Student Seminar: Classical and Quantum Integrable Systems

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system is H = 1 2 n∑ p 2 j + g ∑ 2 i
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) . – 100 –
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Preprint typeset in JHEP style - HY
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7. Homework exercises 95 7.1 Semina
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which can be written as follows {q
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To get more insight on the Liouvill
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4. The equations of motion with Ham
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Let us show that the variables (I i
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We have H = p2 2 p2 + V (x) = 2 + V
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We can now see how the solution can
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Thus, and, therefore, the half-peri
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This is the so-called focal equatio
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to show that ˙⃗ R = 0. The last
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is the bounded kinetic energy). Acc
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One can study the structure of the
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Substituting these formula into the
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Upper−half plane 01 01 0 0 1 1 01
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Thus, the combination ˙θ 2 − 2m
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Then using the eoms ˙q = p and ṗ
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where we have assumed that the eige
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are called the Euler-Arnold equatio
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Thus, g(λ)Q(λ)g(λ) −1 = ( = g
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We see that the there is one more v
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4.1 General remarks Remarkably, the
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We thus obtain u 2 x = k − 2eu +
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Here ɛ 0 = ±1. This solution can
Page 49 and 50: where σ i are the Pauli matrices 9
Page 51 and 52: Let g ≡ g(x, t, λ) be a regular
Page 53 and 54: Indeed, ∂ t L x − ∂ x L t + [
Page 55 and 56: ecomes a differential equation for
Page 57 and 58: • Coordinate Bethe ansatz. This t
Page 59 and 60: The first interesting observation i
Page 61 and 62: This state is an eigenstate of the
Page 63 and 64: as the pseudo-particle with the mom
Page 65 and 66: This allows one to determine the ra
Page 67 and 68: The first problem is to find all po
Page 69 and 70: 5.2 Algebraic Bethe Ansatz Here we
Page 71 and 72: Semi-classical limit and quantizati
Page 73 and 74: Thus, the transfer matrix is also p
Page 75 and 76: To write down the fundamental commu
Page 77 and 78: provided W A n + W D n = 0 for all
Page 79 and 80: and if k < j the contribution will
Page 81 and 82: Now we have to specify which of M p
Page 83 and 84: • Pseudo-orthogonal groups SO(p,
Page 85 and 86: 3. The Jacobi identity [[ξ, η],
Page 87 and 88: • Special linear group SL(n, R) o
Page 89 and 90: where A is an element of the corres
Page 91 and 92: Here the Jacobi identity has been u
Page 93 and 94: Note that both ad tz and ad y are d
Page 95 and 96: if α + β is a root, [e α , e −
Page 97 and 98: • case 3: • case 4: g2 U(q) =
Page 99: 7.4 Seminar 4 Exercise 1 (K. Bohlin
Page 103 and 104: 7.6 Seminar 6 Exercise 1 Prove that
Page 105 and 106: is a representation of the group SU
Page 107 and 108: Exercise 4 Consider the non-linear
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Student Seminar: Classical and Quantum Integrable Systems

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