Symbolic Computation of Lax Pairs of Systems of Partial Difference Equations Using Consistency Around the Cube

Size: px
Start display at page:

Download "Symbolic Computation of Lax Pairs of Systems of Partial Difference Equations Using Consistency Around the Cube"

Transcription

1 Symbolic Computation of Lax Pairs of Systems of Partial Difference Equations Using Consistency Around the Cube Willy Hereman Department of Applied Mathematics and Statistics Colorado School of Mines Golden, Colorado, U.S.A. home/whereman/ Applied Mathematics Mathematical Sciences Stellenbosch University, Stellenbosch, South Africa Thursday, April 24, 2014, 2:00p.m.

2 Collaborators Reinout Quispel & Peter van der Kamp La Trobe University, Melbourne, Australia Terry Bridgman (Ph.D. student, CSM) Research supported in part by NSF under Grant CCF This presentation is made in TeXpower

3 Outline What are nonlinear P Es? Classification of integrable nonlinear P Es in 2D Lax pairs of nonlinear PDEs Lax pair of nonlinear P Es Examples of Lax pair of nonlinear P Es Algorithm (Nijhoff 2001, Bobenko & Suris 2001) Software demonstration Additional examples Conclusions and future work

4 What are nonlinear P Es? Nonlinear maps with two (or more) lattice points Some origins: full discretizations of PDEs discrete dynamical systems in 2 dimensions superposition principle (Bianchi permutability) for Bäcklund transformations between 3 solutions (2 parameters) of a completely integrable PDE Example: discrete potential Korteweg-de Vries (pkdv) equation (u n,m u n+1,m+1 )(u n+1,m u n,m+1 ) p 2 + q 2 = 0

5 u is dependent variable or field (scalar case) n and m are lattice points p and q are parameters Notation: (u n,m, u n+1,m, u n,m+1, u n+1,m+1 ) = (x, x 1, x 2, x 12 ) Alternate notations (in the literature): (u n,m, u n+1,m, u n,m+1, u n+1,m+1 ) = (u, ũ, û, ˆũ) (u n,m, u n+1,m, u n,m+1, u n+1,m+1 ) = (u 00, u 10, u 01, u 11 ) Example: discrete pkdv equation (u n,m u n+1,m+1 )(u n+1,m u n,m+1 ) p 2 + q 2 = 0 Short: (x x 12 )(x 1 x 2 ) p 2 + q 2 = 0

6 (u n,m u n+1,m+1 )(u n+1,m u n,m+1 ) p 2 + q 2 = 0 Short: (x x 12 )(x 1 x 2 ) p 2 + q 2 = 0 x 2 = u n,m+1 p x 12 = u n+1,m+1 q q x = u n,m p x 1 = u n+1,m

7 Systems of P Es Example: Schwarzian-Boussinesq System z 1 y x 1 + x = 0 z 2 y x 2 + x = 0 z y 12 (y 1 y 2 ) y(p y 2 z 1 q y 1 z 2 ) = 0 Note: System has two single-edge equations and one full-face equation.

8 Concept of Consistency Around the Cube Superposition of Bäcklund transformations between 4 solutions x, x 1, x 2, x 3 (3 parameters: p, q, k) x 23 x 123 x 2 q k x 3 x 12 x 13 x p x 1

9 Introduce a third lattice variable l View u as dependent on three lattice points: n, m, l. So, x = u n,m x = u n,m,l Move in three directions: n n + 1 over distance p m m + 1 over distance q l l + 1 over distance k (spectral parameter) Require that the same lattice holds on the front, bottom, and left faces of the cube Require consistency for the computation of x 123 = u n+1,m+1,l+1 (3 ways same answer)

10 x 23 x 123 x 2 q k x 3 x 12 x 13 x p x 1

11 Verification of Consistency Around the Cube Example: discrete pkdv equation Equation on front face of cube: (x x 12 )(x 1 x 2 ) p 2 + q 2 = 0 Solve for x 12 = x p2 q 2 x 1 x 2 Compute x 123 : x 12 x 123 = x 3 p2 q 2 x 13 x 23 Equation on floor of cube: (x x 13 )(x 1 x 3 ) p 2 + k 2 = 0 Solve for x 13 = x p2 k 2 x 1 x 3 Compute x 123 : x 13 x 123 = x 2 p2 k 2 x 12 x 23

12 Equation on left face of cube: (x x 23 )(x 3 x 2 ) k 2 + q 2 = 0 Solve for x 23 = x q2 k 2 x 2 x 3 Compute x 123 : x 23 x 123 = x 1 q2 k 2 x 12 x 13 Verify that all three coincide: x 123 =x 1 q2 k 2 x 12 x 13 =x 2 p2 k 2 x 12 x 23 =x 3 p2 q 2 Upon substitution of x 12, x 13, and x 23 : x 13 x 23 x 123 = p2 x 1 (x 2 x 3 ) + q 2 x 2 (x 3 x 1 ) + k 2 x 3 (x 1 x 2 ) p 2 (x 2 x 3 ) + q 2 (x 3 x 1 ) + k 2 (x 1 x 2 ) Consistency around the cube is satisfied!

13 Tetrahedron property x 123 = p2 x 1 (x 2 x 3 ) + q 2 x 2 (x 3 x 1 ) + k 2 x 3 (x 1 x 2 ) p 2 (x 2 x 3 ) + q 2 (x 3 x 1 ) + k 2 (x 1 x 2 ) is independent of x. Connects x 123 to x 1, x 2 and x 3. x 23 x 123 x 2 x 3 x 12 q k x 13 x p x 1

14 Classification of 2D scalar nonlinear P Es Adler, Bobenko, Suris (ABS) 2003, 2007 Consider a family P Es: Q(x, x 1, x 2, x 12 ; p, q) = 0 Assumptions (ABS 2003): 1. Affine linear Q(x, x 1, x 2, x 12 ; p, q) = a 1 xx 1 x 2 x 12 + a 2 xx 1 x a 14 x 2 + a 15 x 12 + a Invariant under D 4 (symmetries of square) Q(x, x 1, x 2, x 12 ; p, q) = ɛq(x, x 2, x 1, x 12 ; q, p) ɛ, σ = ±1 3. Consistency around the cube = σq(x 1, x, x 12, x 2 ; p, q)

15 Result of the ABS Classification List H H1 (x x 12 )(x 1 x 2 ) + q p = 0 H2 (x x 12 )(x 1 x 2 )+(q p)(x+x 1 +x 2 +x 12 )+q 2 p 2 =0 H3 p(xx 1 + x 2 x 12 ) q(xx 2 + x 1 x 12 ) + δ(p 2 q 2 ) = 0

16 List A A1 p(x+x 2 )(x 1 +x 12 ) q(x+x 1 )(x 2 +x 12 ) δ 2 pq(p q) = 0 A2 (q 2 p 2 )(xx 1 x 2 x ) + q(p 2 1)(xx 2 + x 1 x 12 ) p(q 2 1)(xx 1 + x 2 x 12 ) = 0

17 List Q Q1 p(x x 2 )(x 1 x 12 ) q(x x 1 )(x 2 x 12 )+δ 2 pq(p q) = 0 Q2 p(x x 2 )(x 1 x 12 ) q(x x 1 )(x 2 x 12 )+pq(p q) (x+x 1 +x 2 +x 12 ) pq(p q)(p 2 pq+q 2 )=0 Q3 (q 2 p 2 )(xx 12 +x 1 x 2 )+q(p 2 1)(xx 1 +x 2 x 12 ) p(q 2 1)(xx 2 +x 1 x 12 ) δ2 4pq (p2 q 2 )(p 2 1)(q 2 1)=0

18 Q4 (mother) Hietarinta s Parametrization sn(α + β; k) (x 1 x 2 + xx 12 ) sn(α; k) (xx 1 + x 2 x 12 ) sn(β; k) (xx 2 + x 1 x 12 ) + sn(α; k) sn(β; k) sn(α + β; k)(1 + k 2 xx 1 x 2 x 12 )=0 where sn(α; k) is the Jacobi elliptic sine function with modulus k. Other parametrizations (Adler, Nijhoff, Viallet) are given in the literature.

19 Peter D. Lax (1926-) Seminal paper: Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21 (1968)

20 Refresher: Lax Pairs of Nonlinear PDEs Historical example: Korteweg-de Vries equation u t + α uu x + u xxx = 0 α IR Key idea: Replace the nonlinear PDE with a compatible linear system (Lax pair): ( ) ψ xx αu λ ψ = 0 ψ t + 4ψ xxx + αuψ x αu xψ = 0 ψ is eigenfunction; λ is constant eigenvalue (λ t = 0) (isospectral)

21 Lax Pairs in Operator Form Replace a completely integrable nonlinear PDE by a pair of linear equations (called a Lax pair): Lψ = λψ and D t ψ = Mψ Require compatibility of both equations L t ψ + (LM ML)ψ = 0 Lax equation: L t + [L, M] = O with commutator [L, M] = LM ML. Furthermore, L t ψ = [D t, L]ψ = D t (Lψ) LD t ψ and = means evaluated on the PDE

22 Example: Lax operators for the KdV equation L = D 2 x αu I M = ( ) 4 D 3 x + αud x αu x I Note: L t ψ + [L, M]ψ = 1 6 α (u t + αuu x + u xxx ) ψ

23 Alternate Operator Formulations Define L = L λ I and M = M Dt Then, the Lax pair becomes Lψ = 0 and Mψ = 0 and the Lax equation becomes [ L, M] = O Challenge: Find linear commuting operators modulo the (nonlinear) PDE If S is an arbitrary invertible operator, then ˆL = SLS 1, ˆM = SMS 1, ˆDt = SD t S 1 satisfy ˆL t + [ ˆL, ˆM] = O

