Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1 / 4

Table of contents
- Solution 4

Solution

z = x - c t

and

u (t, x) = Q_{c} (z)

. Substitution into the equation gives

\partial_{z} (Q_{c}^{''} - c Q_{c} + Q_{c}^{p}) = 0.

Decay at infinity makes the integration constant zero. If

Q_{c} (x) = c^{1/ (p - 1)} Q (c x),

then every term in

Q_{c}^{''} - c Q_{c} + Q_{c}^{p}

equals

c^{1 + 1/ (p - 1)}

times the corresponding term in

Q^{''} - Q + Q^{p}

. Thus the required Generalized KdV solitary wave is

Q_{c} (x) = c^{1/ (p - 1)} Q (c x) .

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