Exercises
Q6.9 Sinh-Gordon equation. Show that the choice
i i A,, = — A21 = ^ cosh u, A! 2 = — A 2, = — — sinh u
with </ = r= ,ux (cf. §6.1.5(b)), gives
uxt — sinh u.
Q6.10 Operator K     Ocduee from equation (6.66) the identity
K_(x,z) = F{x,z) + |   K + (x,y)F(y,z)dy
Q6.11 Operator identity. Show that, if
A(/ +J , ) = (/ + J + )A0        and        (/ + J +)(/ + J,) = / where A0 and Jf commute, then
A(/+J.) = (/ + J.)A0. Q6.12 Commuting operators. Verify that the following pairs of operators e     "     c" d      M rm
vl ox ty   m=1 dx
where a, a„,/i and /im are constant scalars;
where x, / and m are constants;
,„   //   ON   r    /-/   0\d ,„   // 0\5
(in) A,' - *- + —, A'02l= —
where 2 is a constant. Q6.13 KdV equation: operators. Show that, if
,,, 5 ^ 0    t)r dx3
then A'0" and JF commute provided
F, + 4(FXXX+FZ„) = 0;
see equation (6.84). Q6.14 Two-dimensional KdV equation. If
v      d3 5 c32
A(1> = —+ 4—, A(2) =
ct     dx3 dy' dx2'
and we write