Soliton solutions to the ABS list
|
|
- Camron Harris
- 5 years ago
- Views:
Transcription
1 to the ABS list Department of Physics, University of Turku, FIN Turku, Finland in collaboration with James Atkinson, Frank Nijhoff and Da-jun Zhang DIS-INI, February 2009
2 The setting CAC The setting Consistency Around a Cube We consider lattice maps defined on an elementary square: x n,m = x 00 = x x n+1,m = x 10 = x [1] = x x n,m+1 = x 01 = x [2] = x x n+1,m+1 = x 11 = x [12] = x x [2] x [12] x x [1] The four corner values are related by a multi-linear equation: Q(x, x [1], x [2], x [12] ; p, q) = 0 This allows propagation from initial data given on a staircase or on a corner.
3 The setting Consistency Around a Cube CAC - Consistency Around a Cube One definition of integrability for such lattices is that they should allow consistent extensions from 2D to 3D (and higher) Adjoin a third direction x n,m x n,m,k and construct a cube. x 101 x 001 x 011 x 111 x 100 x 000 x 010 x 110
4 The setting Consistency Around a Cube CAC - Consistency Around a Cube One definition of integrability for such lattices is that they should allow consistent extensions from 2D to 3D (and higher) Adjoin a third direction x n,m x n,m,k and construct a cube. x 101 x 001 x 011 x 111 x 100 x 000 x 010 x 110 Use the same map (with different parameters) on each side. Given values at black disks, we can compute values at open disks uniquely, but x 111 can be computed in 3 different ways, they must agree.
5 ABS classification CAC The setting Consistency Around a Cube It can be used as a method to classify integrable equation. [Adler, Bobenko and Suris, Commun. Math. Phys (2003)]. Two additional assumptions: symmetry (ε, σ = ±1): Q(x 000, x 100, x 010, x 110 ; p, q) =εq(x 000, x 010, x 100, x 110 ; q, p) =σq(x 100, x 000, x 110, x 010 ; p, q) tetrahedron property : x 111 does not depend on x 000.
6 ABS classification CAC The setting Consistency Around a Cube It can be used as a method to classify integrable equation. [Adler, Bobenko and Suris, Commun. Math. Phys (2003)]. Two additional assumptions: symmetry (ε, σ = ±1): Q(x 000, x 100, x 010, x 110 ; p, q) =εq(x 000, x 010, x 100, x 110 ; q, p) =σq(x 100, x 000, x 110, x 010 ; p, q) tetrahedron property : x 111 does not depend on x 000. Result: complete classification under these assumptions, 9 models.
7 The setting Consistency Around a Cube ABS results: List H: () (x ˆ x)( x ˆx) + q p = 0, () (x ˆ x)( x ˆx) + (q p)(x + x + ˆx + ˆ x) + q 2 p 2 = 0, () p(x x + ˆx ˆ x) q(x ˆx + x ˆ x) + δ(p 2 q 2 ) = 0. List A: (A1) p(x + ˆx)( x + ˆ x) q(x + x)(ˆx + ˆ x) δ 2 pq(p q) = 0, (A2) (q 2 p 2 )(x x ˆx ˆ x +1)+q(p 2 1)(x ˆx + x ˆ x) p(q 2 1)(x x +ˆx ˆ x) = 0.
8 The setting Consistency Around a Cube Main list: (Q1) p(x ˆx)( x ˆ x) q(x x)(ˆx ˆ x) + δ 2 pq(p q) = 0, (Q2) p(x ˆx)( x ˆ x) q(x x)(ˆx ˆ x) + pq(p q)(x + x + ˆx + ˆ x) pq(p q)(p 2 pq + q 2 ) = 0, (Q3) (q 2 p 2 )(x ˆ x + x ˆx) + q(p 2 1)(x x + ˆx ˆ x) p(q 2 1)(x ˆx + x ˆ x) δ 2 (p 2 q 2 )(p 2 1)(q 2 1)/(4pq) = 0, (Q4) (the root model from which others follow) a 0 x x ˆx ˆ x + a 1 (x x ˆx + x ˆx ˆ x + ˆx ˆ xx + ˆ xx x) + a 2 (x ˆ x + x ˆx)+ ā 2 (x x + ˆx ˆ x) + ã 2 (x ˆx + x ˆ x) + a 3 (x + x + ˆx + ˆ x) + a 4 = 0, where the a i depend on the lattice directions and are given in terms of Weierstrass elliptic functions. This was first derived by Adler as a superposition rule of BT s for the Krichever-Novikov equation. [Adler, Intl. Math. Res. Notices, # 1 (1998) 1-4]
9 The setting Consistency Around a Cube J.H., JNMP 12 Suppl 2, 223 (2005). a simpler Jacobi form for (Q4) of ABS: (h 1 f 2 h 2 f 1 )[(xx [1] x [12] x [2] + 1)f 1 f 2 (xx [12] + x [1] x [2] )] + (f 2 1 f 2 2 1)[(xx [1] + x [12] x [2] )f 1 (xx [2] + x [1] x [12] )f 2 ] = 0, hi 2 = fi 4 + δfi 2 + 1, parametrizable by Jacobi elliptic functions.
