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 Incoming links: Generalized KdV solitary wave

p = 2, 3, 4

and

c > 0

, the positive decaying Generalized KdV solitary wave is orbitally stable in

H^{1} (R)

modulo spatial translations. The variational slope has the stable sign because

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

is strictly increasing in

c

. Conserved mass and energy then furnish a Lyapunov functional transverse to the translation orbit.

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}

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