N-soliton solutions of two-dimensional soliton cellular automata

Transcription

1 N-soliton solutions of two-dimensional soliton cellular automata Kenichi Maruno Department of Mathematics, The University of Texas - Pan American Joint work with Sarbarish Chakravarty (University of Colorado) International Workshop on Nonlinear and Modern Mathematical Physics, Beijing, China July 15-21, 2009 K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

2 1 Soliton Cellular Automata and Ultradiscretization 2 KP equation, τ -function and Wronskian 3 Ultradiscrete Soliton Equations and Total Non-Negativity 4 Grammian and Wronskian K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

3 Motivation Soliton solutions of continuous and discrete soliton equations Grammian Wronskian Hirota form (perturbation form) Ultradiscretization????? Q1: Ultradiscrete (tropical) analogue of Wronskian and Grammian including all types of line soliton solutions? Q2: Grammian form for general line soliton solutions for 2-dimensional soliton systems?? K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

4 Soliton Cellular Automata Soliton Cellular Automata (SCA)= Box and Ball System D. Takahashi & J. Satsuma (1989) J. Phys. Soc. Japan K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

5 Soliton Cellular Automata 3 soliton interaction T t+1 j j 1 = max(tj t 1, (T t+1 i i= T t i )) K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

6 Ultradiscretization T.Tokihiro, D.Takahashi, J.Matsukidaira, J.Satsuma: Phys. Rev. Lett. 76 (1996) The relationship between Soliton equations and Soliton Cellular Automata Key formula lim ɛ ɛ 0+ ln(ea/ɛ + e B/ɛ ) = max(a, B) KdV eq. semi-discrete KdV eq. discrete KdV eq. SCA space discretization time discretization ultradiscretization K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

7 Ultradiscretization KdV equation V t + 6V V x + V xxx = 0 Space discretization Semi-discrete KdV (Lotka-Volterra) equation dvn = v dt n (v n 1 v n+1 ) Time discretization Discrete KdV (discrete Lotka-Volterra) equation u t+1 n u t n δ = u t n ut n 1 ut+1 n ut+1 n+1 Ultradiscretization Ultradiscrete KdV equation Un t+1 Un t = max(0, U n 1 t t+1 1) max(0, Un+1 1) Un t = n+1 j= T j t n+1 j= T t+1 j Box-Ball system T t+1 j = max(t t j 1, j 1 i= (T t+1 i T t i )) K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

8 Ultradiscretization 1 soliton solution of the Toda lattice a + b max(a, B), ab A + B K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

9 Question How to obtain exact solutions? Take the ultradiscrete limit of exact solutions of difference equations. However, we can take ultradiscrete limit of exact solutions only in which all terms are non-negative! It is not easy to find which type of exact solutions exists in ultradiscrete limit. Many works use the Hirota form (which is equivalent to Gram type) of N-soliton solutions. Some soliton solutions were missed in previous studies! K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

10 Question R. Hirota & D. Takahashi(J.Phys.Soc.Japan, 2007): They proposed a new type of solutions of ultradiscrete soliton equations. It is similar to Permanent. Q1. Show the root of permanent type solutions in ultradiscrete soliton systems. Derive permanent type solutions systematically from the Determinant solutions. Q2. Systematic method to construct simple forms of soliton solutions of 2D and 1D ultradiscrete soliton equations. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

11 Determinant and Permanent Determinant Permanent where det(a) = sgn(σ) σ S n perm(a) = n σ S n i=1 n i=1 a i σ(i) a i σ(i) A = (a ij ) 1 i,j N is a matrix. e.g. 2 2-matrix det(a) = a 11 a 22 a 12 a 21, perm(a) = a 11 a 22 + a 12 a 21. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

12 Permanent-type solution of Ultradiscrete Systems Ultradiscrete Lotka-Volterra (ultradiscrete KdV) equation Vn m+1 Vn m = max(0, V n 1 m m+1 1) max(0, Vn+1 1) Vn m = F n 1 m + F m+1 n+2 F n m F m+1 n+1 Note F m n = 1 2 max[ s i(m + 2(j 1), n) ] 1 i,j N ( n ) max[a ij ] = max a i,σ(i) σ S n i=1 lim ɛ ɛ 0+ ln(perm[ea ij /ɛ ]) = max[a ij ] Does this solution exist only in ultradiscrete systems??? K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

13 Hirota Bilinear form, τ -function KP (Kadomtsev-Petviashvili) equation x Transformation Hirota bilinear form ( 4 u ) t + 3 u x + 6u u u 3 x y = 0. 2 u(x, y, t) = 2 2 log τ (x, y, t), x2 [ 4D x D t + D 4 x + 3D2 y ]τ τ = 0, D m x f g = ( x x ) m f(x, y, t)g(x, y, t) x=x. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

14 τ = Hirota Bilinear form, τ -function f (0) 1 f (0) N..... f (N 1) 1 f (N 1) N, f (n) i := n x n f i. where f i (x, y, t) is a set of M linearly independent solutions of the linear equations f i y = 2 f i x, f i 2 t = 3 f i x, 3 for 1 i N. A finite dimensional solutions: f i (x, y, t) = M a ij E j (x, y, t), i = 1,, N < M, j=1 E j (x, y, t) := e θ j = exp(k j x + k 2 j y + k3 j t + θ0 j ). K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