10 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case:
11 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case: 1 find a dependent variable transformation into Hirota bilinear form
12 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case: 1 find a dependent variable transformation into Hirota bilinear form 1 find a background or vacuum solution
13 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case: 1 find a dependent variable transformation into Hirota bilinear form 1 find a background or vacuum solution 2 find a 1-soliton-solution (1SS)
14 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case: 1 find a dependent variable transformation into Hirota bilinear form 1 find a background or vacuum solution 2 find a 1-soliton-solution (1SS) 3 use this info to guess the transformation
15 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case: 1 find a dependent variable transformation into Hirota bilinear form 1 find a background or vacuum solution 2 find a 1-soliton-solution (1SS) 3 use this info to guess the transformation 2 construct the first few soliton solutions perturbatively
16 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case: 1 find a dependent variable transformation into Hirota bilinear form 1 find a background or vacuum solution 2 find a 1-soliton-solution (1SS) 3 use this info to guess the transformation 2 construct the first few soliton solutions perturbatively 3 guess the general from (usually a determinant: Wronskian, Pfaffian etc) and prove it
17 Recall Hirota s direct method for constructing multisoliton solutions in the continuous case: 1 find a dependent variable transformation into Hirota bilinear form 1 find a background or vacuum solution 2 find a 1-soliton-solution (1SS) 3 use this info to guess the transformation 2 construct the first few soliton solutions perturbatively 3 guess the general from (usually a determinant: Wronskian, Pfaffian etc) and prove it Hirota s bilinear form is well suited for constructing soliton solutions, because the new dependent variable is a polynomial of exponentials with linear exponents.
18 What is a discrete Hirota bilinear form? The key property of an equation in Hirota s bilinear form is invariance under a gauge transformation or in the discrete case f i f i := e ax+bt f i. f j (n, m) f j (n, m) = An B m f j (n, m).
19 What is a discrete Hirota bilinear form? The key property of an equation in Hirota s bilinear form is invariance under a gauge transformation or in the discrete case f i f i := e ax+bt f i. f j (n, m) f j (n, m) = An B m f j (n, m). We say an equation is in Hirota bilinear (HB) form if it can be written as c j f j (n + ν + j, m + µ + j ) g j (n + ν j, m + µ j ) = 0 j where the index sums ν + j + ν j = ν s, µ + j + µ j = µ s do not depend on j.
20 The background solution First problem in the perturbative approach: What is the background solution?
21 The background solution First problem in the perturbative approach: What is the background solution? Atkinson: Take the CAC cube and insist that the solution is a fixed point of the bar shift. The side -equations are then Q(u, ũ, u, ũ; p, r) = 0, Q(u, û, u, û; q, r) = 0.
22 The background solution First problem in the perturbative approach: What is the background solution? Atkinson: Take the CAC cube and insist that the solution is a fixed point of the bar shift. The side -equations are then Q(u, ũ, u, ũ; p, r) = 0, Q(u, û, u, û; q, r) = 0. The equation is given by then the side-equations are (u ũ)(ũ û) (p q) = 0, (ũ u) 2 = r p, (û u) 2 = r q.
23 The background solution First problem in the perturbative approach: What is the background solution? Atkinson: Take the CAC cube and insist that the solution is a fixed point of the bar shift. The side -equations are then Q(u, ũ, u, ũ; p, r) = 0, Q(u, û, u, û; q, r) = 0. The equation is given by then the side-equations are (u ũ)(ũ û) (p q) = 0, (ũ u) 2 = r p, (û u) 2 = r q. For convenience we reparametrize (p, q) (a, b) by p = r a 2, q = r b 2.
24 The side-equations then factorize as (ũ u a)(ũ u + a) = 0, (û u b)(û u + b) = 0,
25 The side-equations then factorize as (ũ u a)(ũ u + a) = 0, (û u b)(û u + b) = 0, Since the factor that vanishes may depend on n, m we actually have to solve ũ u = ( 1) θ a, û u = ( 1) χ b, where θ, χ Z may depend on n, m.
26 The side-equations then factorize as (ũ u a)(ũ u + a) = 0, (û u b)(û u + b) = 0, Since the factor that vanishes may depend on n, m we actually have to solve ũ u = ( 1) θ a, û u = ( 1) χ b, where θ, χ Z may depend on n, m. From consistency θ, {n, 0},χ, {m, 0}.
27 The side-equations then factorize as (ũ u a)(ũ u + a) = 0, (û u b)(û u + b) = 0, Since the factor that vanishes may depend on n, m we actually have to solve ũ u = ( 1) θ a, û u = ( 1) χ b, where θ, χ Z may depend on n, m. From consistency θ, {n, 0},χ, {m, 0}. The set of possible background solution turns out to be an + bm + γ, 1 2 ( 1)n a + bm + γ, an ( 1)m b + γ, 1 2 ( 1)n a ( 1)m b + γ.
28 1SS CAC [Atkinson, JH and Nijhoff, JPhysA 41 (2008) ] The BT generating 1SS for is (u ũ)(ũ ū) = p κ, (u ū)(ū û) = κ q. u is the background solution an + bm + γ, ū is the new 1SS, κ is the soliton parameter (the parameter in the bar-direction).
29 1SS CAC [Atkinson, JH and Nijhoff, JPhysA 41 (2008) ] The BT generating 1SS for is (u ũ)(ũ ū) = p κ, (u ū)(ū û) = κ q. u is the background solution an + bm + γ, ū is the new 1SS, κ is the soliton parameter (the parameter in the bar-direction). We search for a new solution ū of the form ū = ū 0 + v, where ū 0 is the bar-shifted background solution ū 0 = an + bm + k + λ.
