Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1 / 5

Table of contents
- Solution 5

Solution

For every

p < 5

, the wave

Q_{c}

has Orbital stability of a generalized KdV solitary wave in

H^{1} (R)

modulo translation: for every

ε > 0

there is

δ > 0

such that

∥ u_{0} - Q_{c} ∥_{H^{1}} < δ ⟹ sup_{t} in f_{y \in R} ∥ u (t) - Q_{c} (\cdot - y) ∥_{H^{1}} < ε .

The relevant stability slope has the correct sign because scaling gives

∥ Q_{c} ∥_{2}^{2} = c^{2/ (p - 1) - 1/2} ∥ Q ∥_{2}^{2} = c^{(5 - p) / (2 (p - 1))} ∥ Q ∥_{2}^{2},

which is strictly increasing in

c

precisely for

p < 5

. Together with the constrained variational characterization of

Q_{c}

, the conserved mass and energy provide a coercive Lyapunov function transverse to the translation direction.

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