KdV equation in the form it can be modified as 2) W n (t + δ 2 ) 1 + W n (t + δ 2 ) W n(t δ 2 ) 1 + W n (t δ 2 ) = δ ( 1 1 + W n (t + δ 2 ) 1 1 + W n (t δ 2 ) = δ W n 1 2 ) (t) W n+ 1 (t) 2, (7) (W n+1 (t δ2 ) W n 1(t + δ2 ) ), (8) where the spatial lattice interval is changed from 1/2 to 1 that does not matter. Equation (8) can be written by setting y n = 1/(1 + W n (t δ/2)), Y n = 1/(1 + W n (t + δ/2)) such as ( 1 Y n y n = δ 1 ) or Y n + δ = y n + δ. (9) y n+1 Y n 1 Y n 1 y n+1 On the other hand Saito et al. 3) used the form X 1 X +1 = x 1 x 1, (10) which can be obtained by setting δ = 1 and n = in (9). This implies that (10) is spacetime reversal of (9), which can be approved since the KdV equation u t + uu x + u xxx = 0 itself has such symmetry. To summarize the procedure, the original equations (8) are changed into dimensionless form (10) by setting (δ < 0) x = 1 δ [ 1 + W (t δ 2 )], X = We use the form of (10) by Saito et al. hereafter. 1 [ δ 1 + W (t + δ (11) 2 )]. Saito et al. have examined again the solutions of (10) for N = 3. When we extend such examination to N = 4 and N = 5, we arrive at a guess for the form of solutions such as X = x +N 2 +1, (12) 4/11
where we can rewrite x +N 2 = x 2 due to the periodicity. Substituting (12) into (10) we have linear equations for s x 3 x 2 1 + (1 x 2 x 1 ) +1 = 0, (13) whose determinant of the coefficient matrix again vanishes. Therefore as before we arrive at the solution for 0 without a common factor, 1 x N 1 x 0 1 x N 1 x 0 1 x 0 x 1 1 0 =, (14) x N 5 x N 4 1 x N 4 x N 3 1 x N 4 x N 3 1 x N 3 x N 2 which is again a determinant of tri-diagonal matrix. Others can be written down by shifting the indices as before. We have made a renewed simulation of the original paper by Zabusky and Kruskal in 1965, which has led to the discovery of solitons. We set the initial condition W (0) = ɛ cos(2π/n) by choosing N = 300, δ = 0.1 and ɛ = 0.005. Figure.2 is a snapshot of W near the time when the eighth soliton is formed. The shape is almost the same as Zabusky-Kruskal s. Fig. 2. Discrete KdV equation. A difference of our simulation from the original one by Zabusky-Kruskal is that the continuous limit of our model contains a linear term u 0 u x, where u 0 is a constant depending on the scaling of W, t and x, which causes an additional shift of the wave by constant velocity u 0 to the right direction, which can be partially remedied by shifting back by certain amounts artifitially. 3. Second Order Discrete Integrable Maps 3.1 Discrete Toda Lattice Equation of Type I Difference equation for discrete time Toda lattice of type I is given by Hirota as 5) I n+1 V n+1 = I+1V n n, I n+1 I n = δ 2 (V 1 n+1 V n ), (15) 5/11
which can be reduced to continuous time Toda lattice equation by setting V = exp(q Q +1 ), I = 1 δp and by taking δ 0 limit. Equation (15) can be transformed by redefining δ 2 V n V n into I V = i +1 v, I + V 1 = i + v, (16) where we employed our notation using upper characters for the next step variables. Saito et al. have given N = 3 case solutions for this model too. When we extend their examination to N = 4 and N = 5 we arrive at solutions with the form V = v +1, I = i +1 +1. (17) When we subtitute (17) into (16), we find that the first equation becomes an identity, and the second equation gives linear equations for s v 1 + (i + v ) i +1 +1 = 0, (18) where we assume again the periodic boundary condition. Since the determinant of the coefficient matrix again vanishes, we obtain i 1 + v 1 i 2 v 1 i 2 + v 2 i 3 0 =, (19) v N 3 i N 2 + v N 2 i N 1 v N 2 i N 1 + v N 1 which is once again a determinant of tri-diagonal matrix. Others can be written down by shifting the indices as before. 3.2 Discrete KP Equation The discrete KP (Kadomtsev-Petviashvili) equation is given by 6, 3) X i, U i, = x i, u i+1,, X i, + U i,+1 = x i,+1 + u i+1,, (20) where periodic boundary conditions for direction, x i,+n = x i,, u i,+n = u i,, are assumed. Boundary condition for i direction can be arbitrary. We have as before a solution of the form X i, = x i, i, i,+1, U i, = u i+1, i,+1 i,. (21) When we substitute these into (20), the first equation becomes an identity and the second one becomes linear equations x i, 1 i, 1 + (x i, + u i+1, 1 ) i, u i+1, i,+1 = 0, ( = 0, 1,, N 1). (22) 6/11