30 For v we then find (in the case of ) where ṽ = Ev v + F, v = Gv v + H, E = (a +k), F = (a k), G = (b +k), H = (b k), and k is related to κ by κ = r k 2.
31 For v we then find (in the case of ) where ṽ = Ev v + F, v = Gv v + H, E = (a +k), F = (a k), G = (b +k), H = (b k), and k is related to κ by κ = r k 2. Introducing v = f /g and Φ = (g, f ) T we can write the v-equations as matrix equations Φ(n +1, m) = N(n, m)φ(n, m), where N(n, m) = Λ Φ(n, m +1) = M(n, m)φ(n, m), ( ) ( ) E 0, M(n, m) = Λ G 0, 1 F 1 H In this case E, F, G, H are constants and we can choose Λ = Λ = 1.
32 Since the matrices N, M commute it is easy to find ( E n G m 0 Φ(n, m) = F n H m E n G m F n H m 2k ) Φ(0, 0).
33 Since the matrices N, M commute it is easy to find ( E n G m 0 Φ(n, m) = F n H m If we let ρ n,m = ( E F (ρ takes the role of e η ) E n G m F n H m 2k ) n ( ) G m ρ 0,0 = H ( a + k a k ) Φ(0, 0). ) n ( ) b + k m ρ 0,0, b k
34 Since the matrices N, M commute it is easy to find ( E n G m 0 Φ(n, m) = F n H m If we let ρ n,m = ( E F E n G m F n H m 2k ) n ( ) G m ρ 0,0 = H ( a + k a k (ρ takes the role of e η ) then we obtain v n,m = 2kρ n,m 1 + ρ n,m. Finally we obtain the 1SS for : u (1SS) n,m ) Φ(0, 0). ) n ( ) b + k m ρ 0,0, b k = (an + bm + λ) + k + 2kρ n,m 1 + ρ n,m.
35 Bilinearizing transformation In an explicit form the 1SS is u 1SS n,m = an + bm + λ + k(1 ρn,m) 1+ρ n,m
36 Bilinearizing transformation In an explicit form the 1SS is u 1SS n,m = an + bm + λ + k(1 ρn,m) 1+ρ n,m This suggests the dependent variable transformation u n,m = an + bm + λ g n,m f n,m.
37 Bilinearizing transformation In an explicit form the 1SS is u 1SS n,m = an + bm + λ + k(1 ρn,m) 1+ρ n,m This suggests the dependent variable transformation Then it is easy to show that (u ũ)(ũ û) p + q where u n,m = an + bm + λ g n,m f n,m. = [ H 1 + (a b)f f ][ + (a + b) f f ] /(f f f f ) + (a 2 b 2 ), H 1 ĝ f g f + (a b)( f f f f ) = 0, H 2 g f gf + (a + b)(f f f f ) = 0.
38 Casoratians CAC For given functions ϕ i (n, m, h) we define the column vectors ϕ(n, m, h) = (ϕ 1 (n, m, h), ϕ 2 (n, m, h),, ϕ N (n, m, h)) T,
39 Casoratians CAC For given functions ϕ i (n, m, h) we define the column vectors ϕ(n, m, h) = (ϕ 1 (n, m, h), ϕ 2 (n, m, h),, ϕ N (n, m, h)) T, and then compose the N N Casorati matrix from columns with different shifts h i, and then the determinant C n,m (ϕ; {h i }) = ϕ(n, m, h 1 ), ϕ(n, m, h 2 ),, ϕ(n, m, h N ).
40 Casoratians CAC For given functions ϕ i (n, m, h) we define the column vectors ϕ(n, m, h) = (ϕ 1 (n, m, h), ϕ 2 (n, m, h),, ϕ N (n, m, h)) T, and then compose the N N Casorati matrix from columns with different shifts h i, and then the determinant C n,m (ϕ; {h i }) = ϕ(n, m, h 1 ), ϕ(n, m, h 2 ),, ϕ(n, m, h N ). For example C 1 n,m(ϕ) := ϕ(n, m, 0), ϕ(n, m, 1),, ϕ(n, m, N 1) 0, 1,, N 1 N 1, C 2 n,m(ϕ) := ϕ(n, m, 0),, ϕ(n, m, N 2), ϕ(n, m, N) 0, 1,, N 2, N N 2, N,
41 Main result for CAC The equation (u ũ)(ũ û) (p q) = 0, is bilinearized with the substitution u n,m = an + bm + λ g n,m f n,m, p = r a 2, q = r b 2 and the resulting bilinear equations H 1 ĝ f g f + (a b)( f f f f ) = 0, H 2 g f gf + (a + b)(f f f f ) = 0. are solved by Casoratians f = N 1, g = N 2, N, with ψ given by ψ i (n, m, h; k i ) = ϱ + i k h i (a+k i ) n (b+k i ) m +ϱ i ( k i ) h (a k i ) n (b k i ) m.