24 Lax Pairs in Matrix Form (AKNS Scheme) Express compatibility of D x Ψ = X Ψ where Ψ = ψ 1 ψ 2. ψ N D t Ψ = T Ψ, X and T are N N matrices Lax equation (zero-curvature equation): D t X D x T + [X, T] = 0 with commutator [X, T] = XT TX

25 Example: Lax pair for the KdV equation 0 1 X = λ 1 6 αu 0 T = 1 6 αu x 4λ 1 3 αu 4λ αλu α2 u αu 2x 1 6 αu x Substitution into the Lax equation yields D t X D x T + [X, T] = 1 6 α 0 0 u t + αuu x + u 3x 0

26 Equivalence under Gauge Transformations Lax pairs are equivalent under a gauge transformation: If (X, T) is a Lax pair then so is ( X, T) with X = GXG 1 + D x (G)G 1 T = GTG 1 + D t (G)G 1 G is arbitrary invertible matrix and Ψ = GΨ. Thus, X t T x + [ X, T] = 0

27 Example: For the KdV equation 0 1 X = λ 1 6 αu 0 and X = ik 1 6 αu 1 ik Here, X = GXG 1 and T = GTG 1 with where λ = k 2 G = i k 1 1 0

28 Reasons to Compute a Lax Pair Compatible linear system is the starting point for application of the IST and the Riemann-Hilbert method for boundary value problems Confirm the complete integrability of the PDE Zero-curvature representation of the PDE Compute conservation laws of the PDE Discover families of completely integrable PDEs Question: How to find a Lax pair of a completely integrable PDE? Answer: There is no completely systematic method

29 Lax Pair of Nonlinear P Es Replace the nonlinear P E by ψ 1 = L ψ (recall ψ 1 = ψ n+1,m ) ψ 2 = M ψ (recall ψ 2 = ψ n,m+1 ) For scalar P Es, L, M are 2 2 matrices; ψ = f Express compatibility: g ψ 12 = L 2 ψ 2 = L 2 M ψ ψ 12 = M 1 ψ 1 = M 1 L ψ Lax equation: L 2 M M 1 L = 0

30 Equivalence under Gauge Transformations Lax pairs are equivalent under a gauge transformation If (L, M) is a Lax pair then so is (L, M) with L = G 1 L G 1 M = G 2 M G 1 where G is non-singular matrix and φ = Gψ Proof: Trivial verification that (L 2 M M 1 L) φ = 0 (L 2 M M 1 L) ψ = 0

31 Example 1: Discrete pkdv Equation (x x 12 )(x 1 x 2 ) p 2 + q 2 = 0 Lax pair: L = tl c = t x p2 k 2 xx 1 1 x 1 M = sm c = s x q2 k 2 xx 2 1 x 2 with t = s = 1 or t = 1 DetLc = 1 k 2 p 2 and s = 1 DetMc = 1. Here, t 2 k 2 q 2 t s s 1 = 1.

32 Example 2: Schwarzian-Boussinesq System z 1 y x 1 + x = 0 z 2 y x 2 + x = 0 z y 12 (y 1 y 2 ) y(p y 2 z 1 q y 1 z 2 ) = 0 Lax pair: L = tl c = t y 0 yz 1 kyy 1 pyz 1 0 z z 0 1 y 1

33 y 0 yz 2 M = sm c = s kyy 2 qyz 2 0 z z 0 1 y 2 with t = s = 1 y, or t = 1 y 1 and s = 1 y 2, or t = 3 z y 2 y 1 z 1 and s = z 3 y 2 y 2 z 2. Here, t 2 t s s 1 = y 1 y 2.

34 Lax Pair Algorithm for Scalar P Es (Nijhoff 2001, Bobenko and Suris 2001) Applies to P Es that are consistent around the cube Example: Discrete pkdv Equation Step 1: Verify the consistency around the cube Use the equation on floor (x x 13 )(x 1 x 3 ) p 2 + k 2 = 0 to compute x 13 = x p2 k 2 x 1 x 3 = x 3x xx 1 +p 2 k 2 x 3 x 1 Use the equation on left face (x x 23 )(x 3 x 2 ) k 2 + q 2 = 0 to compute x 23 = x q2 k 2 x 2 x 3 = x 3x xx 2 +q 2 k 2 x 3 x 2

35 x 23 x 123 x 2 x 12 q x 3 x 13 k x p x 1

36 Step 2: Homogenization Numerator and denominator of x 13 = x 3x xx 1 +p 2 k 2 x 3 x 1 and x 23 = x 3x xx 2 +q 2 k 2 x 3 x 2 are linear in x 3. Substitute x 3 = f g x 13 = xf+(p2 k 2 xx 1 )g f x 1 g into x 13 to get On the other hand, x 3 = f g x 13 = f 1 g 1. Thus, x 13 = f 1 g 1 = xf+(p2 k 2 xx 1 )g f x 1 g. Hence, f 1 = t ( xf + (p 2 k 2 xx 1 )g ) and g 1 = t (f x 1 g)

37 In matrix form f 1 = t g 1 x p2 k 2 xx 1 1 x 1 Matches ψ 1 = L ψ with ψ = f g f g Similarly, from x 23 = f 2 g 2 = xf+(q2 k 2 xx 2 )g f x 2 g ψ 2 = f 2 = s g 2 x q2 k 2 xx 2 1 x 2 f = M ψ. g

38 Therefore, L = t L c = t x p2 k 2 xx 1 1 x 1 M = s M c = s x q2 k 2 xx 2 1 x 2

39 Step 3: Determine t and s Substitute L = t L c, M = s M c into L 2 M M 1 L = 0 t 2 s (L c ) 2 M c s 1 t (M c ) 1 L c = 0 Solve the equation from the (2-1)-element for t 2t s s 1 = f(x, x 1, x 2, p, q,... ) Find rational t and s. Apply determinant to get t 2 s det L c = t s 1 det (L c ) 2 det (M c ) 1 det M c Solution: t = 1 det Lc, s = 1 det Mc Always works but introduces roots!

40 The ratio t 2 t s s 1 is invariant under the change t a 1 a t, s a 2 a s, where a(x) is arbitrary. Proper choice of a(x) = Rational t and s. No roots needed!

41 Lax Pair Algorithm Systems of P Es (Nijhoff 2001, Bobenko and Suris 2001) Applies to systems of P Es that are consistent around the cube Example: Schwarzian-Boussinesq System z 1 y x 1 + x = 0 z 2 y x 2 + x = 0 z y 12 (y 1 y 2 ) y(p y 2 z 1 q y 1 z 2 ) = 0 Note: System has two single-edge equations and one full-face equation

42 Edge equations require augmentation of system with additional shifted, edge equations z 12 y 2 x 12 + x 2 = 0 z 12 y 1 x 12 + x 1 = 0 Edge equations will provide additional constraints during homogenization (Step 2).

43 Step 1: Verify the consistency around the cube System on the front face: z 1 y x 1 + x = 0 z 2 y x 2 + x = 0 z 12 y 2 x 12 + x 2 = 0 z 12 y 1 x 12 + x 1 = 0 zy 12 (y 1 y 2 ) y(py 2 z 1 qy 1 z 2 ) = 0

44 Solve for x 12, y 12, and z 12 : x 12 = x 2y 1 x 1 y 2 y 1 y 2 y 12 = (x 2 x 1 )(qy 1 z 2 py 2 z 1 ) z(y 1 y 2 )(z 1 z 2 ) z 12 = x 2 x 1 y 1 y 2 Compute x 123, y 123, and z 123 : x 123 = x 23y 13 x 13 y 23 y 13 y 23 y 123 = py 23y 3 z 13 qy 13 y 3 z 23 z 3 (y 13 y 23 ) z 123 = x 23 x 13 y 13 y 23

45 System on the bottom face: z 1 y x 1 + x = 0 z 3 y x 3 + x = 0 zy 13 (y 1 y 3 ) y(py 3 z 1 ky 1 z 3 ) = 0 Solve for x 13, y 13, and z 13 : x 13 = x 3y 1 x 1 y 3 y 1 y 3 z 13 y 3 x 13 + x 3 = 0 z 13 y 1 x 13 + x 1 = 0 y 13 = (x 2 x 1 )(ky 1 z 3 py 3 z 1 ) z(y 1 y 3 )(z 1 z 2 ) z 13 = x 3 x 1 y 1 y 3

46 Compute x 123, y 123, and z 123 : x 123 = x 23y 12 x 12 y 23 y 12 y 23 y 123 = py 2y 23 z 12 ky 12 y 2 z 23 z 2 (y 12 y 23 ) z 123 = x 23 x 12 y 12 y 23

47 System on the left face: z 3 y x 3 + x = 0 z 2 y x 2 + x = 0 z 23 y 2 x 23 + x 2 = 0 z 23 y 3 x 23 + x 3 = 0 zy 23 (y 3 y 2 ) y(py 2 z 3 qy 3 z 2 ) = 0 Solve for x 23, y 23, and z 23 : x 23 = x 3y 2 x 2 y 3 y 2 y 3 y 23 = (x 2 x 1 )(ky 2 z 3 qy 3 z 2 ) z(y 2 y 3 )(z 1 z 2 ) z 23 = x 3 x 2 y 2 y 3

48 Compute x 123, y 123, and z 123 : x 123 = x 13y 12 x 12 y 13 y 12 y 13 y 123 = qy 1y 13 z 12 ky 1 y 12 z 13 z 1 (y 12 y 13 ) z 123 = x 13 x 12 y 12 y 13 Substitute x 12, y 12, y 12, x 13, y 13, z 13, x 23, y 23, z 23 into the above to get

49 x 123 = (pz 1(x 3 y 2 x 2 y 3 )+ qz 2 (x 1 y 3 x 3 y 1 )+ kz 3 (x 2 y 1 x 1 y 2 ) (pz 1 (y 2 y 3 ) + qz 2 (y 3 y 1 ) + kz 3 (y 1 y 2 ) y 123 = pqy 3z 1 (x 2 x 1 )+kpy 2 z 1 (x 1 x 3 )+kqy 1 (x 3 z 2 x 1 z 2 +x 1 z 3 x 2 z 3 ) z 1 (pz 1 (y 2 y 3 ) + qz 2 (y 3 y 1 ) + kz 3 (y 1 y 2 )) z 123 = z(z 1 z 2 )(x 3 (y 1 y 2 ) + x 1 (y 2 y 3 )+ x 2 (y 3 y 1 )) (x 1 x 2 )(pz 1 (y 2 y 3 ) + qz 2 (y 3 y 1 ) + kz 3 (y 1 y 2 )) Answer is unique and independent of x and y. Consistency around the cube is satisfied!