And as before their solutions are given by x i,1 + u i+1,0 u i+1,1 x i,1 x i,2 + u i+1,1 u i+1,2 i,0 =, x i,n 3 x i,n 2 + u i+1,n 3 u i+1,n 2 x i,n 2 x i,n 1 + u i+1,n 2 (23) and other i, s are given by permutating the second indices cyclically. 3.3 Tropical R Model A very similar map with the discrete Toda equation, but a little bit different integrable map has recently attracted mathematicians interest, which is called tropical R model. 7, 8) It is given by X Y = x y, 1 + 1 = 1 + 1, ( = 1, 2,, N) (24) X Y +1 x y +1 where periodic boundary condition x N+ = x, y N+ = y etc. are assumed. By combinatorial procedure the solution of (24) is found in the form where periodic P s are given by P = X = y N N k=1 l=k P P 1, x +l l=1 Y = x P 1 P, (25) k y +l, ( = 1, 2,, N) (26) which is an order (N + 1) homogeneous polynomial. Since the signs of each monomial of P are positive (and equal to one), the model is called also as subtraction free (i.e. tropical) model. One can easily notice that P is by definition a product of x N+ y +1 = x y +1 and an order (N 1) primary (unfactorizable) polynomial. When we substitute (25) into (24), the first equation becomes an identity and the second one becomes linear equations If we introduce further by x y +1 y P 1 + (x + y +1 )P y +1 x x +1 P +1 = 0. (27) P = x y +1, (28) which is a product of an obvious factor before mentioned and the rest, we find that s are 7/11
again given by determinants of tri-diagonal matrices. In fact we have x x 1 + y 2 y 1 2 x 2 y x 3 x 2 y 2 x 2 + y 3 y 2 3 x 3 0 =, (29) y x N 1 x N 2 y N 2 x N 2 + y N 1 y N 2 N 1 x N 1 y x N N 1 x N 1 + y N y N 1 and others by changing the indices cyclically. Although above expression contains denominators in matrix elements, one can transform it into a simpler form x 1 + y 2 1 x 1 y 3 x 2 + y 3 1 0 = x N 3 y N 1 x N 2 + y N 1 1 x N 2 y N x N 1 + y N, (30) which is manifestly an order (N 1) polynomial. The obtained results agree with the known tropical polynomials (26). In other words we have derived another expression of P s in terms of the determinants of tri-diagonal matrices. It should be noted here that a similar determinant expression is obtained by Yamada, 9) where the case of discrete Toda equation is also discussed. The author thanks an anonymous referee for this information. 4. Summary and Concluding Remarks Explicit formulas of time developments for several integrable maps are obtained, which are originally written in implicit forms. All the examined models, such as the discrete Lotka- Volterra, the discrete KdV, the discrhete Toda lattice etc., have a common property that the explicit formulas are written as quotient of two determinants of tri-diagonal matrices. Effectiveness of these explicit formulas is satisfactorily demonstrated by numerical simulations. Some remarks are in order. The first one is on the paper by Saito et al. 3), which stimulated the present work. The main purpose of their work was to examine the q-deformation of the discrete maps. Although we have not concerned with any deformation in this paper, we can expect that our determinant expression survives also when q-deformation is considered, which will be discussed in future publication. The second one is on Hirota s dependent variables. Let us take the discrete Lotka-Volterra equation for an example. The variable x n variables f n s as (we write time step n now) is written by Hirota x n = a f 1 n f +2 n+1 f nf +1 n+1, (31) 8/11