42 The background solution/ The equation is (u ˆũ)(ũ û) + (q p)(u + ũ + û + ˆũ) + q 2 p 2 = 0. After reparametrization p = r a 2, q = r b 2, u = y r the side equations lead to the background solution u (0ss) nm = [an + bm + γ] r,
43 The 1SS/ CAC The 1SS is contructed as in the previous case using the cube. The result is where u 1SS n,m = U 2 n,m + 2kU n,m 1 ρ n,m 1 + ρ n,m + k 2, U n,m := an + bm + λ, ρ n,m = ( a + k a k ) n ( ) b + k m ρ 0,0, b k Note that in addition to the implicit dependency on the soliton parameter k through ρ, the 1SS also depends on k explicitly.
44 Bilinearizing transformation/ The bilinearization is given by u NSS n,m = U 2 n,m 2U n,m g f + h + s, f where f = N 1, g = N 2, N, s = N 3, N 1, N, h = N 2, N + 1, with the same entries as before. These Casoratians satisfy h s = f ( N i=1 k i 2 ) and H 1 ĝ f g f + (a b)( f f f f ) = 0, H 2 g f gf + (a + b)(f f f f ) = 0, H 3 (a + b) f g + a f g + bf g + f h f h = 0, H 4 (a b)f g + a f ĝ b f g + f ĥ f h = 0, H 5 b( f g f ĝ) + f ĥ + f s gĝ = 0.
45 In terms of these bilinear equations, can be given as = 5 H i P i, i=1 where [ P 1 = 4(a + b) (Ũ Ũ a 2 + b 2 ) f f Ũ f g (a + b)f g ], [ P 2 = 4 (a b)(û Ũ a 2 + b 2 ) f f + (Ũ Ũ a 2 + b 2 ) f ĝ [ P 3 = 4 [ P 4 = 4 Ũ Ũ f g (a b)ũf g ], (a b)u f f + Û f ĝ Ũ f g (a + b)(ûf f f g) + Ũ( f g f g) P 5 = 4(a 2 b 2 ) f f, ] f ĥ + f h, ],
46 The background solution/ The equation is δ p(uũ + û ũ) q(uû + ũ ũ) δ(q 2 p 2 ) = 0. The side equation for T (x) = x read r(u 2 +ũ 2 ) 2puũ = δ(p 2 r 2 ), r(u 2 +û 2 ) 2quû = δ(q 2 r 2 ).
47 The background solution/ The equation is δ p(uũ + û ũ) q(uû + ũ ũ) δ(q 2 p 2 ) = 0. The side equation for T (x) = x read r(u 2 +ũ 2 ) 2puũ = δ(p 2 r 2 ), r(u 2 +û 2 ) 2quû = δ(q 2 r 2 ). The parametrization u nm = Ae ynm + Be ynm, p = r 1 + α2 2α, q = r 1 + β2 2β, then leads to the 0SS: u (0SS) = Aα n β m + Bα n β m, AB = rδ/4.
48 The 1SS/ CAC The algorithmic procedure given before leads to the 1SS u 1SS n,m = Aαn β m (1 + κ 2 ρ n,m ) + Bα n β m (1 + κ 2 ρ n,m ) 1 + ρ n,m, where κ is the soliton parameter and ρ is defined by ( ) S n ( ) T m ( α 2 κ 2 ) 1 n ( β 2 κ 2 ) 1 m ρ n,m = ρ 0,0 = Ω α 2 κ 2 β 2 κ 2 ρ 0,0,
49 The 1SS/ CAC The algorithmic procedure given before leads to the 1SS u 1SS n,m = Aαn β m (1 + κ 2 ρ n,m ) + Bα n β m (1 + κ 2 ρ n,m ) 1 + ρ n,m, where κ is the soliton parameter and ρ is defined by ( ) S n ( ) T m ( α 2 κ 2 ) 1 n ( β 2 κ 2 ) 1 m ρ n,m = ρ 0,0 = Ω α 2 κ 2 β 2 κ 2 ρ 0,0, This suggests the reparametrization α 2 = a c a+c, β2 = b c b+c, κ2 = k c k+c, p2 = r 2 c 2 c 2 a 2, q 2 = r 2 c 2 c 2 b 2. with a new parameter c, after which ( ) a + k n ( ) b + k m ρ n,m = ρ 0,0. a k b k
50 Bilinearizing transformation/ The bilinearization is given by u NSS n,m = A α n β m f f + B α n β m f f, 4AB = rδ, where f = N 1 as before but its etries are given by ψ i (n, m, h) = ϱ + i (a+k i ) n (b+k i ) m (c+k i ) h +ϱ i (a k i ) n (b k i ) m (c k i ) h, and the bar-shift is in the h-index, associated with c.
51 There are (at least) two bilinearizations of, one in terms of another in terms of B 1 2cf f + (a c) f f (a + f = 0, c) f B 2 2cf f + (b c) f f (b + f = 0. c) f B 1 (b + c) f f + (a c)f f (a + b) f f = 0, B 2 (c b) f f (a + c)f + (a + f = 0, f b) f B 3 (c a)(b + f + (a + c)(b f + 2c(a b)f f = 0. c) f c) f These equations can be proved with the usual Laplace expansion techniques of the Casoratians.