50 Step 2: Homogenization Observed that x 3, y 3 and z 3 appear linearly in numerators and denominators of x 13 = y 1(x + yz 3 ) y 3 (x + yz 1 ) y 1 y 3 y 13 = pyy 3z 1 kyy 1 z 3 z(y 1 y 3 ) z 13 = y(z 3 z 1 ) y 1 y 3 x 23 = y 2(x + yz 3 ) y 3 (x + yz 2 ) y 2 y 3 y 23 = qyy 3z 2 kyy 2 z 3 z(y 2 y 3 ) z 23 = y(z 3 z 2 ) y 2 y 3

51 Substitute x 3 = h H, y 3 = g G, and z 3 = f F. Use constraints (from left face edges) z 1 y x 1 + x = 0, z 2 y x 2 + x = 0 z 3 y x 3 + x = 0 Solve for x 3 = z 3 y + x (since x 3 appears linearly). Thus, x 3 = fy+f x F, y 3 = g G, and z 3 = f F.

52 Substitute x 3, y 3, z 3 into x 13, y 13, z 13 : x 13 = F gx F Gxy 1 fgyy 1 + F gyz 1 F (g Gy 1 ) y 13 = y(fgky 1 F gpz 1 ) F (g Gy 1 )z z 13 = Gy(F z 1 f) F (g Gy 1 ) Require that numerators and denominators are linear in f, F, g, and G. That forces G = F. Hence, x 3 = fy+f x F, y 3 = g F, and z 3 = f F.

53 Compute Hence, x 3 = fy + F x F y 3 = g F y 13 = g 1 F 1 z 3 = f F z 13 = f 1 F 1 x 13 = f 1y 1 + F 1 x 1 F 1 x 13 = fyy 1 + g(x + yz 1 ) F xy 1 g F y 1 = f 1y 1 + F 1 x 1 F 1 y 13 = y(fky 1 gpz 1 ) (g F y 1 )z = g 1 F 1 z 13 = y( f + F z 1) g F y 1 = f 1 F 1

54 Note that x 13 = fyy 1 + g(x + yz 1 ) F xy 1 g F y 1 = f 1y 1 + F 1 x 1 F 1 is automaticlly satisfied as a result of the relation x 3 = z 3 y + x. Write in matrix form: f 1 y 0 yz 1 ψ 1 = g 1 = t kyy 1 pyz 1 0 z z 0 1 y 1 F 1 f g F = L ψ Repeat the same steps for x 23, y 23, z 23 to obtain

55 ψ 2 = f 2 g 2 F 2 = t y 0 yz 2 kyy 2 qyz 2 0 z z 0 1 y 2 f g F = M ψ Therefore, y 0 yz 1 L = tl c = t kyy 1 pyz 1 0 z z 0 1 y 1 y 0 yz 2 M = sm c = s kyy 2 qyz 2 0 z z 0 1 y 2

56 Step 3: Determine t and s Substitute L = t L c, M = s M c into L 2 M M 1 L = 0 t 2 s (L c ) 2 M c s 1 t (M c ) 1 L c = 0 Solve the equation from the (2-1)-element: t 2t s s 1 = y 1 y 2. Thus, t = s = 1 y, or t = 1 y 1 and s = 1 y 2, or t = 3 z y 2 y 1 z 1 and s = 3 z y 2 y 2 z 2.

57 Example 3: Lattice due to Hietarinta (2011) x 1 z y 1 x = 0 x 2 z y 2 x = 0 z 12 y x 1 ( px1 qx ) 2 = 0 x z 1 z 2 Note: System has two single-edge equations and one full-face equation. Lax pair: L = t yz x k x kx px 1 z yzz 1 x x 1 z z 1 xz 1 z 0 zz 1

58 and M = s yz x k x kx qx 2 z yzz 2 x x 2 z z 2 xz 2, z 0 zz 2 where t = s = 1 z, or t = 1 z 1, s = 1 z 2, or t = 3 x x 1 z 2 z 1, s = 3 x x 2 z 2 z 2. Here, t 2 t s s 1 = z 1 z 2.

59 Example 4: Discrete Boussinesq System (Tongas and Nijhoff 2005) z 1 xx 1 + y = 0 z 2 xx 2 + y = 0 (x 2 x 1 )(z xx 12 + y 12 ) p + q = 0 Lax pair: L = t x y p k xy 1 + x 1 z z x

60 x M = s y q k xy 2 + x 2 z z x with t = s = 1, or t = 1 3 p k and s = 1 Note: x 3 = f F, y 3 = g F, and ψ = 3 q k. f F, and t 2 t g s s 1 = 1.

61 Example 5: System of pkdv Lattices (Xenitidis and Mikhailov 2009) (x x 12 )(y 1 y 2 ) p 2 + q 2 = 0 (y y 12 )(x 1 x 2 ) p 2 + q 2 = 0 Lax pair: 0 0 tx t(p 2 k 2 xy 1 ) 0 0 t ty L = 1 T y T (p 2 k 2 x 1 y) 0 0 T T x 1 0 0

62 M = 0 0 sx s(q 2 k 2 xy 2 ) 0 0 s sy 2 Sy S(q 2 k 2 x 2 y) 0 0 S Sx with t = s = T = S = 1, or t T = 1 DetLc = 1 p k and s S = 1 DetMc = 1 q k. Note: t 2 T or T 2 T S s 1 = 1 and T 2 t s S 1 = 1, S S 1 = 1, with T = t T, S = s S. Here, x 3 = f F, y 3 = g G, and ψ = [f F g G]T.

63 Example 6: Discrete NLS System (Xenitidis and Mikhailov 2009) y 1 y 2 y ( (x 1 x 2 )y + p q ) = 0 ( x 1 x 2 + x 12 (x1 x 2 )y + p q ) = 0 Lax pair: L = t 1 x 1 y k p yx 1 M = s 1 x 2 y k q yx 2

64 with t = s = 1, or t= 1 DetLc = 1 α k and s= 1 β k. Note: x 3 = f F, ψ = f F, and t 2 t s s 1 = 1.

65 Example 7: Schwarzian Boussinesq Lattice (Nijhoff 1999) z 1 y x 1 + x = 0 z 2 y x 2 + x = 0 z y 12 (y 1 y 2 ) y(p y 2 z 1 q y 1 z 2 ) = 0 Lax pair: L = t y 0 yz 1 pyz 1 0 z z 0 1 y 1 kyy 1

66 M = s y 0 yz 2 pyz 2 0 z z 0 1 y 2 kyy 2 with t = s = 1 y, or t = 1 y 1 and s = 1 y 2, or t = z 3 y 2 y 1 z 1 and s = z 3 y 2 y 2 z 2. Note: x 3 = fy+f x F, y 3 = g F, z 3 = f F, ψ = f g F, and t 2 t s s 1 = y 1 y 2.

67 Example 8: Toda modified Boussinesq System (Nijhoff 1992) y 12 (p q + x 2 x 1 ) (p 1)y 2 + (q 1)y 1 = 0 y 1 y 2 (p q z 2 + z 1 ) (p 1)yy 2 + (q 1)yy 1 = 0 y(p + q z x 12 )(p q + x 2 x 1 ) (p 2 + p + 1)y 1 + (q 2 + q + 1)y 2 = 0 Lax pair: 1+k+k k + p z 2 y L=t 0 p 1 (1 k)y p k x 1 k 2 y y 1 p 2 (y 1 y) ky(x 1 z)+yzx 1 y p(y 1+yx 1 +yz) y

68 1+k+k k + q z 2 y M =s 0 q 1 (1 k)y q k x 2 k 2 y y 2 q 2 (y 2 y) ky(x 2 z)+yzx 2 y q(y 2+yx 2 +yz) y with t = s = 1, or t = 3 y1 y and s = 3 y2 f Note: x 3 = f F, y 3 = g F, ψ = g F y. Here, t 2 t s s 1 = 1.

69 Software Demonstration

70 Conclusions and Future Work Mathematica code works for scalar P Es in 2D defined on quad-graphs (quadrilateral faces). Mathematica code has been extended to systems of P Es in 2D defined on quad-graphs. Code can be used to test (i) consistency around the cube and compute or test (ii) Lax pairs. Consistency around cube = P E has Lax pair. P E has Lax pair consistency around cube. Indeed, there are P Es with a Lax pair that are not consistent around the cube. Example: discrete sine-gordon equation.

71 Avoid the determinant method to avoid square roots! Factorization plays an essential role! Hard cases: Q4 equation (elliptic curves, Weierstraß functions) (Nijhoff 2001) and Q5 Future Work: Extension to more complicated systems of P Es. P Es in 3D: Lax pair will be expressed in terms of tensors. Consistency around a hypercube. Examples: discrete Kadomtsev-Petviashvili (KP) equations.

