Phase matching for a quadratic wave interaction

A resonant interaction requires matching both wavenumber and angular frequency:

k_{3} = k_{1} + k_{2}

and

ω_{3} = ω_{1} + ω_{2}

for a sum interaction, with corresponding sign changes for differences. A common phase velocity alone does not guarantee that a generated harmonic lies on the dispersion relation.

Table of contents
- Two-to-one resonance of dispersive waves Phase matching for a quadratic wave interaction
  - Explosive two-to-one amplitude system Two-to-one resonance of dispersive waves

Two-to-one resonance of dispersive waves ( $ω (2 k) = 2 ω (k)$ )

 0  0

A fundamental wave and its second harmonic satisfy phase matching for a quadratic wave interaction when

ω (2 k) = 2 ω (k)

. The square of the fundamental drives the second harmonic, while its product with the conjugate second harmonic drives the fundamental. Projecting these terms onto the resonant modes gives coupled amplitude equations rather than a uniformly valid correction at fixed amplitude.

Explosive two-to-one amplitude system ( $A_{T} = - \overline{A} B /2, B_{T} = - A^{2} /4$ )

 0  0

For

A_{T} = - \overline{A} B /2

and

B_{T} = - A^{2} /4

, write

A = R e^{i θ}

B = P e^{i ϕ}

and

δ = ϕ - 2 θ

. The polar equations are

R_{T} = - \frac{RP}{2} cos δ, θ_{T} = - \frac{P}{2} sin δ, P_{T} = - \frac{R ^{2}}{4} cos δ, ϕ_{T} = \frac{R ^{2}}{4 P} sin δ .

(1)

Where the polar phases are defined, direct differentiation gives the first integrals

R^{2} - 2 P^{2}

and

R^{2} P sin δ

. In particular

R^{2} θ_{T}

and

P^{2} ϕ_{T}

are constant. If

δ = π

and

R^{2} - 2 P^{2} = C > 0

, then

P_{T} = P^{2} /2 + C /4

, a Riccati equation whose positive solution develops a finite-time pole. This is finite-time blowup of the reduced amplitude system; it does not establish blowup of the full wave equation beyond the domain of the weakly nonlinear expansion.