52 Conclusions CAC We have considered the construction of multisoliton solution and bilinearization for the quadrilateral lattice equations in the ABS list. The technique presented in this talk has been applied for all but Q3,Q4. The structure of the soliton solution is similar to those of the Hirota-Miwa equation
53 Conclusions CAC We have considered the construction of multisoliton solution and bilinearization for the quadrilateral lattice equations in the ABS list. The technique presented in this talk has been applied for all but Q3,Q4. The structure of the soliton solution is similar to those of the Hirota-Miwa equation Explicit NSS has also been derived for Q3, using a connection to the NQC equation. (Atkinson, JH and Nijhoff, JPhysA 41 (2008) )
54 Conclusions CAC We have considered the construction of multisoliton solution and bilinearization for the quadrilateral lattice equations in the ABS list. The technique presented in this talk has been applied for all but Q3,Q4. The structure of the soliton solution is similar to those of the Hirota-Miwa equation Explicit NSS has also been derived for Q3, using a connection to the NQC equation. (Atkinson, JH and Nijhoff, JPhysA 41 (2008) ) The result for Q4 is less explicit. This is expected since elliptic functions enter big time. (Atkinson, JH and Nijhoff, JPhysA 40 (2007) F1.)
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 informationarxiv: 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 informationarxiv: 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 informationIntegrability 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 informationElliptic 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 informationSymbolic 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 informationWeak 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 informationIntroduction 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 informationSymmetry 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 informationFrom 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 informationarxiv: v1 [nlin.si] 2 Dec 2011
DIRECT LINEARIZATION OF EXTENDED LATTICE BSQ SYSTEMS DA-JUN ZHANG & SONG-LIN ZHAO AND FRANK W NIJHOFF arxiv:1112.0525v1 [nlin.si] 2 Dec 2011 Abstract. The direct linearization structure is presented of
More informationGEOMETRY 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 informationSymbolic Computation of Lax Pairs of Systems of Partial Difference Equations Using Consistency Around the Cube
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,
More informationOn 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 informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationOn universality of critical behaviour in Hamiltonian PDEs
Riemann - Hilbert Problems, Integrability and Asymptotics Trieste, September 23, 2005 On universality of critical behaviour in Hamiltonian PDEs Boris DUBROVIN SISSA (Trieste) 1 Main subject: Hamiltonian
More informationModeling Interactions of Soliton Trains. Effects of External Potentials. Part II
Modeling Interactions of Soliton Trains. Effects of External Potentials. Part II Michail Todorov Department of Applied Mathematics and Computer Science Technical University of Sofia, Bulgaria Work done
More informationWRT in 2D: Poisson Example
WRT in 2D: Poisson Example Consider 2 u f on [, L x [, L y with u. WRT: For all v X N, find u X N a(v, u) such that v u dv v f dv. Follows from strong form plus integration by parts: ( ) 2 u v + 2 u dx
More informationClassification 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 informationThe Solitary Wave Solutions of Zoomeron Equation
Applied Mathematical Sciences, Vol. 5, 011, no. 59, 943-949 The Solitary Wave Solutions of Zoomeron Equation Reza Abazari Deparment of Mathematics, Ardabil Branch Islamic Azad University, Ardabil, Iran
More informationMultisoliton Interaction of Perturbed Manakov System: Effects of External Potentials
Multisoliton Interaction of Perturbed Manakov System: Effects of External Potentials Michail D. Todorov Faculty of Applied Mathematics and Computer Science Technical University of Sofia, Bulgaria (Work
More informationExact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially Integrable Equations
Thai Journal of Mathematics Volume 5(2007) Number 2 : 273 279 www.math.science.cmu.ac.th/thaijournal Exact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially
More informationgroup: 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 informationA variational perspective on continuum limits of ABS and lattice GD equations
A variational perspective on continuum limits of ABS and lattice GD equations Mats Vermeeren Technische Universität Berlin vermeeren@mathtu-berlinde arxiv:80855v [nlinsi] 5 Nov 08 Abstract A pluri-lagrangian
More informationarxiv: 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 informationCost-extended control systems on SO(3)
Cost-extended control systems on SO(3) Ross M. Adams Mathematics Seminar April 16, 2014 R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 1 / 34 Outline 1 Introduction 2 Control systems
More informationA Finite Genus Solution of the Veselov s Discrete Neumann System
Commun. Theor. Phys. 58 202 469 474 Vol. 58, No. 4, October 5, 202 A Finite Genus Solution of the Veselov s Discrete Neumann System CAO Ce-Wen and XU Xiao-Xue Æ Department of Mathematics, Zhengzhou University,
More informationMultisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system
arxiv:407.7743v3 [math-ph] 3 Jan 205 Multisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system L. Cortés Vega*, A. Restuccia**, A. Sotomayor* January 5,
More informationSemiclassical spin coherent state method in the weak spin-orbit coupling limit
arxiv:nlin/847v1 [nlin.cd] 9 Aug Semiclassical spin coherent state method in the weak spin-orbit coupling limit Oleg Zaitsev Institut für Theoretische Physik, Universität Regensburg, D-934 Regensburg,
More informationSymbolic 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 informationPainlevé 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 informationExact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation
Exact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation Francesco Demontis (based on a joint work with C. van der Mee) Università degli Studi di Cagliari Dipartimento di Matematica e Informatica
More informationDispersionless integrable systems in 3D and Einstein-Weyl geometry. Eugene Ferapontov
Dispersionless integrable systems in 3D and Einstein-Weyl geometry Eugene Ferapontov Department of Mathematical Sciences, Loughborough University, UK E.V.Ferapontov@lboro.ac.uk Based on joint work with
More informationGeneralized Burgers equations and Miura Map in nonabelian ring. nonabelian rings as integrable systems.