72 Thank You

73 Additional Examples Example 9: (H1) Equation (ABS classification) (x x 12 )(x 1 x 2 ) + q p = 0 Lax pair: L = t x p k xx 1 1 x 1 M = s x q k xx 2 1 x 2 with t = s = 1 or t = 1 k p and s = 1 k q Note: t 2 s t s 1 = 1.

74 Example 10: (H2) Equation (ABS 2003) (x x 12 )(x 1 x 2 )+(q p)(x+x 1 +x 2 +x 12 )+q 2 p 2 =0 Lax pair: L = t p k + x p2 k 2 + (p k)(x + x 1 ) xx 1 1 (p k + x 1 ) M = s with t = Note: t 2 t q k + x q2 k 2 + (q k)(x + x 2 ) xx 2 1 (q k + x 2 ) 1 and s = 1 2(k p)(p+x+x1 ) 2(k q)(q+x+x2 ) s s 1 = p+x+x 1 q+x+x 2.

75 Example 11: (H3) Equation (ABS 2003) p(xx 1 + x 2 x 12 ) q(xx 2 + x 1 x 12 ) + δ(p 2 q 2 ) = 0 Lax pair: L = t kx ( δ(p 2 k 2 ) ) + pxx 1 p kx 1 M = s kx ( δ(q 2 k 2 ) ) + qxx 2 q kx 2 with t = Note: t 2 t 1 and s = 1 (p 2 k 2 )(δp+xx 1 ) (q 2 k 2 )(δq+xx 2 ) s s 1 = δp+xx 1 δq+xx 2.

76 Example 12: (H3) Equation (δ = 0) (ABS 2003) p(xx 1 + x 2 x 12 ) q(xx 2 + x 1 x 12 ) = 0 Lax pair: L = t kx pxx 1 p kx 1 M = s kx qxx 2 q kx 2 with t = s = 1 x or t = 1 x 1 and s = 1 x 2 Note: t 2 s t s 1 = x x 1 x x 2 = x 1 x 2.

77 Example 13: (Q1) Equation (ABS 2003) p(x x 2 )(x 1 x 12 ) q(x x 1 )(x 2 x 12 )+δ 2 pq(p q) = 0 Lax pair: L = t (p k)x 1 + kx p ( δ 2 ) k(p k) + xx 1 p ( ) (p k)x + kx 1 M = s (q k)x 2 + kx q ( δ 2 ) k(q k) + xx 2 q ( ) (q k)x + kx 2 1 with t = δp±(x x 1 ) and s = 1 δq±(x x 2 ), 1 or t = k(p k)((δp+x x 1 )(δp x+x 1 )) and s = 1 k(q k)((δq+x x 2 )(δq x+x 2 )) Note: t 2 t s s 1 = q (δp+(x x 1 ))(δp (x x 1 )) p(δq+(x x 2 ))(δq (x x 2 )).

78 Example 14: (Q1) Equation (δ = 0) (ABS 2003) p(x x 2 )(x 1 x 12 ) q(x x 1 )(x 2 x 12 ) = 0 which is the cross-ratio equation (x x 1 )(x 12 x 2 ) (x 1 x 12 )(x 2 x) = p q Lax pair: L = t (p k)x 1 + kx pxx 1 p ( ) (p k)x + kx 1 M = s (q k)x 2 + kx qxx 2 q ( ) (q k)x + kx 2

79 Here, t 2 t or t = s s 1 = q(x x 1) 2 p(x x 2. So, t = 1 ) 2 x x 1 and s = 1 x x 2 1 and s = 1. k(k p)(x x1 ) k(k q)(x x2 )

80 Example 15: (Q2) Equation (ABS 2003) p(x x 2 )(x 1 x 12 ) q(x x 1 )(x 2 x 12 )+pq(p q) (x+x 1 +x 2 +x 12 ) pq(p q)(p 2 pq+q 2 )=0 Lax pair: (k p)(kp x 1 )+kx L=t p ( k(k p)(k 2 kp+p 2 ) x x 1 )+xx 1 p ( ) (k p)(kp x)+kx 1 (k q)(kq x 2 )+kx M =s q ( k(k q)(k 2 kq+q 2 ) x x 2 )+xx 2 q ( ) (k q)(kq x)+kx 2

81 with t = 1 k(k p)((x x 1 ) 2 2p 2 (x+x 1 )+p 4 ) and s = 1 k(k q)((x x 2 ) 2 2q 2 (x+x 2 )+q 4 ) Note: t 2 t s = q ( (x x1 ) 2 2p 2 (x + x 1 ) + p 4) s 1 p ( (x x 2 ) 2 2q 2 (x + x 2 ) + q 4) = p ( (X + X 1 ) 2 p 2) ( (X X 1 ) 2 p 2) q ( (X + X 2 ) 2 q 2) ( (X X 2 ) 2 q 2) with x = X 2, and, consequently, x 1 = X 2 1, x 2 = X 2 2.

82 Example 16: (Q3) Equation (ABS 2003) (q 2 p 2 )(xx 12 +x 1 x 2 )+q(p 2 1)(xx 1 +x 2 x 12 ) p(q 2 1)(xx 2 +x 1 x 12 ) δ2 4pq (p2 q 2 )(p 2 1)(q 2 1)=0 Lax pair: 4kp ( p(k 2 1)x+(p 2 k 2 ) )x 1 L=t (p 2 1)(δ 2 k 2 δ 2 k 4 δ 2 p 2 +δ 2 k 2 p 2 4k 2 pxx 1 ) 4k 2 p(p 2 1) 4kp ( p(k 2 1)x 1 +(p 2 k 2 )x ) 4kq ( q(k 2 1)x+(q 2 k 2 ) )x 2 M=s (q 2 1)(δ 2 k 2 δ 2 k 4 δ 2 q 2 +δ 2 k 2 q 2 4k 2 qxx 2 ) 4k 2 q(q 2 1) 4kq ( q(k 2 1)x 2 +(q 2 k 2 )x )

83 with t= 1 2k p(k 2 1)(k 2 p 2 )(4p 2 (x 2 +x 2 1 ) 4p(1+p2 )xx 1 +δ 2 (1 p 2 ) 2 ) and s= 2k 1 q(k 2 1)(k 2 q 2 )(4q 2 (x 2 +x 2 2 ) 4q(1+q2 )xx 2 +δ 2 (1 q 2 ) 2 ).

84 Note: t 2 t s s 1 = q(q2 1) ( 4p 2 (x 2 +x 2 1 ) 4p(1+p2 )xx 1 +δ 2 (1 p 2 ) 2) p(p 2 1) ( 4q 2 (x 2 +x 2 2 ) 4q(1+q2 )xx 2 +δ 2 (1 q 2 ) 2) = q(q2 1) ( 4p 2 (x x 1 ) 2 4p(p 1) 2 xx 1 +δ 2 (1 p 2 ) 2) p(p 2 1) ( 4q 2 (x x 2 ) 2 4q(q 1) 2 xx 2 +δ 2 (1 q 2 ) 2) = q(q2 1) ( 4p 2 (x+x 1 ) 2 4p(p+1) 2 xx 1 +δ 2 (1 p 2 ) 2) p(p 2 1) ( 4q 2 (x+x 2 ) 2 4q(q+1) 2 xx 2 +δ 2 (1 q 2 ) 2)

85 where 4p 2 (x 2 +x 2 1) 4p(1+p 2 )xx 1 +δ 2 (1 p 2 ) 2 = δ 2 (p e X+X 1 )(p e (X+X1) )(p e X X 1 )(p e (X X1) ) = δ 2 (p cosh(x + X 1 )+sinh(x + X 1 )) (p cosh(x + X 1 ) sinh(x + X 1 )) (p cosh(x X 1 )+sinh(x X 1 )) (p cosh(x X 1 ) sinh(x X 1 )) with x = δ cosh(x), and, consequently, x 1 = δ cosh(x 1 ), x 2 = δ cosh(x 2 ).

86 Example 17: (Q3) Equation (δ) = 0 (ABS 2003) (q 2 p 2 )(xx 12 + x 1 x 2 ) + q(p 2 1)(xx 1 + x 2 x 12 ) p(q 2 1)(xx 2 + x 1 x 12 ) = 0 Lax pair: L=t (p2 k 2 )x 1 +p(k 2 1)x k(p 2 1)xx 1 (p 2 1)k ( (p 2 k 2 )x+p(k 2 ) 1)x 1 M =s (q2 k 2 )x 2 +q(k 2 1)x k(q 2 1)xx 2 (q 2 1)k ( (q 2 k 2 )x+q(k 2 ) 1)x 2

87 with t = 1 px x 1 and s = 1 qx x 2 or t = 1 px 1 x and s = 1 qx 2 x or t = 1 (k 2 1)(p 2 k 2 )(px x 1 )(px 1 x) and s = 1. (k 2 1)(q 2 k 2 )(qx x 2 )(qx 2 x) Note: t 2 t s s 1 = (q2 1)(px x 1 )(px 1 x) (p 2 1)(qx x 2 )(qx 2 x).