where we put a constant a defined by the conserved quantity N 1 =0 xn = an. Then, if we assume a = 1 for the simplicity, eqs. (2) become f n f n+1 +1 + f n 1f n+1 +2 = 2f n +1f n+1, ( = 0, 1,, N 1) (32) which are the Hirota s bilinear equations. Then the assumption a = 1 is so restrictive that it makes the map for N = 3 periodic. For example x 0 transforms like x 0 x 1 1 x 2 x 1 0 x 1 x 1 2 x 0, (33) thus returns itself after six steps. For larger N there is no such exact periodical behavior although they change values quasi periodically. The third remark is on geometrical implications of discrete maps with periodic boundary. When we view eqs. (32) as linear equations for f0 n+1,, f n+1 N 1, for the existence of non-trivial solutions the determinant of coefficient matrix must vanish. For the case of N = 3 it becomes f 3 0 + f 3 1 + f 3 2 3f 0f 1 f 2 = 0, which is an algebraic surface in three dimensional space. For general N it is an algebraic hypersurface in N dimensional space. And the map gives an discrete automorphism of such surface. It may be interesting to consider the discrete maps from such viewpoint. The last remark is on exact solutions of the discrete maps. Since we know that the soliton solutions of continuous time are written by hyperelliptic or Riemann s theta functions, the discrete map can be interpreted as an addition formula for such functions. The author has discussed it on the discrete sine-gordon model. 10) And according to some evidences in the second order maps the number of solitons p which is equal to the genus of the corresponding algebraic curve seems to satisfy 0 p N 1 depending on degeneracy or ramification of the curve. These problems will open new perspective in soliton theory. Appendix Here we summarize briefly an algorithm to compute the determinant of tri-diagonal M M matrix d 0 f 0 e 1 d 1 f 1 A =. (34) e M 2 d M 2 f M 2 e M 1 d M 1 Our method is due to the so called LU decomposition. Suppose that the matrix A is decomposed as a product A = LU with lower triangle matrix L and upper triangle matrix U such 9/11
as 1 0 a 0 u 0 l 1 1 0 0 a 1 u 1 L =, U = l M 2 1 0 0 a M 2 u M 2 l M 1 1 0 a M 1 Comparing matrix elements of both sides of A = LU we have. (35) d 0 = a 0, f 0 = u 0, e = l a 1, d = l u 1 + a, f = u, ( = 1, 2,, M 2) (36) e M 1 = l M 1 a M 2, d M 1 = l M 1 u M 2 + a M 1, which can be solved iteratively as a 0 = d 0, u 0 = f 0, l = e /a 1, a = d l u 1, u = f, ( = 1, 2,, M 2) (37) l M 1 = e M 1 /a M 2, a M 1 = d M 1 l M 1 u M 2. Then the determinant A is readily computed as A = a 1 a 2 a M 1, (38) since L = 1 and U = a 1 a 2 a M 1. 10/11
References 1) N.J. Zabusky and M.D. Kruskal. Phys. Rev. Lett. 15 (1965) 240. 2) R. Hirota and S. Tsuimoto, J. Phys. Soc. Jpn 64 (1995) 3125. 3) S. Saito, N. Saitoh, J. Yamamoto and K. Yoshida, J. Phys. Soc. Jpn 70 (2001) 3517. 4) R. Hirota, J. Phys. Soc. Jpn 43 (1977) 1424. 5) R. Hirota, J. Phys. Soc. Jpn 43 (1977) 2074. 6) R. Hirota, S. Tsuimoto and T. Imai, Difference Scheme of Soliton Equations, in Future Directions of Nonlinear Dynamics in Physical and Biological Systems, ed. by P.L. Christiansen et al. (Plenum Press, New York, 1993) p. 7. 7) K. Kaiwara, M. Noumi and Y. Yamada, Lett. Math. Phys. 60 (2002) 211. 8) A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Commun. Math. Phys. 245 (2004) 491. 9) Y. Yamada, Physics and Combinatorics 2000 ed. by A. N. Kirillov and N. Liskova, Proceedings of the Nagoya 2000 International Workshop, (World Scientific, Singapore, 2001) p. 305. 10) K. Sogo, J. Phys. Soc. Jpn 75 (2006) to be published. 11/11