Generalized Burgers equations and Miura Map in nonabelian rings as integrable systems. Sergey Leble Gdansk University of Technology 05.07.2015 Table of contents 1 Introduction: general remarks 2 Remainders
More informationFinsler Structures and The Standard Model Extension
Finsler Structures and The Standard Model Extension Don Colladay New College of Florida Talk presented at Miami 2011 work done in collaboration with Patrick McDonald Overview of Talk Modified dispersion
More informationarxiv: v1 [math.ap] 20 Nov 2007
Long range scattering for the Maxwell-Schrödinger system with arbitrarily large asymptotic data arxiv:0711.3100v1 [math.ap] 20 Nov 2007 J. Ginibre Laboratoire de Physique Théorique Université de Paris
More informationSome Elliptic Traveling Wave Solutions to the Novikov-Veselov Equation
Progress In Electromagnetics Research Symposium 006, Cambridge, USA, March 6-9 59 Some Elliptic Traveling Wave Solutions to the Novikov-Veselov Equation J. Nickel, V. S. Serov, and H. W. Schürmann University
More informationMATH 387 ASSIGNMENT 2
MATH 387 ASSIGMET 2 SAMPLE SOLUTIOS BY IBRAHIM AL BALUSHI Problem 4 A matrix A ra ik s P R nˆn is called symmetric if a ik a ki for all i, k, and is called positive definite if x T Ax ě 0 for all x P R
More informationTalk 3: Examples: Star exponential
Akira Yoshioka 2 7 February 2013 Shinshu Akira Yoshioka 3 2 Dept. Math. Tokyo University of Science 1. Star exponential 1 We recall the definition of star exponentials. 2 We show some construction of star
More informationCanonicity of Bäcklund transformation: r-matrix approach. I. arxiv:solv-int/ v1 25 Mar 1999
LPENSL-Th 05/99 solv-int/990306 Canonicity of Bäcklund transformation: r-matrix approach. I. arxiv:solv-int/990306v 25 Mar 999 E K Sklyanin Laboratoire de Physique 2, Groupe de Physique Théorique, ENS
More informationCh 5.4: Regular Singular Points
Ch 5.4: Regular Singular Points! Recall that the point x 0 is an ordinary point of the equation if p(x) = Q(x)/P(x) and q(x)= R(x)/P(x) are analytic at at x 0. Otherwise x 0 is a singular point.! Thus,
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationNILPOTENT QUANTUM MECHANICS AND SUSY
Ó³ Ÿ. 2011.. 8, º 3(166).. 462Ä466 ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ. ˆŸ NILPOTENT QUANTUM MECHANICS AND SUSY A. M. Frydryszak 1 Institute of Theoretical Physics, University of Wroclaw, Wroclaw, Poland Formalism where
More informationExact Solutions of Supersymmetric KdV-a System via Bosonization Approach
Commun. Theor. Phys. 58 1 617 6 Vol. 58, No. 5, November 15, 1 Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach GAO Xiao-Nan Ô é, 1 YANG Xu-Dong Êü, and LOU Sen-Yue 1,, 1 Department
More informationSpin foam vertex and loop gravity
Spin foam vertex and loop gravity J Engle, R Pereira and C Rovelli Centre de Physique Théorique CNRS Case 907, Université de la Méditerranée, F-13288 Marseille, EU Roberto Pereira, Loops 07 Morelia 25-30
More informationa = ( a σ )( b σ ) = a b + iσ ( a b) mω 2! x + i 1 2! x i 1 2m!ω p, a = mω 2m!ω p Physics 624, Quantum II -- Final Exam
Physics 624, Quantum II -- Final Exam Please show all your work on the separate sheets provided (and be sure to include your name). You are graded on your work on those pages, with partial credit where
More informationDiagonalization of the Coupled-Mode System.
Diagonalization of the Coupled-Mode System. Marina Chugunova joint work with Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Collaborators: Mason A. Porter, California Institute
More informationRecursion Systems and Recursion Operators for the Soliton Equations Related to Rational Linear Problem with Reductions
GMV The s Systems and for the Soliton Equations Related to Rational Linear Problem with Reductions Department of Mathematics & Applied Mathematics University of Cape Town XIV th International Conference
More informationPositive entries of stable matrices
Positive entries of stable matrices Shmuel Friedland Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago Chicago, Illinois 60607-7045, USA Daniel Hershkowitz,
More information7-7 Multiplying Polynomials
Example 1: Multiplying Monomials A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) (6 3)(y 3 y 5 ) 18y 8 Group factors with like bases together. B. (3mn 2 ) (9m 2 n) Example 1C: Multiplying Monomials Group factors with
More informationAN ELEMENTARY APPROACH TO THE MACDONALD IDENTITIES* DENNIS STANTON
AN ELEMENTARY APPROACH TO THE MACDONALD IDENTITIES* DENNIS STANTON Abstract. Elementary proofs are given for the infinite families of Macdonald identities. The reflections of the Weyl group provide sign-reversing
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationRelativistic 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 informationLECTURE 3: Quantization and QFT
LECTURE 3: Quantization and QFT Robert Oeckl IQG-FAU & CCM-UNAM IQG FAU Erlangen-Nürnberg 14 November 2013 Outline 1 Classical field theory 2 Schrödinger-Feynman quantization 3 Klein-Gordon Theory Classical
More informationThe Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases
arxiv:8.976v [math.rt] 3 Dec 8 The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases Central European University Amadou Keita (keita amadou@student.ceu.edu December 8 Abstract The most famous
More information1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...