88 Example 18: (α, β)-equation (Quispel 1983) ( (p α)x (p+β)x1 ) ( (p β)x2 (p+α)x 12 ) ( (q α)x (q+β)x 2 ) ( (q β)x1 (q+α)x 12 ) =0 Lax pair: L=t (p α)(p β)x+(k2 p 2 )x 1 (k α)(k β)xx 1 (k+α)(k+β) ( (p+α)(p+β)x 1 +(k 2 p 2 )x ) M =s (q α)(q β)x+(k2 q 2 )x 2 (k α)(k β)xx 2 (k+α)(k+β) ( (q+α)(q+β)x 2 +(k 2 q 2 )x )

89 with t = 1 (α p)x+(β+p)x 1 ) and s = 1 (α q)x+(β+q)x 2 ) or t = or t = 1 (β p)x+(α+p)x 1 ) and s = 1 (β q)x+(α+q)x 2 ) 1 (p 2 k 2 )((β p)x+(α+p)x 1)((α p)x+(β+p)x 1) and s = Note: t 2 t 1 (q 2 k 2 )((β q)x+(α+q)x 2)((α q)x+(β+q)x 2) s s 1 = ( (β p)x+(α+p)x 1)((α p)x+(β+p)x 1) ((β q)x+(α+q)x 2)((α q)x+(β+q)x 2).

90 Example 19: (A1) Equation (ABS 2003) p(x+x 2 )(x 1 +x 12 ) q(x+x 1 )(x 2 +x 12 ) δ 2 pq(p q) = 0 (Q1) if x 1 x 1 and x 2 x 2 Lax pair: L = t (k p)x 1 + kx p ( δ 2 ) k(k p) + xx 1 p ( ) (k p)x + kx 1 M = s (k q)x 2 + kx q ( δ 2 ) k(k q) + xx 2 q ( ) (k q)x + kx 2

91 with t = 1 k(k p)((δp+x+x 1 )(δp x x 1 )) and s = 1 k(k q)((δq+x+x 2 )(δq x x 2 )) Note: t 2 t s s 1 = q (δp+(x+x 1 ))(δp (x+x 1 )) p(δq+(x+x 2 ))(δq (x+x 2 )). Question: Rational choice for t and s?

92 Example 20: (A2) Equation (ABS 2003) (q 2 p 2 )(xx 1 x 2 x ) + q(p 2 1)(xx 2 + x 1 x 12 ) p(q 2 1)(xx 1 + x 2 x 12 ) = 0 (Q3) with δ = 0 via Möbius transformation: x x, x 1 1 x 1, x 2 1 x 2, x 12 x 12, p p, q q Lax pair: k(p L=t 2 1)x ( p 2 k 2 +p(k 2 ) 1)xx 1 p(k 2 1)+(p 2 k 2 )xx 1 k(p 2 1)x 1 k(q M =s 2 1)x ( q 2 k 2 +q(k 2 ) 1)xx 2 q(k 2 1)+(q 2 k 2 )xx 2 k(q 2 1)x 2

93 with t = and s = 1 (k 2 1)(k 2 p 2 )(p xx 1 )(pxx 1 1) 1 (k 2 1)(k 2 q 2 )(q xx 2 )(qxx 2 1) Note: t 2 t s s 1 = (q2 1)(p xx 1 )(pxx 1 1) (p 2 1)(q xx 2 )(qxx 2 1). Question: Rational choice for t and s?

94 Example 21: Discrete sine-gordon Equation xx 1 x 2 x 12 pq(xx 12 x 1 x 2 ) 1 = 0 (H3) with δ = 0 via extended Möbius transformation: x x, x 1 x 1, x 2 1 x 2, x 12 1 x 12, p 1 p, q q Discrete sine-gordon equation is NOT consistent around the cube, but has a Lax pair! Lax pair: L = p kx 1 M = k x px 1 x qx 2 x 1 kx x 2 k q

Symbolic Computation of Lax Pairs of Integrable Nonlinear Partial Difference Equations on Quad-Graphs

Symbolic Computation of Lax Pairs of Integrable Nonlinear Partial Difference Equations on Quad-Graphs Symbolic Computation of Lax Pairs of Integrable Nonlinear Partial Difference Equations on Quad-Graphs Willy Hereman & Terry Bridgman Department of Mathematical and Computer Sciences Colorado School of

More information

Symbolic Computation of Lax Pairs of Nonlinear Systems of Partial Difference Equations using Multidimensional Consistency

Symbolic Computation of Lax Pairs of Nonlinear Systems of Partial Difference Equations using Multidimensional Consistency Symbolic Computation of Lax Pairs of Nonlinear Systems of Partial Difference Equations using Multidimensional Consistency Willy Hereman Department of Applied Mathematics and Statistics Colorado School

More information

SYMBOLIC COMPUTATION OF LAX PAIRS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS

SYMBOLIC COMPUTATION OF LAX PAIRS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS SYMBOLIC COMPUTATION OF LAX PAIRS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS by Terry J. Bridgman c Copyright by Terry J. Bridgman, 2018 All Rights Reserved A thesis submitted to the Faculty and the Board

More information

Symmetry Reductions of Integrable Lattice Equations

Symmetry Reductions of Integrable Lattice Equations Isaac Newton Institute for Mathematical Sciences Discrete Integrable Systems Symmetry Reductions of Integrable Lattice Equations Pavlos Xenitidis University of Patras Greece March 11, 2009 Pavlos Xenitidis

More information

ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS

ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS Proceedings of the International Conference on Difference Equations, Special Functions and Orthogonal Polynomials, World Scientific (2007 ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS ANASTASIOS

More information

arxiv: v1 [nlin.si] 25 Mar 2009

arxiv: v1 [nlin.si] 25 Mar 2009 Linear quadrilateral lattice equations and multidimensional consistency arxiv:0903.4428v1 [nlin.si] 25 Mar 2009 1. Introduction James Atkinson Department of Mathematics and Statistics, La Trobe University,

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

SYMBOLIC VERIFICATION OF OPERATOR AND MATRIX LAX PAIRS FOR SOME COMPLETELY INTEGRABLE NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

SYMBOLIC VERIFICATION OF OPERATOR AND MATRIX LAX PAIRS FOR SOME COMPLETELY INTEGRABLE NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS SYMBOLIC VERIFICATION OF OPERATOR AND MATRIX LAX PAIRS FOR SOME COMPLETELY INTEGRABLE NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS by Jennifer Larue A thesis submitted to the Faculty and the Board of Trustees

More information

L bor y nnd Union One nnd Inseparable. LOW I'LL, MICHIGAN. WLDNHSDA Y. JULY ), I8T. liuwkll NATIdiNAI, liank

L bor y nnd Union One nnd Inseparable. LOW I'LL, MICHIGAN. WLDNHSDA Y. JULY ), I8T. liuwkll NATIdiNAI, liank G k y $5 y / >/ k «««# ) /% < # «/» Y»««««?# «< >«>» y k»» «k F 5 8 Y Y F G k F >«y y

More information

Soliton solutions to the ABS list

Soliton solutions to the ABS list to the ABS list Department of Physics, University of Turku, FIN-20014 Turku, Finland in collaboration with James Atkinson, Frank Nijhoff and Da-jun Zhang DIS-INI, February 2009 The setting CAC The setting

More information

LOWELL WEEKLY JOURNAL.

LOWELL WEEKLY JOURNAL. Y 5 ; ) : Y 3 7 22 2 F $ 7 2 F Q 3 q q 6 2 3 6 2 5 25 2 2 3 $2 25: 75 5 $6 Y q 7 Y Y # \ x Y : { Y Y Y : ( \ _ Y ( ( Y F [ F F ; x Y : ( : G ( ; ( ~ x F G Y ; \ Q ) ( F \ Q / F F \ Y () ( \ G Y ( ) \F

More information

LOWELL WEEKLY JOURNAL.

LOWELL WEEKLY JOURNAL. Y k p p Y < 5 # X < k < kk

More information

arxiv:nlin/ v1 [nlin.si] 17 Sep 2006

arxiv:nlin/ v1 [nlin.si] 17 Sep 2006 Seed and soliton solutions for Adler s lattice equation arxiv:nlin/0609044v1 [nlin.si] 17 Sep 2006 1. Introduction James Atkinson 1, Jarmo Hietarinta 2 and Frank Nijhoff 1 1 Department of Applied Mathematics,

More information

From discrete differential geometry to the classification of discrete integrable systems

From discrete differential geometry to the classification of discrete integrable systems From discrete differential geometry to the classification of discrete integrable systems Vsevolod Adler,, Yuri Suris Technische Universität Berlin Quantum Integrable Discrete Systems, Newton Institute,

More information

Symmetries and Group Invariant Reductions of Integrable Partial Difference Equations

Symmetries and Group Invariant Reductions of Integrable Partial Difference Equations Proceedings of 0th International Conference in MOdern GRoup ANalysis 2005, 222 230 Symmetries and Group Invariant Reductions of Integrable Partial Difference Equations A. TONGAS, D. TSOUBELIS and V. PAPAGEORGIOU

More information

Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multiple Space Dimensions

Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multiple Space Dimensions Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multiple Space Dimensions Willy Hereman and Douglas Poole Department of Mathematical and Computer Sciences Colorado

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y G y G Y 87 y Y 8 Y - $ X ; ; y y q 8 y $8 $ $ $ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 $ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G

More information

OWELL WEEKLY JOURNAL

OWELL WEEKLY JOURNAL Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --

More information

GEOMETRY OF DISCRETE INTEGRABILITY. THE CONSISTENCY APPROACH

GEOMETRY OF DISCRETE INTEGRABILITY. THE CONSISTENCY APPROACH GEOMETRY OF DISCRETE INTEGRABILITY. THE CONSISTENCY APPROACH Alexander I. Bobenko Institut für Mathematik, Fakultät 2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany 1 ORIGIN

More information

Lowell Dam Gone Out. Streets Turned I n t o Rivers. No Cause For Alarm Now However As This Happened 35 Years A&o

Lowell Dam Gone Out. Streets Turned I n t o Rivers. No Cause For Alarm Now However As This Happened 35 Years A&o V ()\\ ))? K K Y 6 96 Y - Y Y V 5 Z ( z x z \ - \ - - z - q q x x - x 5 9 Q \ V - - Y x 59 7 x x - Y - x - - x z - z x - ( 7 x V 9 z q &? - 9 - V ( x - - - V- [ Z x z - -x > -) - - > X Z z ( V V V

