Student Seminar: Classical and Quantum Integrable Systems
Student Seminar: Classical and Quantum Integrable Systems
Student Seminar: Classical and Quantum Integrable Systems
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Exercise 4<br />
Consider the non-linear Schrodinger equation:<br />
i ∂ψ<br />
∂t = −∂2 ψ<br />
∂x 2 + 2κ|ψ|2 ψ ,<br />
where ψ ≡ ψ(x, t) is a complex function. Show that this equation admits the following<br />
zero-curvature representation<br />
where<br />
<strong>and</strong><br />
U = U 0 + λU 1 , V = V 0 + λV 1 + λ 2 V 2 ,<br />
U 0 = √ κ( ¯ψσ + + ψσ − ) , U 1 = 1 2i σ 3<br />
V 0 = iκ|ψ| 2 σ 3 − i √ κ(∂ x ¯ψσ+ − ∂ x ψσ − ) , V 1 = −U 0 , V 2 = −U 1 .<br />
Using the abelianization procedure around λ = ∞ find the first four local integrals<br />
of motion. What is the physical meaning of the first three integrals?<br />
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