Contents 1 Geometry of R 1.1 Conic Sections............................................ 1. Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................
More informationKac-Moody Algebras. Ana Ros Camacho June 28, 2010
Kac-Moody Algebras Ana Ros Camacho June 28, 2010 Abstract Talk for the seminar on Cohomology of Lie algebras, under the supervision of J-Prof. Christoph Wockel Contents 1 Motivation 1 2 Prerequisites 1
More informationDiscontinuous Galerkin methods for fractional diffusion problems
Discontinuous Galerkin methods for fractional diffusion problems Bill McLean Kassem Mustapha School of Maths and Stats, University of NSW KFUPM, Dhahran Leipzig, 7 October, 2010 Outline Sub-diffusion Equation
More informationStarting Methods for Two-Step Runge Kutta Methods of Stage-Order 3 and Order 6
Cambridge International Science Publishing Cambridge CB1 6AZ Great Britain Journal of Computational Methods in Sciences and Engineering vol. 2, no. 3, 2, pp. 1 3 ISSN 1472 7978 Starting Methods for Two-Step
More informationVortex knots dynamics and momenta of a tangle:
Lecture 2 Vortex knots dynamics and momenta of a tangle: - Localized Induction Approximation (LIA) and Non-Linear Schrödinger (NLS) equation - Integrable vortex dynamics and LIA hierarchy - Torus knot
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationD-branes in non-abelian gauged linear sigma models
D-branes in non-abelian gauged linear sigma models Johanna Knapp Vienna University of Technology Bonn, September 29, 2014 Outline CYs and GLSMs D-branes in GLSMs Conclusions CYs and GLSMs A Calabi-Yau
More informationPeriod Domains. Carlson. June 24, 2010
Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes
More informationMatrix KP: tropical limit and Yang-Baxter maps.
Matrix KP: tropical limit and Yang-Baxter maps Aristophanes Dimakis a and Folkert Müller-Hoissen b a Dept of Financial and Management Engineering University of the Aegean Chios Greece E-mail: dimakis@aegeangr
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationSuperintegrable 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 informationCasimir elements for classical Lie algebras. and affine Kac Moody algebras
Casimir elements for classical Lie algebras and affine Kac Moody algebras Alexander Molev University of Sydney Plan of lectures Plan of lectures Casimir elements for the classical Lie algebras from the
More informationON 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 informationPerturbative Static Four-Quark Potentials
arxiv:hep-ph/9508315v1 17 Aug 1995 Perturbative Static Four-Quark Potentials J.T.A.Lang, J.E.Paton, Department of Physics (Theoretical Physics) 1 Keble Road, Oxford OX1 3NP, UK A.M. Green Research Institute
More informationDynamic and Thermodynamic Stability of Black Holes and Black Branes
Dynamic and Thermodynamic Stability of Black Holes and Black Branes Robert M. Wald with Stefan Hollands arxiv:1201.0463 Commun. Math. Phys. 321, 629 (2013) (see also K. Prabhu and R.M. Wald, Commun. Math.
More informationQuantum Dynamics. March 10, 2017
Quantum Dynamics March 0, 07 As in classical mechanics, time is a parameter in quantum mechanics. It is distinct from space in the sense that, while we have Hermitian operators, X, for position and therefore
More informationThe 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 information7. The classical and exceptional Lie algebras * version 1.4 *
7 The classical and exceptional Lie algebras * version 4 * Matthew Foster November 4, 06 Contents 7 su(n): A n 7 Example: su(4) = A 3 4 7 sp(n): C n 5 73 so(n): D n 9 74 so(n + ): B n 3 75 Classical Lie
More informationErrata 1. p. 5 The third line from the end should read one of the four rows... not one of the three rows.
Errata 1 Front inside cover: e 2 /4πɛ 0 should be 1.44 10 7 ev-cm. h/e 2 should be 25800 Ω. p. 5 The third line from the end should read one of the four rows... not one of the three rows. p. 8 The eigenstate
More informationDark energy constraints using matter density fluctuations
Dark energy constraints using matter density fluctuations Ana Pelinson Astronomy Department, IAG/USP Collaborators: R. Opher (IAG/USP), J. Solà, J. Grande (Dept. ECM/UB) I Campos do Jordão, April 5-30,
More informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More information8.821 F2008 Lecture 8: Large N Counting
8.821 F2008 Lecture 8: Large N Counting Lecturer: McGreevy Scribe: Swingle October 4, 2008 1 Introduction Today we ll continue our discussion of large N scaling in the t Hooft limit of quantum matrix models.
