where F is a solution of the pair of equations
F„-F„-(x-z)F, 3tF, -F + Fxxx + Fzzz = xFx + zFt,
then
where
u
u, +--6uu, + u„r = 0,
It
u{x, t) =------- - — K(X, X; t) with X = - '
(\2t)2icX   ' (12f)' (ii) Hence show that a solution for F is
F(x, z; !) =       /(y(1/3)Ai(x + y)M{y + z)dy.
J - CO
where / is an arbitrary function and Ai is the Airy function. Q4.3 Two-soliton solution of the KdV equation. Obtain a solution for F(x, z; t) (see Q4.1) which depends upon x + z (but not x — z since this would become trivial on z ■= x), and which is exponential in both x + z and t (cf. examples (i) and (ii), §4.5).
Hence write down a solution for F which is the sum of two exponential terms, and construct the two-soliton solution of the KdV equation. Q4.4 Some initial-value problems. Use the inverse scattering transform to find the solution of the KdV equation
u, - 6uux + uxxx = 0
which satisfies u(x,0) =/(.x), — oo < x < oo, where
(') /(x)= - fsech2(|x); (ii) /(x) = - 12 sech2 x;
- V for - Kx< 1, 0 otherwise,
*(iii) f(x) =
where V > 0 is a constant.
[Case (iii) is too difficult to solve explicitly, so just give a qualitative description of the solution for various K] Q4.5 The character of two-soliton solutions. Show that a special case of the two-soliton solution obtained in Q4.3 gives a sech2 pulse at t = 0 (seeexample (ii),§4.5). Show also that, for suitably defined x, the pulse at t = 0 may have either one or two local maxima.
[You will find it convenient to define x so that a symmetric profile occurs at t = 0.]
(Lax, 1968)
Q4.6 Three-soliton solution. Find the asymptotic form of the three-soliton solution (see Q4.4, (ii)) as t-> ± oo, and hence determine the phase shifts. Q4.7 Connection with Fourier transforms. Consider the initial-value problem