On the Model AC=BD and Trigram" Structures of the Soliton - - PowerPoint PPT Presentation
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship On the Model AC=BD and Trigram" Structures of the Soliton Theory Shou-Fu Tian joint work with Prof.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Shou-Fu Tian joint work with Prof. Hong-Qing Zhang (My PhD Supervisor)
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P . R. China.
Workshop on the tenth ASCM in Beijing, 2012
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Outline
1
The problem of soliton theory
2
The model AC=BD and its applications The model AC=BD Algebraic-geometry solution Sato theory
3
The Trigram structures and Trigram identities The Trigram structures and Trigram identities Exterior decomposition Trigram identities Interior decomposition Trigram identities
4
The relationship between Tau function and Theta function
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
It is well known that searching for soliton solutions of the nonlinear differential equations is one of the most important topics in soliton theory; therefore, various methods of finding soliton solutions have been developed. Some of the most important methods are
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
It is well known that searching for soliton solutions of the nonlinear differential equations is one of the most important topics in soliton theory; therefore, various methods of finding soliton solutions have been developed. Some of the most important methods are the inverse scattering transformation (IST), Algebraic geometry solution, Lie group method, Sato theory and Hirota’s bilinear method, Determinant technique, and so on.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Question What is the unified and fundamental structure of all soliton equations?
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Question What is the unified and fundamental structure of all soliton equations? Recently, Hirota (2004) indicates that soliton solutions expressed by the Pfaffans and soliton equations (or the bilinear equations) are just equivalent to the Pfaffan identities.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
In 1978, one of authors Zhang proposed his “AC=BD" theory to get general solutions of linear partial differential equations. After that, he extend the “AC=BD" theory to solve some nonlinear partial differential equations.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
In 1978, one of authors Zhang proposed his “AC=BD" theory to get general solutions of linear partial differential equations. After that, he extend the “AC=BD" theory to solve some nonlinear partial differential equations. In this talk further investigate AC=BD theory. present compositively our “Trigram" theory. present a new method to construct the relationship between Tau- and Theta- function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD
In this section, by virtue of Zhang (1978) and (2008), we recall the following results.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD
In this section, by virtue of Zhang (1978) and (2008), we recall the following results. Definition 2.1 Suppose X is a linear space and assume that A, B, C and D are some operators from X to X. For an arbitrary v ∈ X satisfying ACv = B · Dv (2.1) where AC(v) = A(Cv), BD(v) = B · (Dv), then ACv = B · Dv is called AC=B·D model.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD
Definition 2.2 For an arbitrary operator A, if there exist operators B, C, D, satisfying AC = B · D, C ker D=Ker A, where Ker A = {u|Au = 0}, Ker D = {v|Dv = 0}, then Au = 0 is called C-D integrable system. If C ker D Ker A, then Au = 0 is called local integrable system.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD
Definition 2.2 For an arbitrary operator A, if there exist operators B, C, D, satisfying AC = B · D, C ker D=Ker A, where Ker A = {u|Au = 0}, Ker D = {v|Dv = 0}, then Au = 0 is called C-D integrable system. If C ker D Ker A, then Au = 0 is called local integrable system. Theorem 2.3 Suppose X is a linear space and assume that A, B, C and D are some operators from X to X. If AC = B · D, B0 = 0, C Ker D ⊇ Ker A, then u = Cv is a general solution of Au = 0 where v satisfies Dv = 0.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The model AC=BD
Theorem 2.4 Suppose X is a Banach space, AC : X → R is a function, and D is a monodromy operator. If ACD−1 is Gateaux differential, and ACv = 0 is derived by Dv = 0, then there exist an operator B satisfying ACv = BvDv.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution
Trace formulas and Dubrovin equations Step 1. Au=0: Here we take KdV hierarchy for example, whose Lax equation and associate zero-curvature equation , respectively, given by ˙ L = [L, M], Utn − Vn+1,x + [U, Vn+1] = 0, (2.2) where [L, M] = LM − ML, and U = U(z) =
−z + u
Vn+1 = Vn+1(z) =
Fn(z) −Hn+1(z) −Gn−1(z)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution
Step 2. u=Cv: Trace formula u = 2n
m=0 Em − 2 n j=1 µj, where µj are
zeros of Fn(z), and Em are zeros of R2n+1(z) satisfying the hyperelliptic curve
Kr : y2−R2r+1(z) = y2− 1 2
Fr(z) − 1 4
Fr(z)2
m=0 Em is
an arbitrary constant by comparison of z2n for R2n+1(z). From z = −2 n
j=1 µj, we obtain
ACv = −2
n
µj,tr + 2
n
µj,x Fr(µj) + 1 2
2n
Em Fr,x(µj).