More information

New Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation

New Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation Applied Mathematical Sciences, Vol. 6, 2012, no. 12, 579-587 New Exact Solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli Equation Ying Li and Desheng Li School of Mathematics and System Science

More information

Integrability of P Es

Integrability of P Es Lecture 3 Integrability of P Es Motivated by the lattice structure emerging from the permutability/superposition properties of the Bäcklund transformations of the previous Lecture, we will now consider

More information

A Memorial. Death Crash Branch Out. Symbol The. at Crossing Flaming Poppy. in Belding

A Memorial. Death Crash Branch Out. Symbol The. at Crossing Flaming Poppy. in Belding - G Y Y 8 9 XXX G - Y - Q 5 8 G Y G Y - - * Y G G G G 9 - G - - : - G - - ) G G- Y G G q G G : Q G Y G 5) Y : z 6 86 ) ; - ) z; G ) 875 ; ) ; G -- ) ; Y; ) G 8 879 99 G 9 65 q 99 7 G : - G G Y ; - G 8

More information

2. Examples of Integrable Equations

2. Examples of Integrable Equations Integrable equations A.V.Mikhailov and V.V.Sokolov 1. Introduction 2. Examples of Integrable Equations 3. Examples of Lax pairs 4. Structure of Lax pairs 5. Local Symmetries, conservation laws and the

More information

group: A new connection

group: A new connection Schwarzian integrable systems and the Möbius group: A new connection March 4, 2009 History The Schwarzian KdV equation, SKdV (u) := u t u x [ uxxx u x 3 2 uxx 2 ] ux 2 = 0. History The Schwarzian KdV equation,

More information

Solving Nonlinear Wave Equations and Lattices with Mathematica. Willy Hereman

Solving Nonlinear Wave Equations and Lattices with Mathematica. Willy Hereman Solving Nonlinear Wave Equations and Lattices with Mathematica Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, USA http://www.mines.edu/fs home/whereman/

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL KY Y 872 K & q $ < 9 2 q 4 8 «7 K K K «> 2 26 8 5 4 4 7»» 2 & K q 4 [«5 «$6 q X «K «8K K88 K 7 ««$25 K Q ««q 8 K K Y & 7K /> Y 8«#»«Y 87 8 Y 4 KY «7««X & Y» K ) K K 5 KK K > K» Y Y 8 «KK > /» >» 8 K X

More information

Symbolic Computation of Conserved Densities and Symmetries of Nonlinear Evolution and Differential-Difference Equations WILLY HEREMAN

Symbolic Computation of Conserved Densities and Symmetries of Nonlinear Evolution and Differential-Difference Equations WILLY HEREMAN . Symbolic Computation of Conserved Densities and Symmetries of Nonlinear Evolution and Differential-Difference Equations WILLY HEREMAN Joint work with ÜNAL GÖKTAŞ and Grant Erdmann Graduate Seminar Department

More information

HEAGAN & CO., OPP. f>, L. & W. DEPOT, DOYER, N. J, OUR MOTTO! ould Iwv ia immediate vltlui. VEEY BEST NEW Creamery Butter 22c ib,

HEAGAN & CO., OPP. f>, L. & W. DEPOT, DOYER, N. J, OUR MOTTO! ould Iwv ia immediate vltlui. VEEY BEST NEW Creamery Butter 22c ib, #4 NN N G N N % XX NY N Y FY N 2 88 N 28 k N k F P X Y N Y /» 2«X ««!!! 8 P 3 N 0»9! N k 25 F $ 60 $3 00 $3000 k k N 30 Y F00 6 )P 0» «{ N % X zz» «3 0««5 «N «XN» N N 00/ N 4 GN N Y 07 50 220 35 2 25 0

More information

Continuous and Discrete Homotopy Operators with Applications in Integrability Testing. Willy Hereman

Continuous and Discrete Homotopy Operators with Applications in Integrability Testing. Willy Hereman Continuous and Discrete Homotopy Operators with Applications in Integrability Testing Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, USA http://www.mines.edu/fs

More information

Two Posts to Fill On School Board

Two Posts to Fill On School Board Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

More information

SYMBOLIC SOFTWARE FOR SOLITON THEORY: INTEGRABILITY, SYMMETRIES CONSERVATION LAWS AND EXACT SOLUTIONS. Willy Hereman

SYMBOLIC SOFTWARE FOR SOLITON THEORY: INTEGRABILITY, SYMMETRIES CONSERVATION LAWS AND EXACT SOLUTIONS. Willy Hereman . SYMBOLIC SOFTWARE FOR SOLITON THEORY: INTEGRABILITY, SYMMETRIES CONSERVATION LAWS AND EXACT SOLUTIONS Willy Hereman Dept. of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado

More information

Introduction of Partial Differential Equations and Boundary Value Problems

Introduction of Partial Differential Equations and Boundary Value Problems Introduction of Partial Differential Equations and Boundary Value Problems 2009 Outline Definition Classification Where PDEs come from? Well-posed problem, solutions Initial Conditions and Boundary Conditions

More information

Introduction to the Hirota bilinear method

Introduction to the Hirota bilinear method Introduction to the Hirota bilinear method arxiv:solv-int/9708006v1 14 Aug 1997 J. Hietarinta Department of Physics, University of Turku FIN-20014 Turku, Finland e-mail: hietarin@utu.fi Abstract We give

More information

Department of mathematics MA201 Mathematics III

Department of mathematics MA201 Mathematics III Department of mathematics MA201 Mathematics III Academic Year 2015-2016 Model Solutions: Quiz-II (Set - B) 1. Obtain the bilinear transformation which maps the points z 0, 1, onto the points w i, 1, i

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y G q G Y Y 29 8 $ 29 G 6 q )

More information

Conservation laws for difference equations. Peter Hydon (Surrey) Elizabeth Mansfield (Kent) Olexandr Rasin (Bar-Ilan) Tim Grant (Surrey)

Conservation laws for difference equations. Peter Hydon (Surrey) Elizabeth Mansfield (Kent) Olexandr Rasin (Bar-Ilan) Tim Grant (Surrey) Conservation laws for difference equations Peter Hydon (Surrey) Elizabeth Mansfield (Kent) Olexandr Rasin (Bar-Ilan) Tim Grant (Surrey) Introduction: What is a conservation law? What is a conservation

More information

Multiple-Soliton Solutions for Extended Shallow Water Wave Equations

Multiple-Soliton Solutions for Extended Shallow Water Wave Equations Studies in Mathematical Sciences Vol. 1, No. 1, 2010, pp. 21-29 www.cscanada.org ISSN 1923-8444 [Print] ISSN 1923-8452 [Online] www.cscanada.net Multiple-Soliton Solutions for Extended Shallow Water Wave

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL G $ G 2 G ««2 ««q ) q «\ { q «««/ 6 «««««q «] «q 6 ««Z q «««Q \ Q «q «X ««G X G ««? G Q / Q Q X ««/«X X «««Q X\ «q «X \ / X G XX «««X «x «X «x X G X 29 2 ««Q G G «) 22 G XXX GG G G G G G X «x G Q «) «G

More information

Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations

Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations Willy Hereman Department of Applied Mathematics and Statistics Colorado School of Mines Golden, Colorado whereman@mines.edu

More information

Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225

Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225 Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225 Dr. Asmaa Al Themairi Assistant Professor a a Department of Mathematical sciences, University of Princess Nourah bint Abdulrahman,

More information

Module 2: First-Order Partial Differential Equations

Module 2: First-Order Partial Differential Equations Module 2: First-Order Partial Differential Equations The mathematical formulations of many problems in science and engineering reduce to study of first-order PDEs. For instance, the study of first-order

More information

Classification of integrable equations on quad-graphs. The consistency approach

Classification of integrable equations on quad-graphs. The consistency approach arxiv:nlin/0202024v2 [nlin.si] 10 Jul 2002 Classification of integrable equations on quad-graphs. The consistency approach V.E. Adler A.I. Bobenko Yu.B. Suris Abstract A classification of discrete integrable

More information

Educjatipnal. L a d ie s * COBNWALILI.S H IG H SCHOOL. I F O R G IR L S A n B k i n d e r g a r t e n.

Educjatipnal. L a d ie s * COBNWALILI.S H IG H SCHOOL. I F O R G IR L S A n B k i n d e r g a r t e n. - - - 0 x ] - ) ) -? - Q - - z 0 x 8 - #? ) 80 0 0 Q ) - 8-8 - ) x ) - ) -] ) Q x?- x - - / - - x - - - x / /- Q ] 8 Q x / / - 0-0 0 x 8 ] ) / - - /- - / /? x ) x x Q ) 8 x q q q )- 8-0 0? - Q - - x?-

More information

The elliptic sinh-gordon equation in the half plane

The elliptic sinh-gordon equation in the half plane Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 25), 63 73 Research Article The elliptic sinh-gordon equation in the half plane Guenbo Hwang Department of Mathematics, Daegu University, Gyeongsan

More information

FASHIONABLE VISITORS RECORD AND GUIDE

FASHIONABLE VISITORS RECORD AND GUIDE BB k k B QY G p xp 00 - b BX B G B0 k Q Y B G p xp 00 - b XK BK Y? G K 4 >pg - G - g kg p b q b 862 B v y- YG 5 B p b b ( G v ) B py By? b pp p b g ( y p yg Gg k b g B B GB B () Q GB ( v) v By " x p 350

More information

LOWELL WEEKLY JOURNAL. ^Jberxy and (Jmott Oao M d Ccmsparftble. %m >ai ruv GEEAT INDUSTRIES