15 A-matrix determines the type of soliton & non- KP τ -function negativity Hirota Bilinear form, τ -function τ (x, y, t) = det(aθk) = V m1,...,m N A m1,...,m N exp (θ m1,,m N ), 1 m 1 < <m N M A = (a n,m ) is the N M coefficient matrix, Θ = diag(e θ 1,, e θ M ), the M N matrix K is given by K = (km n 1), θ m1,,m N (x, y, t) = θ m1 + + θ mn, k 1 < k 2 < < k M V m1,...,m N is the vandermonde determinant V m1,...,m N = 1 j<l N (k m j k ml ), and A m1,...,m N is determined by columns the N N-minor of A. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

16 Hirota Bilinear form, τ -function A-matrix for non-singular soliton solution τ -function is totally non-negative if A-matrix is the following: ( ) ( A O =, A P = A T = ( ) ( ) ( ) A I = a 1 0 a a, A II =, ( ) ( ) A III = a 1 0 a 1, A IV = , Y. Kodama (2004), G. Biondini & S. Chakravarty (2006), S. Chakravarty & Y. Kodama (2008), K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33 ),

17 O-type (2143) P-type (4321) T-type (3412) K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

18 The 2-dimensional Toda lattice 2 x t Q n(x, t) = e Q n +1(x,t) 2e Q n (x,t) + e Q n 1(x,t) 2 τ n x t τ n τ n τ n t x = τ n+1 τ n 1 τn 2 ) Q n (x, t) = log (1 + 2 x t log τ n(x, t). K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

19 Discretization of 2D Toda lattice + l m Q l,m,n = V l,m 1,n+1 V l+1,m 1,n V l,m,n + V l+1,m,n 1, V l,m,n = (δκ) 1 log[1 + δκ (exp Q l,m,n 1)], + l f l,m,n = f l+1,m,n f l,m,n, m δ f l,m,n = f l,m,n f l,m 1,n. κ ( + l m τ l,m,n) τ l,m,n ( + l τ l,m,n) m τ l,m,n = τ l,m 1,n+1 τ l+1,m,n 1 τ l+1,m 1,n τ l,m,n, V l,m,n = + l m log τ l,m,n, Q l,m,n = log τ l+1,m+1,n 1τ l,m,n+1 τ l+1,m,n τ l,m+1,n. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

20 Ultradiscretization of 2D Toda lattice Using lim ɛ 0+ ɛ ln(e A/ɛ + e B/ɛ ) = max(a, B), we can create the ultradiscrete 2D Toda lattice + l + m v l,m,n = max(0, v l,m,n r s), f l,m,n f l+1,m+1,n 1 + f l,m,n+1 f l+1,m,n f l,m+1,n. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

21 Construction of soliton solutions of ultradiscrete 2D Toda lattice The past research : Some special cases. Take ultradiscrete limit of some special soliton solutions. The solution is not in determinant or permanent. Idea Determinant solution of fully discrete 2D Toda ultradiscretization Determinant-tye solution of ultradiscrete 2D Toda? KM & G. Biondini (2004): resonant type determinant-type (actually, permanent-type) solution Difficulty: Ultradiscretization is available only in which all terms in τ -function are non-negative. (Resonant-type) determinant solution has total non-negativity. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

22 Soliton solution of ultradiscrete 2D Toda Theorem The tau-function of N line soliton solutions of the ultradiscrete 2D Toda lattice, i.e. the tropical tau-function, is ρ(l, m, n) = lim ɛ 0 + ɛ log τ ɛ l,m,n = lim ɛ 0 + ɛ log AΦ ɛ K ɛ where A = (a ij ) 1 i N,1 j M, Φ ɛ = diag(φ 1, φ 2,..., φ M ), φ j = k n j (1 + δk j) l (1 + κk 1 j ) m φ j,0, k j = e K j /ɛ, δ = e r/ɛ, κ = e s/ɛ, φ j,0 = e ψ j, 0/ɛ and K ɛ = (k j 1 i ) 1 i,j N. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

23 Soliton solution of ultradiscrete 2D Toda Moreover, the tropical tau-function ρ(l, m, n) is a tropical determinant ρ(l, m, n) = AΦ trop K trop trop where Φ trop, K trop are tropical matrices and trop is a tropical determinant, these are obtained by taking ultradiscrete limit of matrices Φ ɛ, K ɛ and a determinant. Here a matrix A must be chosen for satsfying that each term of tau-function τl,m,n ɛ is non-negative. In other words, the tropical τ -function can be considered as ultradiscrete limit of permanent if an appropriate A is chosen because all terms are non-negative. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

24 Soliton solution of ultradiscrete 2D Toda Theorem By using Binet-Cauchy formula, a tropical tau-function is expressed in the form of N ρ(l, m, n) = max φ(h 1,, h N ) + (j 1)K hj where the phase combination is defined by φ(h 1,, h N ) = N, the phase is j=1 φtrop h j φ trop j = nk j + l max(0, K j r) m max(0, K j s) + ψ j,0. Note that each term in the tropical tau-function ρ corresponds to non-zero minor A(h 1, h 2,, h N ) which denotes the N N minor of a matrix A obtained by selecting columns h 1, h 2,, h N. j=1 K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