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution
Step 3. Dv=0: Dubrovin equations
Dx
j v = 0 :
µj,x = −2iy( µj)
k=j
(µj − µk)−1, Bx
j = 2
Fr(µj), Dt
j v = 0 :
µj,tn = −2i Fr(µj)y( µj)
k=j
(µj − µk)−1, Bt
j = −2,
where ( µj, νj) is a new point in hyperelliptic curve Kn, that is
(2.3)
Now we obtain ACv = n
j=1 Bx j vDx j v + n j=1 Bt j vDt j v, i.e.,
AC = 2n
j=1 BjDj.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution
Its-Matveev formula Step 1. Au=0: Here we take a generalized Lax equation for example given by ˙ L = [L, M] with its general form : N (u, ut, ux1, . . . , uxN, . . .) = 0, (2.4) where X=(x1, . . . , xN), N is a differential polynomial.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution
Its-Matveev formula Step 1. Au=0: Here we take a generalized Lax equation for example given by ˙ L = [L, M] with its general form : N (u, ut, ux1, . . . , uxN, . . .) = 0, (2.4) where X=(x1, . . . , xN), N is a differential polynomial. Step 2. u=Cv: Its-Matveev formula: u = u0 + a∂m
Λ ln ϑ(ξ), where u0 is
an initial solution, a is a constant, Λ = xm1
1 xm2 2
. . . xmN
N ,
m = m1 + m2 + · · · + mN, ϑ(ξ) = ∞
n=−∞ eπiτn,n+2πiξ,n with
ξ = (ξ1, . . . , ξn), ξi = kix1 + lix2 + . . . + ρixN + ωit + εi, i = 1, 2, . . . , n, is a Riemann theta function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution
Step 3. Dv=0: Algebraic equations (See Tian & Zhang (2010)) Quasi-periodic wave: Dv=0:
1, . . . , m′ i, . . . , m′ N) =
L (0, . . . , 0, . . . , 0) = 0, m′
i is even,
1, . . . , m′ i, . . . , m′ N) =
L (0, . . . , 1, . . . , 0) = 0, m′
i is odd,
Bm′ = e2πiξ,m′+πiτ,m′, m′ = (m′
1, . . . , m′ N), k, ρ, ω ∈ CN, (2.5)
where L (m′) =
n∈Zn L (2πi2n − m′, k, . . . , 2πi2n −
m′, ω)eπi[τ(n−m′),n−m′+τn,n], k, . . . , ω ∈ Cn, and “” implies inner product, L is a bilinear form of Au=0: L (Dx1, . . . , DxN, Dt, c)ϑ(ξ) · ϑ(ξ) = 0.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Algebraic-geometry solution
Remark 3.1 Extending the Its-Matveev formula to superspace, we can show that the super-Its-Matveev formula is (see Tian & Zhang (2012)) u = u0 + aDm
Λ ln ϑ(ξ, ε, s),
(2.6) where u0 is an initial solution, a is a constant, Λ = xm1
1 xm2 2
. . . xmN
N , m = m1 + m2 + · · · + mN,
ϑ(ξ) = ∞
n=−∞ eπiτn,n+2πiξ,n with ξ = (ξ1, . . . , ξn),
ξi = kix1 + lix2 + · · · + ρixN + ωit + αiσ1θ1 + · · · γiσMθM + εi, i = 1, . . . , n, is a Riemann theta function [σi, θi are Grassmann
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Sato theory
Generalized Lax equation and Sato equation Step 1. Au=0: Here we take a generalized Lax equation for example given by ∂L ∂tn = [En, L] = EnL − LEn, (2.7) where En = (W∂nW −1)+, W is a microdifferntial operator, and ()+ denotes the differential part of the operator.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Sato theory
