(a) Verify, using symbolic-manipulation computer software (e.g., Maple, Matlab, or Mathematica) that the two-soliton expression (16.36) satisfies the dimensionless KdV equation.

(b) Verify analytically that the two-soliton solution (16.36) has the properties claimed in the text. First consider the solution at early times in the spatial region where f₁ ∼ 1, f₂ ≪ 1. Show that the solution is approximately that of the single soliton described by Eq. (16.33). Demonstrate that the amplitude is ζ₀₁ = 3α²₁, and find the location of its peak. Repeat the exercise for the second wave and for late times.

(c) Use a computer to follow numerically the evolution of this two-soliton solution as time η passes (thereby filling in timesteps between those shown in Fig. 16.7).

Figure 16.7.

1/2 [ (9) ( - - Sor)]. 12 5 = o sech (16.33)