where F is a solution of the pair of equations
F„-F„-(x-z)F, 3tF, -F + Fxxx + Fzzz = xFx + zF„
then
u, + — - 6uux + uxxx = 0,
where
2
F(x,z;f)=
/(yt"3)Ai(x + y)Ai(y+z)dy,
- OO
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 f (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
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 m Q4.3 gives a sech2 pulse at t = 0 (see example (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
u, - 6uux + uxxx = 0
which satisfies u(x,0) =/(x). — oo < x < oo, where
0) /(x)= - fsech2(|x); (ii) /(x)= - 12 sech2 x;