Generalized Lax equation and Sato equation Step 1. Au=0: Here we take a generalized Lax equation for example given by ∂L ∂tn = [En, L] = EnL − LEn, (2.7) where En = (W∂nW −1)+, W is a microdifferntial operator, and ()+ denotes the differential part of the operator. Step 2. u=Cv: Plant-harvesting transformation : L = W∂W −1.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Sato theory
Step 3. Dv=0: Sato equation B1vD1vB2v + B1vD1vB2v = 0 with B1 = I, D1 = ∂W ∂tn − EnW + W∂n = 0, B2 = ∂W −1, B3 = −W∂W −1, D1 = ∂W ∂tn − EnW + W∂n = 0, B2 = W −1, (2.8) which implies that ACv = B1vD1vB2v + B3vD2vB4v, where D1v = D2v = 0 denotes Sato equation.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Sato theory
Sato equation and Hirota’s bilinear equation Step 1. Au=0: Sato equation ∂W
∂tn = BnW − W∂n with
W = 1 + w1∂−1 + w2∂−2 + · · · ,
wj =
m−1
· · · −h(1)
m
· · · h(1) . . . · · · . . . · · · . . . h(m)
m−1
· · · −h(m)
m
· · · h(m)
m−1
· · · −h(1)
m−j
· · · h(1) . . . · · · . . . · · · . . . h(m)
m−1
· · · −h(m)
m−j
· · · h(m)
(2.9)
where Bn=(W∂nW −1)+, W is a microdifferntial operator, and ()+ denotes the differential part of the operator.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Sato theory
Step 2. u=Cv: wj = (−1)j 1
τ S
( ∂t)τ, where ∂t = ( ∂
∂t1 , 1 2 ∂ ∂t2 , 1 3 ∂ ∂t3 , . . .),
τ = τ(x, y) = |0, 1, . . . , m − 1| is a tau function, S is the Schur function satisfying S ( ∂t) = 1 j!
hρχρ ∂ ∂t1 a1 ∂ ∂t2 a2 · · · ∂ ∂tj aj , (2.10) where is the Young diagram.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Sato theory
Step 3. Dv=0: Hirota’s bilinear equation S ( ∂t)τ = |0, 1, . . . , m − j − 1, m − j + 1, . . . , m| B = I which implies ACv = BvDv, where Dv = 0 implies Hirota’s bilinear equation. Here the symbol |l1, l2, . . . , lm| denotes
|l1, l2, . . . , lm| =
l1
h(2)
l1
· · · h(m)
l1
h(1)
l2
h(2)
l2
· · · h(m)
l2
· · · · · · · · · · · · h(1)
lm
h(2)
lm
· · · h(m)
lm
n (x, t) = ∂hj 0(x, t)
∂tn = ∂nhj
0(x, t)
∂xn .
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
The Trigram structures
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
The Trigram structures Definition 3.1 Suppose there are n positions, one can put the Yang Yao “—" (
(or “—") in the rest of the positions. A diagram of this form is called 2n Trigram diagram For a 2n Trigram diagram, (i) putting the same Yao in j-th position to n − j + 1-th position yields a new 2n Trigram diagram, then we call the inverse (or zong) Trigram diagram of the 2n Trigram diagram, such as for a 23 Trigram , its inverse Trigram is ;
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Definition 3.1 (continue) (ii) putting another Yao in the same position yields a new Trigram diagram, then we call the adjugate (or cuo) Trigram diagram of the 2n Trigram diagram, such as for a 23 Trigram , its adjugate Trigram is ; (iii) if it can be constituted by the 2n1 Trigram, 2n2 Trigram, . . ., 2nN Trigram satisfying n = n1 + · · · + nN, then we call the repetition Trigram diagram of the 2n Trigram diagram, such as a 26 repetition Trigram is constituted by two 23 Trigrams and .
