代写data编程、代写C/C+，Java程序

日期：2023-05-29 08:28

Worksheet 7

Toda equation

Definition. Consider Painlev´e-II equation. Since momentum and Hamiltonian are expressed in terms of

q(x), the action of B¨acklund transformation B+ can be extended from pairs (q(x), α) to pairs (p(x), α) and

(H(x), α). We denote

(qn(x), α + n) = B

n

+(q(x), α) (1)

(pn(x), α + n) = B

n

+(p(x), α) (2)

(Hn(x), α + n) = B

n

+(H(x), α) (3)

Definition. Define tau function for Painlev´e-II equation using formula

d

dx ln(τn(x)) = Hn(x) (4)

1. Show that Hn+1(x) = Hn(x) − qn+1(x).

2. Show that the Toda equation holds

d

2

dx2

ln(τn(x)) = C

τn+1(x)τn−1(x)

τ

2

n

(x)

(5)

for some constant C.

Definition. Consider Painlev´e-III equation. Fix ε = ±1. Define auxiliary Hamiltonian using formula

h(x) = 1

2



H(x) + q(x)p(x) − εx2 +

1

4

(β − 4ε)(β + ε(α − 2))

(6)

Definition. Consider Painlev´e-III equation. Choose ε = 1. Since momentum and Hamiltonian are expressed

in terms of q(x), the action of B¨acklund transformation B1 can be extended from pairs (q(x), α, β) to pairs

(p(x), α, β) and (H(x), α, β). We denote

(qn(x), α + 2n, β + 2n) = B

n

1

(q(x), α, β) (7)

(pn(x), α + 2n, β + 2n) = B

n

1

(p(x), α, β) (8)

(Hn(x), α + 2n, β + 2n) = B

n

1

(H(x), α, β) (9)

(hn(x), α + 2n, β + 2n) = B

n

1

(h(x), α, β) (10)

Definition. Define tau function for Painlev´e-III equation using formula

x

d

dx ln(τn(x)) = hn(x) (11)

3. Show that hn+1(x) = hn(x) − pnqn(x) −

3

2 +

α

4 +

3β

4 + 2n.

4. Show that the Toda equation holds

x

d

dxx

d

dx ln(τn(x)) = C

τn+1(x)τn−1(x)

τ

2

n

(x)

(12)

for some constant C.

Definition. Consider Painlev´e-III equation. Choose ε = −1. Since momentum and Hamiltonian are expressed in terms of q(x), the action of B¨acklund transformation B3 can be extended from pairs (q(x), α, β)

to pairs (p(x), α, β) and (H(x), α, β). We denote

(qn(x), α + 2n, β − 2n) = B

n

3

(q(x), α, β) (13)

(pn(x), α + 2n, β − 2n) = B

n

3

(p(x), α, β) (14)

(Hn(x), α + 2n, β − 2n) = B

n

3

(H(x), α, β) (15)

(hn(x), α + 2n, β − 2n) = B

n

3

(h(x), α, β) (16)

Definition. Define tau function for Painlev´e-III equation using formula

x

d

dx ln(τn(x)) = hn(x) (17)

5. Show that hn+1(x) = hn(x) − pnqn(x) −

3

2 +

α

4 −

3β

4 + 2n.

6. Show that the Toda equation holds

x

d

dxx

d

dx ln(τn(x)) = C

τn+1(x)τn−1(x)

τ

2

n

(x)

(18)

for some constant C.