More informationProjective Bivector Parametrization of Isometries Part I: Rodrigues' Vector
Part I: Rodrigues' Vector Department of Mathematics, UACEG International Summer School on Hyprcomplex Numbers, Lie Algebras and Their Applications, Varna, June 09-12, 2017 Eulerian Approach The Decomposition
More informationModels with fundamental length 1 and the finite-dimensional simulations of Big Bang
Models with fundamental length 1 and the finite-dimensional simulations of Big Bang Miloslav Znojil Nuclear Physics Institute ASCR, 250 68 Řež, Czech Republic 1 talk in Dresden (June 22, 2011, 10.50-11.35
More informationOblique derivative problems for elliptic and parabolic equations, Lecture II
of the for elliptic and parabolic equations, Lecture II Iowa State University July 22, 2011 of the 1 2 of the of the As a preliminary step in our further we now look at a special situation for elliptic.
More informationA COMPLETENESS STUDY ON CERTAIN 2 2 LAX PAIRS Mike Hay. University of Sydney.
Lax COMPLETENESS STUDY ON CERTIN 2 2 LX PIRS University of Sydney email: michaelh@maths.usyd.edu.au Lax pair is a pair of linear problems who s compatibility is associated with a nonlinear equation. φ
More informationLinear degree growth in lattice equations
Linear degree growth in lattice equations Dinh T. Tran and John A. G. Roberts School of Mathematics and Statistics, University of New South Wales, Sydney 2052 Australia February 28, 2017 arxiv:1702.08295v1
More informationNumerical Approximation of Phase Field Models
Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School
More informationRelativistic Waves and Quantum Fields
Relativistic Waves and Quantum Fields (SPA7018U & SPA7018P) Gabriele Travaglini December 10, 2014 1 Lorentz group Lectures 1 3. Galileo s principle of Relativity. Einstein s principle. Events. Invariant
More informationOn Hamiltonian perturbations of hyperbolic PDEs
Bologna, September 24, 2004 On Hamiltonian perturbations of hyperbolic PDEs Boris DUBROVIN SISSA (Trieste) Class of 1+1 evolutionary systems w i t +Ai j (w)wj x +ε (B i j (w)wj xx + 1 2 Ci jk (w)wj x wk
More informationGauge Invariant Variables for SU(2) Yang-Mills Theory
Gauge Invariant Variables for SU(2) Yang-Mills Theory Cécile Martin Division de Physique Théorique, Institut de Physique Nucléaire F-91406, Orsay Cedex, France. Abstract We describe a nonperturbative calculation
More informationarxiv:hep-th/ v1 7 Feb 1992
CP N 1 MODEL WITH A CHERN-SIMONS TERM arxiv:hep-th/9202026v1 7 Feb 1992 G. Ferretti and S.G. Rajeev Research Institute for Theoretical Physics Siltavuorenpenger 20 C SF-00170 Helsinki, Finland Abstract
More informationPrime Numbers and Irrational Numbers
Chapter 4 Prime Numbers and Irrational Numbers Abstract The question of the existence of prime numbers in intervals is treated using the approximation of cardinal of the primes π(x) given by Lagrange.
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationTravelling Wave Solutions and Conservation Laws of Fisher-Kolmogorov Equation
Gen. Math. Notes, Vol. 18, No. 2, October, 2013, pp. 16-25 ISSN 2219-7184; Copyright c ICSRS Publication, 2013 www.i-csrs.org Available free online at http://www.geman.in Travelling Wave Solutions and
More informationFinite Elements for Elastic Shell Models in
Elastic s in Advisor: Matthias Heinkenschloss Computational and Applied Mathematics Rice University 13 April 2007 Outline Elasticity in Differential Geometry of Shell Geometry and Equations The Plate Model
More informationSoliton resonance and web structure: the Davey-Stewartson equation
Soliton resonance and web structure: the Davey-Stewartson equation Gino Biondini State University of New York at Buffalo joint work with Sarbarish Chakravarty, University of Colorado at Colorado Springs
More informationCLASSIFICATION OF NOVIKOV ALGEBRAS
CLASSIFICATION OF NOVIKOV ALGEBRAS DIETRICH BURDE AND WILLEM DE GRAAF Abstract. We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification
More informationand in each case give the range of values of x for which the expansion is valid.
α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else P Kraft Further Maths A (MFPD)
More informationTRIANGLE CENTERS DEFINED BY QUADRATIC POLYNOMIALS
Math. J. Okayama Univ. 53 (2011), 185 216 TRIANGLE CENTERS DEFINED BY QUADRATIC POLYNOMIALS Yoshio Agaoka Abstract. We consider a family of triangle centers whose barycentric coordinates are given by quadratic
More informationStability of Black Holes and Black Branes. Robert M. Wald with Stefan Hollands arxiv: Commun. Math. Phys. (in press)
Stability of Black Holes and Black Branes Robert M. Wald with Stefan Hollands arxiv:1201.0463 Commun. Math. Phys. (in press) Stability It is of considerable interest to determine the linear stablity of
More informationNew Formal Solutions of Davey Stewartson Equation via Combined tanh Function Method with Symmetry Method
Commun. Theor. Phys. Beijing China 7 007 pp. 587 593 c International Academic Publishers Vol. 7 No. April 5 007 New Formal Solutions of Davey Stewartson Equation via Combined tanh Function Method with
More informationShow, for infinitesimal variations of nonabelian Yang Mills gauge fields:
Problem. Palatini Identity Show, for infinitesimal variations of nonabelian Yang Mills gauge fields: δf i µν = D µ δa i ν D ν δa i µ..) Begin by considering the following form of the field strength tensor
More informationQuantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams
Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The
More informationN-soliton solutions of two-dimensional soliton cellular automata
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)
More information