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Definition 3.2 A function f is constituted under a corresponding rule by the 2n Trigram, then we call f is a Trigram function. Suppose A, C, Bi and Di, (i = 1, 2, . . . , n) are some Trigram functions and assume that B∗
i are the adjugate Trigrams of Bi, then the
following identities
AC = B1D1B∗
1 + B2D2B∗ 2 + · · · + BnDnB∗ n,
(3.1) λ1B1D1B∗
1 + λ2B2D2B∗ 2 + · · · + λnBnDnB∗ n = λ0I,
λj ∈ {+1, −1, 0}, j = 0, 1, . . . , n, (3.2)
are called Trigram identities. Let S be a linear space of Trigram function, all of the elements of the space S satisfy the Trigram identities, then S is called a Trigram space.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Example Here we take a simple determinant for example
f =
a1 a2 a3 b0 b1 b2 b3 a1 a2 a3 b1 b2 b3
(3.3)
where ci = (ai, bi)T. In fact, only the indices are important in (3.3), and so we may also express it as f = (0 1)(2 3) − (0 2)(1 3) + (0 3)(1 2), (3.4) which may also be written in Maya diagrams
. (3.5)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Example The function f (3.5) can be changed into the following expression of Trigram , (3.6) which is equivalent to Chinese traditional 8 Trigram identity , (3.7) by deleting the topmost Yao of each 24 Trigram. Then the determinant f is a Trigram function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Definition 3.3 Plant-harvesting (PH) transformation is of the form X = gX0g−1, where X0 denotes a seed, g−1 denotes spring planting and g denotes autumn harvesting.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Definition 3.3 Plant-harvesting (PH) transformation is of the form X = gX0g−1, where X0 denotes a seed, g−1 denotes spring planting and g denotes autumn harvesting. Definition 3.4 Suppose τ satisfies Hirota’s bilinear equation, then τ is called Tau function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Lemma 3.5 Tau function is a Trigram function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Lemma 3.5 Tau function is a Trigram function. Theorem 3.6 Lax equation, Sato equation, Zakharov-Shabat equation, IST scheme and Hirota’s bilinear equation are all generated by Trigram identities.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship The Trigram structures and Trigram identities
Conjecture 3.7 The general structure of soliton equations can be constructed by 23 Trigram identity Trigram symbol identity: , Chinese language identity: . (3.8)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Exterior decomposition Trigram identities
◮ The Schur functions of Young diagram Schur function SY(x) of Young diagram Y is determined by
SY(x) =
· · · plm · · · · · · · · · pl1−m+1 · · · plm−m+1
xν1
1 xν2 2 xν3 3 · · ·
ν1!ν2!ν3! · · · , (3.9)
where the suffix Y means the Young diagram corresponding to the set of numbers (l1, l2, . . . , lm). Taking
= Sφ( ∂x)τ(x), = S( ∂x)τ(x), = S ( ∂x)τ(x), = S ( ∂x)τ(x) = S ( ∂x)τ(x), = S ( ∂x)τ(x),
2∂x2, 1 3∂x3, . . .
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Exterior decomposition Trigram identities
into the following exterior decomposition Trigram identity , (3.10) which is equivalent to Chinese traditional 8 Trigram identity , (3.11) by deleting the topmost Yao of each 24 Trigram, we obtain
1 12τ(x)(∂4
x4
1 − 4∂2
x1x3 + 3∂2 x2
2 )τ(x) − 1
3∂x1τ(x)(∂3
x3
1 − ∂x3)τ(x)
+ 1 4(∂2
x2
1 + ∂x2)τ(x)(∂2
x2
1 − ∂x2)τ(x) = 0,
(3.12)
which is essentially the same as the bilinear form of the KP
D-operator form (4Dx1Dx3 − D4
x1 − 3D2 x2)τ(x) · τ(x) = 0.
