Student Seminar: Classical and Quantum Integrable Systems

7.5 Seminar 5
Exercise 1 (Open Toda chain)
Consider a system of n interacting particles described by coordinates q j and the
corresponding conjugate momenta p j , where j = 1, . . . , n. The Hamiltonian of the
system has the form
H = 1 n∑ ∑n−1
p 2 j + exp[2(q j − q j+1 )] .
2
1
j=1
Show that equations of motion are equivalent to the Lax equation ˙L = [L, M], where
n∑ ∑n−1
L = p j E jj + exp[(q j − q j+1 )](E j,j+1 + E j+1,j ) ,
j=1
j=1
∑n−1
M = exp[(q j − q j+1 )](E j,j+1 − E j+1,j ) .
j=1
Here E jk is a matrix which has only one non-zero matrix element equal to 1 standing
in the intersection of j’s row with k’s column.
Exercise 2 (Differential equations for Jacobi elliptic functions).
Using the differential equation for the Jacobi elliptic function sn(x, k):
(sn ′ (x, k)) 2 = (1 − sn(x, k) 2 )(1 − k 2 sn(x, k) 2 ) .
and the identities relating sn(x, k) with other two functions cn(x, k) and dn(x, k)
derive the differential equations for cn(x, k) and dn(x, k).
Exercise 3 (The cnoidal wave and soliton of the NLS equation)
Consider the non-linear Schrodinger (NLS) equation
i ∂ψ
∂t = −ψ xx + 2κ|ψ| 2 ψ ,
where ψ = ψ(x, t) is a complex-valued function and we assume that κ < 0. By making
single-wave propagating ansatze for the modulus and the phase of ψ(x, t) determine
the cnoidal wave solution of the NLS equation. Show that upon degeneration of the
elliptic modulus the cnoidal wave turns into a soliton solution of the NLS equation.
Exercise 4 (Hamiltonian formulation of KdV equation)
Consider the following two Poisson brackets on the space of Schwarzian functions
u(x):
{u(x), u(y)} = −∂ x δ(x − y) .
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