LOWELL WEEKLY JOURNAL. ^Jberxy and (Jmott Oao M d Ccmsparftble. %m >ai ruv GEEAT INDUSTRIES ? (») /»» 9 F ( ) / ) /»F»»»»»# F??»»» Q ( ( »»» < 3»» /» > > } > Q ( Q > Z F 5

More information

LOWELL WEEKI.Y JOURINAL

LOWELL WEEKI.Y JOURINAL / $ 8) 2 {!»!» X ( (!!!?! () ~ x 8» x /»!! $?» 8! ) ( ) 8 X x /! / x 9 ( 2 2! z»!!»! ) / x»! ( (»»!» [ ~!! 8 X / Q X x» ( (!»! Q ) X x X!! (? ( ()» 9 X»/ Q ( (X )!» / )! X» x / 6!»! }? ( q ( ) / X! 8 x»

More information

Alexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011

Alexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011 Direct Method of Construction of Conservation Laws for Nonlinear Differential Equations, its Relation with Noether s Theorem, Applications, and Symbolic Software Alexei F. Cheviakov University of Saskatchewan,

More information

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI http://afni.nimh.nih.gov/afni/ Next lecture, DTI For this lecture, think in terms of a single voxel We re still

More information

Generalized bilinear differential equations

Generalized bilinear differential equations Generalized bilinear differential equations Wen-Xiu Ma Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Abstract We introduce a kind of bilinear differential

More information

SIGNATURES OVER FINITE FIELDS OF GROWTH PROPERTIES FOR LATTICE EQUATIONS

SIGNATURES OVER FINITE FIELDS OF GROWTH PROPERTIES FOR LATTICE EQUATIONS SIGNATURES OVER FINITE FIELDS OF GROWTH PROPERTIES FOR LATTICE EQUATIONS JOHN A. G. ROBERTS 1, DINH T. TRAN 1,2 Abstract. We study integrable lattice equations and their perturbations over finite fields.

More information

Second Order Lax Pairs of Nonlinear Partial Differential Equations with Schwarzian Forms

Second Order Lax Pairs of Nonlinear Partial Differential Equations with Schwarzian Forms Second Order Lax Pairs of Nonlinear Partial Differential Equations with Schwarzian Forms Sen-yue Lou a b c, Xiao-yan Tang a b, Qing-Ping Liu b d, and T. Fukuyama e f a Department of Physics, Shanghai Jiao

More information

Relativistic Collisions as Yang Baxter maps

Relativistic Collisions as Yang Baxter maps Relativistic Collisions as Yang Baxter maps Theodoros E. Kouloukas arxiv:706.0636v2 [math-ph] 7 Sep 207 School of Mathematics, Statistics & Actuarial Science, University of Kent, UK September 9, 207 Abstract

More information

E-001 ELECTRICAL SYMBOL LEGEND SCIENCE BUILDING RENOVATION H PD SEISMIC REQUIREMENTS FOR ELECTRICAL SYSTEMS PER IBC-2012/ASCE 7-10

E-001 ELECTRICAL SYMBOL LEGEND SCIENCE BUILDING RENOVATION H PD SEISMIC REQUIREMENTS FOR ELECTRICAL SYSTEMS PER IBC-2012/ASCE 7-10 8 9 0 G H G H G H H Y Z H H, H Z H F H XP: F, X, Q G H G 0/0 H XG 00' GH F P FH FX H 0 /" GH-, H G HGH F H H H, K HP G, XG H F F, Y H H, '-0" Y H H H F F- H H H GH- H Q, G P G /8" HGH K F Y Z H F Y Y-

More information

arxiv: v2 [math-ph] 24 Feb 2016

arxiv: v2 [math-ph] 24 Feb 2016 ON THE CLASSIFICATION OF MULTIDIMENSIONALLY CONSISTENT 3D MAPS MATTEO PETRERA AND YURI B. SURIS Institut für Mathemat MA 7-2 Technische Universität Berlin Str. des 17. Juni 136 10623 Berlin Germany arxiv:1509.03129v2

More information

Lax Pairs Graham W. Griffiths 1 City University, UK. oo0oo

Lax Pairs Graham W. Griffiths 1 City University, UK. oo0oo Lax Pairs Graham W. Griffiths 1 City University, UK. oo0oo Contents 1 Introduction 1 Lax pair analysis.1 Operator form........................................... Operator usage..........................................

More information

Weak Lax pairs for lattice equations

Weak Lax pairs for lattice equations Weak Lax pairs for lattice equations Jarmo Hietarinta 1,2 and Claude Viallet 1 1 LPTHE / CNRS / UPMC, 4 place Jussieu 75252 Paris CEDEX 05, France 2 Department of Physics and Astronomy, University of Turku,

More information

T k b p M r will so ordered by Ike one who quits squuv. fe2m per year, or year, jo ad vaoce. Pleaie and THE ALTO SOLO

T k b p M r will so ordered by Ike one who quits squuv. fe2m per year, or year, jo ad vaoce. Pleaie and THE ALTO SOLO q q P XXX F Y > F P Y ~ Y P Y P F q > ##- F F - 5 F F?? 5 7? F P P?? - - F - F F - P 7 - F P - F F % P - % % > P F 9 P 86 F F F F F > X7 F?? F P Y? F F F P F F

More information

A. H. Hall, 33, 35 &37, Lendoi

A. H. Hall, 33, 35 &37, Lendoi 7 X x > - z Z - ----»»x - % x x» [> Q - ) < % - - 7»- -Q 9 Q # 5 - z -> Q x > z»- ~» - x " < z Q q»» > X»? Q ~ - - % % < - < - - 7 - x -X - -- 6 97 9

More information

On Miura Transformations and Volterra-Type Equations Associated with the Adler Bobenko Suris Equations

On Miura Transformations and Volterra-Type Equations Associated with the Adler Bobenko Suris Equations Symmetry, Integrability and Geometry: Methods and Applications On Miura Transformations and Volterra-Type Equations Associated with the Adler Bobenko Suris Equations SIGMA 4 (2008), 077, 14 pages Decio

More information

Elliptic integrable lattice systems

Elliptic integrable lattice systems Elliptic integrable lattice systems Workshop on Elliptic integrable systems, isomonodromy problems, and hypergeometric functions Hausdorff Center for Mathematics, Bonn, July 21-25, 2008 Frank Nijhoff University

More information

The first three (of infinitely many) conservation laws for (1) are (3) (4) D t (u) =D x (3u 2 + u 2x ); D t (u 2 )=D x (4u 3 u 2 x +2uu 2x ); D t (u 3

The first three (of infinitely many) conservation laws for (1) are (3) (4) D t (u) =D x (3u 2 + u 2x ); D t (u 2 )=D x (4u 3 u 2 x +2uu 2x ); D t (u 3 Invariants and Symmetries for Partial Differential Equations and Lattices Λ Ünal Göktaοs y Willy Hereman y Abstract Methods for the computation of invariants and symmetries of nonlinear evolution, wave,

More information

IBVPs for linear and integrable nonlinear evolution PDEs

IBVPs for linear and integrable nonlinear evolution PDEs IBVPs for linear and integrable nonlinear evolution PDEs Dionyssis Mantzavinos Department of Applied Mathematics and Theoretical Physics, University of Cambridge. Edinburgh, May 31, 212. Dionyssis Mantzavinos

More information

E S T A B L IS H E D. n AT Tnn G.D.O. r.w.-bal'eu. e d n e s d a y. II GRANVILLE HOUSE. GATJDICK ROAD. MEADS. EASTBOUENk

E S T A B L IS H E D. n AT Tnn G.D.O. r.w.-bal'eu. e d n e s d a y. II GRANVILLE HOUSE. GATJDICK ROAD. MEADS. EASTBOUENk K q X k K 5 ) ) 5 / K K x x) )? //? q? k X z K 8 5 5? K K K / / $8 ± K K K 8 K / 8 K K X k k X ) k k /» / K / / / k / ] 5 % k / / k k? Z k K ] 8 K K K )» 5 ) # 8 q»)kk q»» )88{ k k k k / k K X 8 8 8 ]

More information

a s*:?:; -A: le London Dyers ^CleanefSt * S^d. per Y ard. -P W ..n 1 0, , c t o b e e d n e sd *B A J IllW6fAi>,EB. E D U ^ T IG r?

a s*:?:; -A: le London Dyers ^CleanefSt * S^d. per Y ard. -P W ..n 1 0, , c t o b e e d n e sd *B A J IllW6fAi>,EB. E D U ^ T IG r? ? 9 > 25? < ( x x 52 ) < x ( ) ( { 2 2 8 { 28 ] ( 297 «2 ) «2 2 97 () > Q ««5 > «? 2797 x 7 82 2797 Q z Q (

More information

Pithy P o i n t s Picked I ' p and Patljr Put By Our P e r i p a tetic Pencil Pusher VOLUME X X X X. Lee Hi^h School Here Friday Ni^ht

Pithy P o i n t s Picked I ' p and Patljr Put By Our P e r i p a tetic Pencil Pusher VOLUME X X X X. Lee Hi^h School Here Friday Ni^ht G G QQ K K Z z U K z q Z 22 x z - z 97 Z x z j K K 33 G - 72 92 33 3% 98 K 924 4 G G K 2 G x G K 2 z K j x x 2 G Z 22 j K K x q j - K 72 G 43-2 2 G G z G - -G G U q - z q - G x) z q 3 26 7 x Zz - G U-

More information

Superintegrable 3D systems in a magnetic field and Cartesian separation of variables

Superintegrable 3D systems in a magnetic field and Cartesian separation of variables Superintegrable 3D systems in a magnetic field and Cartesian separation of variables in collaboration with L. Šnobl Czech Technical University in Prague GSD 2017, June 5-10, S. Marinella (Roma), Italy