(3.13)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Exterior decomposition Trigram identities
Following the similar method, and taking Trigram identity (3.11) to be
Sφ( ∂x)τ(x)S ( ∂x)τ(x) − S( ∂x)τ(x)S ( ∂x)τ(x) + S ( ∂x)τ(x)S ( ∂x)τ(x) = 0, (3.14) Sφ( ∂x)τ(x)S ( ∂x)τ(x) − S ( ∂x)τ(x)S ( ∂x)τ(x) + S ( ∂x)τ(x)S ( ∂x)τ(x) = 0, (3.15) S( ∂x)τ(x)S ( ∂x)τ(x) − S ( ∂x)τ(x)S ( ∂x)τ(x) + S ( ∂x)τ(x)S ( ∂x)τ(x) = 0, (3.16) S ( ∂x)τ(x)S ( ∂x)τ(x) − S ( ∂x)τ(x)S ( ∂x)τ(x) + S ( ∂x)τ(x)S ( ∂x)τ(x) = 0, (3.17)
respectively, yield the subset of the KP hierarchy given by
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Exterior decomposition Trigram identities
(D3
x1Dx2 − 3Dx1Dx4 + 2Dx2Dx3)τ(x) · τ(x) = 0,
(3.18) (D6
x1 + 4D3 x1Dx3 − 9D2 x1D2 x2 + 36Dx2Dx4 − 32D2 x3)τ(x) · τ(x) = 0,
(3.19) Dx1(D2
x1Dx4 − 2Dx1Dx2Dx3 + D3 x2)τ(x) · τ(x) = 0,
(3.20) (D8
x1 − 20D5 x1Dx3 + 15D4 x1D2 x2 − 72D2 x1Dx2Dx4 + 64D2 x1D2 x3
+ 48Dx1D2
x2Dx3 − 36D4 x2)τ(x) · τ(x) = 0.
(3.21)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Exterior decomposition Trigram identities
◮ The characteristic polynomials of Young diagram Character polynomial of Y is given by
χY = det(hminj(x)), hmn(x) = (−1)n
l≥0
pl+m+1(x)pn−l(−x) =(−1)n+1
l<0
pl+m+1(x)pn−l(−x), (3.22)
where pi(x) is defined as that in (3.9) [pi(x) = 0 if i < 0]. Taking
= χφ( ∂x)τ(x), = χ( ∂x)τ(x), = χ ( ∂x)τ(x), = χ ( ∂x)τ(x), = χ ( ∂x)τ(x), = χ ( ∂x)τ(x),
2∂x2, 1 3∂x3, . . .
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Exterior decomposition Trigram identities
into the following exterior decomposition Trigram identity , (3.23) which is equivalent to Chinese traditional 8 Trigram identity , (3.24) by deleting the topmost Yao of each 24 Trigram, we obtain one
the same way of Schur function, one can also yield the other subset of bilinear forms of the KP equation (3.19)-(3.21).
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
◮ Fermions, Clifford algebra and its Fock representation
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
◮ Fermions, Clifford algebra and its Fock representation Suppose some symbols ψn, ψ∗
n satisfy the anticommutator,
[X, Y]+ =def XY + YX, let n run through the half-integers Z + 1/2, we have [ψn, ψm]+ = 0, [ψ∗
n, ψ∗ m]+ = 0, [ψn, ψ∗ m]+ = δm+n,0.
(3.25) Then ψn and ψ∗
n are called Fermions;
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
◮ Fermions, Clifford algebra and its Fock representation Suppose some symbols ψn, ψ∗
n satisfy the anticommutator,
[X, Y]+ =def XY + YX, let n run through the half-integers Z + 1/2, we have [ψn, ψm]+ = 0, [ψ∗
n, ψ∗ m]+ = 0, [ψn, ψ∗ m]+ = δm+n,0.
(3.25) Then ψn and ψ∗
n are called Fermions; the algebra A they
generate, with the defining relation (3.25), is called the Clifford algebra.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
◮ Fermions, Clifford algebra and its Fock representation Suppose some symbols ψn, ψ∗
n satisfy the anticommutator,
[X, Y]+ =def XY + YX, let n run through the half-integers Z + 1/2, we have [ψn, ψm]+ = 0, [ψ∗
n, ψ∗ m]+ = 0, [ψn, ψ∗ m]+ = δm+n,0.
(3.25) Then ψn and ψ∗
n are called Fermions; the algebra A they
generate, with the defining relation (3.25), is called the Clifford
combination of monomials of the form ψm1 · · · ψmr ψ∗
n1 · · · ψ∗ ns, where m1 < · · · < mr and n1 < · · · < ns.