25 Soliton interaction of ultradiscrete 2D Toda 2-soliton (T-type, (3412)) A = ( ), K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

26 Soliton interaction of ultradiscrete 2D Toda 2-soliton (O-type, (2143)) A = ( ), K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

27 Soliton interaction of ultradiscrete 2D Toda 2-soliton (P-type, (4321)) A = ( ), K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

28 Grammian Question: Are there general line soliton solutions in Grammian form? Grammian solution for the KP equation τ = det(i + BF ˆB T ) = det(i + CF ), where B is an N r matrix and ˆB is an N N matrix, C = ˆB T B is an N r matrix of constant coefficients and F is an r N matrix whose entries are given by ) φ(qn ) eφ(pm F mn =, (m = 1, 2,..., r, n = 1, 2,..., N). p m q n The O-type N-soliton solution is obtained by setting r = N and C = I. In this case the resulting τ -function τ = det(i + F ) is positive for all x, y, t if the parameters are ordered as p N < p N 1 <... < p 1 < q 1 < q 2 <... < q N. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

29 Grammian form of general line soliton solutions Wronskian form τ = det(aek), E = diag(e θ i ) M i=1, K is vandermonde matrix, A is N M coefficient matrix. Grammian form of general line soliton solutions ˆτ = det(i + JÊ 2 χê 1 1 ), with Ê 1 = E 1 D 1 = diag(e ˆφ(q i ) ) N i=1 and Ê 2 = E 2 D 2 = diag(e ˆφ(p i ) ) M N 1 i=1, (χ) ij = p i q j for i = 1,..., M N, j = 1,..., N. D 1 = diag( j=1,j i (q i q j )) N i=1, D 2 = diag( j=1 (p i q j )) M N i=1, E 1 = diag(e φ(q i ) ) N i=1, E 2 = diag(e φ(p i ) ) M N i=1, A = (I, J)P, I is N N submatrix of the pivot columns of A, J is N (M N) submatrix of the non-pivot columns of A, P is M M permutation matrix. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

30 Grammian:Proof τ = AEK = (I, J)P EK = (I, J)(P EP 1 )P K ( ) ( ) E 1 0 K 1 = (I, J) 0 E 2 = IE 1K 1 + JE 2 K 2 K 2 = E 1 K 1 (I + JE 2 K 2 K 1 1 E 1 1 ) = E 1K 1 ˆτ ˆτ = I + JE 2 K 2 K 1 1 E 1 1 ), E 1, E 2 are respectively N N and (M N) (M N) block diagonal matrices whose elements are permutations of the set {e θ 1, e θ 2,, e θ M }. K 1, K 2 are respectively N N and (M N) (M N) matrices obtained by permuting the rows of the vandermonde matrix K by P. P induces a permutation π of the ordered set {k 1,..., k M }: π({k 1, k 2,..., k M }) = {q 1, q 2,..., q N, p 1, p 2,..., p M N }, K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

31 Grammian:Proof Using a formula we obtain K 2 K 1 1 = D 2 χd 1 1, ˆτ = I + JE 2 K 2 K 1 1 E 1 1 = I + JE 2 D 2 χd 1 1 E 1 1 = I + JÊ 2 χê 1 1. where (χ) ij = 1 p i q j for i = 1,..., M N, j = 1,..., N, D 1 = diag( j=1,j i (q i q j )) N i=1, D 2 = diag( j=1 (p i q j )) M N i=1, K 1 = (q j 1 i ), K 2 = (p j 1 i ), E 1 = diag(e φ(q i ) ) N i=1, E 2 = diag(e φ(p i ) ) M N i=1. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

32 Grammian and Wronskian type solutions We can also take ultradiscrete limit of Grammian solution. This will give another form of soliton solutions of ultradiscrete systems. Soliton solutions of 1D soliton cellular automata (CA) are obtained by using the reduction technique from line soliton solutions of ultradiscrete 2D-Toda or KP. We can recover known solutions for 1D soliton CA. It is easy to find possible soliton-type solutions of ultradiscrete soliton equations if we start from our formula. K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33

33 Conclusions We proposed a systematic method to obtain soliton solutions of ultra-discrete soliton systems. Determinant solutions of discrete integrable systems lead to tropical determinant solutions (permanent-type solution) in ultradiscrete limit. Why do soliton solutions look like permanent in ultradiscrete? Because of loss of vandermonde determinant in ultradiscrete limit. We can also do same computation in the ultradiscrete KP equation. N-soliton solutions of 1D soliton cellular automata can be obtained from solutions of ultradiscrete 2D Toda and KP equations. We found the corresspondence between Wronskian form and Grammian form of general line soliton solutions. This correspondence also survive in the ultradiscrete systems. B-type, C-type, D-type 2D Cellular Automata? K.Maruno (UTPA) Soliton Cellular Automata July 15-21, / 33