More information

Hamiltonian partial differential equations and Painlevé transcendents

Hamiltonian partial differential equations and Painlevé transcendents The 6th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics March 22-26, 2014 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN

More information

Linearization of Second-Order Ordinary Dif ferential Equations by Generalized Sundman Transformations

Linearization of Second-Order Ordinary Dif ferential Equations by Generalized Sundman Transformations Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 051, 11 pages Linearization of Second-Order Ordinary Dif ferential Equations by Generalized Sundman Transformations Warisa

More information

S.Novikov. Singular Solitons and Spectral Theory

S.Novikov. Singular Solitons and Spectral Theory S.Novikov Singular Solitons and Spectral Theory Moscow, August 2014 Collaborators: P.Grinevich References: Novikov s Homepage www.mi.ras.ru/ snovikov click Publications, items 175,176,182, 184. New Results

More information

Logistic function as solution of many nonlinear differential equations

Logistic function as solution of many nonlinear differential equations Logistic function as solution of many nonlinear differential equations arxiv:1409.6896v1 [nlin.si] 24 Sep 2014 Nikolai A. Kudryashov, Mikhail A. Chmykhov Department of Applied Mathematics, National Research

More information

Linearization of Two Dimensional Complex-Linearizable Systems of Second Order Ordinary Differential Equations

Linearization of Two Dimensional Complex-Linearizable Systems of Second Order Ordinary Differential Equations Applied Mathematical Sciences, Vol. 9, 2015, no. 58, 2889-2900 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121002 Linearization of Two Dimensional Complex-Linearizable Systems of

More information

Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations

Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado whereman@mines.edu

More information

Symbolic Computation of Recursion Operators of Nonlinear Differential-Difference Equations

Symbolic Computation of Recursion Operators of Nonlinear Differential-Difference Equations Symbolic Computation of Recursion Operators of Nonlinear Differential-Difference Equations Ünal Göktaş and Willy Hereman Department of Computer Engineering Turgut Özal University, Keçiören, Ankara, Turkey

More information

(EccL HUt, Mosheim, p. 40.) AGC3. Rgnres. We shall now proceed to show two things:

(EccL HUt, Mosheim, p. 40.) AGC3. Rgnres. We shall now proceed to show two things: P K K P / P V -K VX - x K D K GFF G D G DKX X P P F D K 8 6 Y X X X P 9 ] K V Y 0 - D 00 D P F G K K PXVK P > G K K V v v v v PK P v D > v Y v D P > P v v - v x V D - ()zx q - & 9 K x K > % x v - - P x

More information

Two dimensional manifolds

Two dimensional manifolds Two dimensional manifolds We are given a real two-dimensional manifold, M. A point of M is denoted X and local coordinates are X (x, y) R 2. If we use different local coordinates, (x, y ), we have x f(x,

More information

THE LAX PAIR FOR THE MKDV HIERARCHY. Peter A. Clarkson, Nalini Joshi & Marta Mazzocco

THE LAX PAIR FOR THE MKDV HIERARCHY. Peter A. Clarkson, Nalini Joshi & Marta Mazzocco Séminaires & Congrès 14, 006, p. 53 64 THE LAX PAIR FOR THE MKDV HIERARCHY by Peter A. Clarkson, Nalini Joshi & Marta Mazzocco Abstract. In this paper we give an algorithmic method of deriving the Lax

More information

Partial Differential Equations

Partial Differential Equations Partial Differential Equations Lecture Notes Dr. Q. M. Zaigham Zia Assistant Professor Department of Mathematics COMSATS Institute of Information Technology Islamabad, Pakistan ii Contents 1 Lecture 01

More information

Rogue periodic waves for mkdv and NLS equations

Rogue periodic waves for mkdv and NLS equations Rogue periodic waves for mkdv and NLS equations Jinbing Chen and Dmitry Pelinovsky Department of Mathematics, McMaster University, Hamilton, Ontario, Canada http://dmpeli.math.mcmaster.ca AMS Sectional

More information

P A L A C E P IE R, S T. L E O N A R D S. R a n n o w, q u a r r y. W WALTER CR O TC H, Esq., Local Chairman. E. CO O PER EVANS, Esq.,.

P A L A C E P IE R, S T. L E O N A R D S. R a n n o w, q u a r r y. W WALTER CR O TC H, Esq., Local Chairman. E. CO O PER EVANS, Esq.,. ? ( # [ ( 8? [ > 3 Q [ ««> » 9 Q { «33 Q> 8 \ \ 3 3 3> Q»«9 Q ««« 3 8 3 8 X \ [ 3 ( ( Z ( Z 3( 9 9 > < < > >? 8 98 ««3 ( 98 < # # Q 3 98? 98 > > 3 8 9 9 ««««> 3 «>

More information

Painlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations. Abstract

Painlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations. Abstract Painlevé Test for the Certain (2+1)-Dimensional Nonlinear Evolution Equations T. Alagesan and Y. Chung Department of Information and Communications, Kwangju Institute of Science and Technology, 1 Oryong-dong,

More information

MACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

MACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS . MACSYMA PROGRAM FOR THE PAINLEVÉ TEST FOR NONLINEAR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Willy Hereman Mathematics Department and Center for the Mathematical Sciences University of Wisconsin at

More information

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

More information

Dispersive nonlinear partial differential equations

Dispersive nonlinear partial differential equations Dispersive nonlinear partial differential equations Elena Kartashova 16.02.2007 Elena Kartashova (RISC) Dispersive nonlinear PDEs 16.02.2007 1 / 25 Mathematical Classification of PDEs based on the FORM

More information

arxiv: v1 [nlin.si] 2 May 2017

arxiv: v1 [nlin.si] 2 May 2017 On decomposition of the ABS lattice equations and related Bäcklund transformations arxiv:705.00843v [nlin.si] 2 May 207 Danda Zhang, Da-jun Zhang Department of Mathematics, Shanghai University, Shanghai

More information

PanHomc'r I'rui;* :".>r '.a'' W"»' I'fltolt. 'j'l :. r... Jnfii<on. Kslaiaaac. <.T i.. %.. 1 >

PanHomc'r I'rui;* :.>r '.a'' W»' I'fltolt. 'j'l :. r... Jnfii<on. Kslaiaaac. <.T i.. %.. 1 > 5 28 (x / &» )»(»»» Q ( 3 Q» (» ( (3 5» ( q 2 5 q 2 5 5 8) 5 2 2 ) ~ ( / x {» /»»»»» (»»» ( 3 ) / & Q ) X ] Q & X X X x» 8 ( &» 2 & % X ) 8 x & X ( #»»q 3 ( ) & X 3 / Q X»»» %» ( z 22 (»» 2» }» / & 2 X

More information

A Generalization of the Sine-Gordon Equation to 2+1 Dimensions

A Generalization of the Sine-Gordon Equation to 2+1 Dimensions Journal of Nonlinear Mathematical Physics Volume 11, Number (004), 164 179 Article A Generalization of the Sine-Gordon Equation to +1 Dimensions PGESTÉVEZ and J PRADA Area de Fisica Teórica, Facultad de

More information

Lump solutions to dimensionally reduced p-gkp and p-gbkp equations

Lump solutions to dimensionally reduced p-gkp and p-gbkp equations Nonlinear Dyn DOI 10.1007/s11071-015-2539- ORIGINAL PAPER Lump solutions to dimensionally reduced p-gkp and p-gbkp equations Wen Xiu Ma Zhenyun Qin Xing Lü Received: 2 September 2015 / Accepted: 28 November

More information

Theorem 6.1 The addition defined above makes the points of E into an abelian group with O as the identity element. Proof. Let s assume that K is

Theorem 6.1 The addition defined above makes the points of E into an abelian group with O as the identity element. Proof. Let s assume that K is 6 Elliptic curves Elliptic curves are not ellipses. The name comes from the elliptic functions arising from the integrals used to calculate the arc length of ellipses. Elliptic curves can be parametrised

More information

Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations.

Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Alexey Shevyakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon,

More information

i r-s THE MEMPHIS, TENN., SATURDAY. DEGfMBER

i r-s THE MEMPHIS, TENN., SATURDAY. DEGfMBER N k Q2 90 k ( < 5 q v k 3X3 0 2 3 Q :: Y? X k 3 : \ N 2 6 3 N > v N z( > > :}9 [ ( k v >63 < vq 9 > k k x k k v 6> v k XN Y k >> k < v Y X X X NN Y 2083 00 N > N Y Y N 0 \ 9>95 z {Q ]k3 Q k x k k z x X

More information

Lecture II Search Method for Lax Pairs of Nonlinear Partial Differential Equations

Lecture II Search Method for Lax Pairs of Nonlinear Partial Differential Equations Lecture II Search Method for Lax Pairs of Nonlinear Partial Differential Equations Usama Al Khawaja, Department of Physics, UAE University, 24 Jan. 2012 First International Winter School on Quantum Gases

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch.

Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch. HOMEWORK From Dieter -3, -4, 3-7 Module #3 Transformation of stresses in 3-D READING LIST DIETER: Ch., pp. 7-36 Ch. 3 in Roesler Ch. in McClintock and Argon Ch. 7 in Edelglass The Stress Tensor z z x O

More information

Presentation for the Visiting Committee & Graduate Seminar SYMBOLIC COMPUTING IN RESEARCH. Willy Hereman

Presentation for the Visiting Committee & Graduate Seminar SYMBOLIC COMPUTING IN RESEARCH. Willy Hereman Presentation for the Visiting Committee & Graduate Seminar SYMBOLIC COMPUTING IN RESEARCH Willy Hereman Dept. of Mathematical and Computer Sciences Colorado School of Mines Golden, CO 80401-1887 Monday,

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information