(3.26)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
By writing m1, m2, . . . for the positions of the black stones, we can describe a Maya diagram as an increasing sequence of half-integers
m = {mj}j≥1 with m1 < m2 < · · · , and mj+1 = mj + 1, (3.27)
for all sufficiently large j.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
By writing m1, m2, . . . for the positions of the black stones, we can describe a Maya diagram as an increasing sequence of half-integers
m = {mj}j≥1 with m1 < m2 < · · · , and mj+1 = mj + 1, (3.27)
for all sufficiently large j. The Fermionic Fock space F can be defined as
F = A|vac = {a|vac|a ∈ A} with |vac is the vacuum state. (3.28)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
Maya diagram and Trigram diagram of vacuum state given by
← → (3.29)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
Maya diagram and Trigram diagram of vacuum state given by
← → (3.29)
The action of the Fermions on Fock space and their Trigram diagrams by the following rules are presented as follows
ψn|m =
(3.30) if mi = −n for some i :
(3.31)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
ψ∗
n|m =
(3.32) if mi < n < mi + 1 for some i :
(3.33)
except that in the case i=1. From (3.26), (3.28) and (3.30)-(3.33), the action of the Clifford algebra on Fock space is given by the following rules
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
ψp1 · · · ψpr ψ∗
q1 · · · ψ∗ qs|m =
(−1)
r
j=1 ipj +s k=1 iqk −r(r+1)/2+s(s−2r−1)/2
mip1−1, mip1+1, . . . , mipr −1, mipr +1, . . . , miq1−1, miq1, mip1+1, . . . , miqs −1, miqs , mips +1, . . .
for j = 1, . . . , r, k = 1, . . . , s,
and if ipj = −pj, miqk < qk < miqk + 1, the Trigram diagram is given by (3.34) whose symbol is (−1)
r
j=1 ipj +s k=1 iqk −r(r+1)/2+s(s−2r−1)/2. At
that time 0 is corresponding to the following Trigram diagram
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
(3.35)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
(3.35) Next, we can show that the action of the Trigram diagram satisfies the anticommutator relation (3.25) [ψp1, ψp2]+|m = 0, [ψ∗
q1, ψ∗ q2]+|m = 0, [ψp1, ψ∗ q2]+|m = δp1+q2,0|m.
(3.36)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
(3.35) Next, we can show that the action of the Trigram diagram satisfies the anticommutator relation (3.25) [ψp1, ψp2]+|m = 0, [ψ∗
q1, ψ∗ q2]+|m = 0, [ψp1, ψ∗ q2]+|m = δp1+q2,0|m.
(3.36) Let p2 < q1, the relationship (3.36) can be described as (3.37)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
(3.38) (3.39) (3.40)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
◮ Bosons, Heisenberg algebra and its Fock representation From Sato’s theory, we can realise the Bosons by following Boson-Fermion correspondence.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
◮ Bosons, Heisenberg algebra and its Fock representation From Sato’s theory, we can realise the Bosons by following Boson-Fermion correspondence. Theorem 3.12 The Fermionic generating functions ψ(k) and ψ∗(k) are realised in the Bosonic Fock space by
Ψ(k) = eξ(X,k)e−ξ(
∂,k−1)eKkH0, Ψ∗(k) = e−ξ(X,k)eξ( ∂,k−1)e−Kk−H0,
(3.41)
where ∂ = (∂x1, 1/2∂x2, 1/3∂x3), ξ( ∂, k−1) = ∞
n=1 1 n∂xnk−n,
and X is an infinitesimal generator. That is, for any |u ∈ F we have
Φ(ψ(k)|u) = Ψ(k)Φ(|u) and Φ(ψ∗(k)|u) = Ψ∗(k)Φ(|u). (3.42)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship Interior decomposition Trigram identities
Remark 3.7 From Theorem 3.15 and Trigram structures of Fermions’s Fock space, we can conjecture that the Fock representation of Heisenberg algebra can be also described by Trigram structures and Trigram identities, which satisfy the canonical commutation relations [am, an]|m = 0, [a∗
m, a∗ n]|m = 0 and [am, a∗ n]|m = δmn|m,
(3.43) which are exterior decomposition Trigram identities. Then Bosons’ Fock space can be mapped into a Trigram space.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
The relationship between Tau function and Theta function From above, one can see that tau function plays an important role in Sato theory, soliton theory, Trigram space and so on. On the other hand, Theta function has the same important position in constructing algebraic-geometry solutions. It is very meaningful to study the relationship between Tau function and Theta function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
From Refs. Novikov (1984), Dubrovin (1976), Mumford (1983) and their co-workers, one can show the following results.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
