Exam Problems: Non-linear waves and soli-tons, Spring 2020
1. Two-soliton solution.  Construct the two-soliton solution of the nonlinear Schrodinger equation
iii + u" + u\u\2 = 0 .
2. Zero curvature representation and conservation laws. Consider the linear system of equations
V = LV,       V = MV ,
where
i/A    u* \       m --(  ^ ~ lM'2 ~iu'* + Xu*
2\2u  -\ J '           2 V 2iu' + 2Am ~a2 + M2
/ Vi \
and V is a two-component vector V =     T.    .   Here L, M, V all
\v2 J
depend on x, t as well as the auxiliary spectral parameter A. Show that the compatibility condition for the linear problem leads to the non-linear Schrodinger equation for u.
Define the functions pk recursively as
, fc-2
Pk =--Pk-1 + Pk-1 + 2 PjPk-2-j ' ^° = M •
U 3=0
It turns out that the charges
/oo -oo
are conserved. Show this for k = 0,1, 2.
3. Sine-Gordon equation. Show that the Sine-Gordon (SG) equation
u" — u = sin u
can be cast in the ZS scheme by taking the 4x4 matrix operators (i) _,dlfl    0  \ d (2)     ( ia2   0 ^ d
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where I denotes the 4x4 and 2x2 unit matrices (in the block matrices 0 denotes the 2x2 zero matrix) and a2 = ^ ^ ^ ^ is one of the Pauli matrices. You can assume that
K =
satisfies [a2, A] = wau B = —ia2D and C = iDa2 where
1 / e«u/2 o
D ~ 4 I    0 e-™/2
and u, w are functions of x, t.
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