From Refs. Novikov (1984), Dubrovin (1976), Mumford (1983) and their co-workers, one can show the following results. Proposition 4.1 Let ΨBA(p) be the Baker-Akhiezer function with divisor D of poles of degree g and singular part ξ(t, z) =
k tkz−k at p∞,
normalized such that: ψBA(P) = eξ(t,z)(1 + O(z)), z ∼ 0. (4.1) Let t − [z] = {tk − 1/kzk}, then we have in the vicinity of P∞: ψBA(P) = eξ(t,z) τ(t − [z]) τ(t) . (4.2)
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Proposition 4.2 Let Ω(S) be the unique Abelian differential of the second kind, normalized with vanishing a-periods, and with singular part at the points Pi of the form dSi(ωi(P)). If D = g
i=1 γi is a generic
divisor of degree g, the following expression defines a Baker-Akhiezer function with D as divisor of poles: ΨBA(P) = ce
p
p0 Ω(S) θ(A(P) + U(S) − ζ)
θ(A(P) − ζ) (4.3) where c is an arbitrary constant, 2πi is the vector of b-periods, whose g components 2πiU(S)
j
are 2πiU(S)
j
=
ζ = A(D) + K, A(·) denotes the Abel map with based point P0, and K is the vector of Riemann’s constants.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Theorem 4.3 The Tau function associated with the algebraic-geometry integrable system is given by τ(t) = eσ(t,t)+ρ(t)θ
tkU(k) − K
(4.4) where σ(t, t) is a bilinear function of t, and ρ(t) is a linear one.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Our method From our Refs. Tian & Zhang (2010), (2011) and (2012), we present the Its-Matveev formula and super-Its-Matveev formula by theta functions, which are derived by Hirota’s bilinear equation and super Hirota’s bilinear equation, respectively. On the other hand, the solutions of bilinear equations can be expressed in terms of Tau function. From this perspective, we can also construct the relationship between Tau function and Theta function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Conclusions and further research
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Conclusions and further research
1
Extend AC=BD theory to investigate a unified framework of solving soliton equations.
2
Propose our “Trigram" theory to study the unified and fundamental structure of all soliton equations.
3
Present a method to construct the relationship between Tau- and Theta- function.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
Conclusions and further research
1
Extend AC=BD theory to investigate a unified framework of solving soliton equations.
2
Propose our “Trigram" theory to study the unified and fundamental structure of all soliton equations.
3
Present a method to construct the relationship between Tau- and Theta- function.
4
Extend the research radius of Trigram theory.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
S.P . Novikov, S.V. Manakov, L.P . Pitaevskii, V.E. Zakharov, Theory of Solitons, the Inverse Scattering Methods, Concultants Bureau, New York, (1984).
. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, (1991). F . Gesztesy and H. Holden, Soliton Equations and their Algebro-Geometric Solutions, Cambridge University Press, Cambridge, (2003).
varieties, Lecture notes taken by M. Noumi, Saint Sophia
elementary introduction to Sato theory, Prog. Theor. Phys. Supplement 94 (1988) 210-241.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
equations, symmetries and infinite dimensional algebras, Cambridge University Press, (2000).
(2004).
nonlinear Schrödinger equation, Phys. Lett. A 99 (1983) 279-280.
systems of elasticity equations, Journal of Dalian University
mechanization, Journal of Systems Science and Mathematical Sciences, 2008, 28(8): 1030-1039.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
methods to partial differential equations in mathematics mechanization and applications, Edited by Xiaoshan Gao, Dongming Wang, Academic Press 2000.
solver for differential equations, Proceedings of the fifth Asian symposium (ASCM 2001) 221-226.
. Tian and H. Q. Zhang, Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations, J. Math. Anal. Appl. 371 (2010) 585-608.
. Tian and H. Q. Zhang, A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations, Commun Nonlinear Sci Numer Simulat 16 (2011) 173-186.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship
. Tian and H. Q. Zhang, Super Riemann theta function periodic wave solutions and rational characteristics for a supersymmetric KdV-Burgers equation, Theor. Math. Phys. 170(3) (2012) 287-314.
The problem of soliton theory The model AC=BD and its applications The Trigram structures and Trigram identities The relationship