Nonlinear stability of oscillatory pulses in the parametric nonlinear Schrödinger equation

Abstract

We extend the renormalization group method, developed for the study of pulse interaction in damped wave equations, to the study of oscillatory motion of supercritical pulses in the parametrically forced nonlinear Schrödinger equation (PNLS). We construct a global manifold which asymptotically attracts the flow into an O(r 4 ) neighbourhood in the H 1 norm, where r is the amplitude of the internal oscillations. The oscillatory and translational dynamics of the pulses are rigorously recovered as a finite-dimensional flow on the manifold. The normal form for the projected dynamics of the oscillatory pulse shows that it is created in a supercritical Poincaré-Hopf bifurcation.

Renormalization group reduction of pulse dynamics in thermally loaded optical parametric oscillators

“…We give a brief summary of results obtained previously [3,7,20] for the PNLS equation when a is constant. The point and essential spectrum of the stable steady state inherit structure from the symmetries inherent in Eq.…”

“…(7). The spectrum is doubly reflection-symmetric about (−1,0) [7], so that we need only consider a single quadrant of the eigenvalue plane. On both branches, translation invariance produces a single eigenvalue λ t at the origin provided γ > 1; if γ = 1, this eigenvalue has multiplicity 2.…”

The semistrong limit of multipulse interaction in a thermally driven optical system

“…Thus the H 1 norm of the difference between Θ and the sum of the local thermal profiles is O(ε) in the rescaled variables for which Θ and each Θ j decay at Region I is the stability region for PNLS pulses, crossing into region II coincides with a Hopf bifurcation, see [1], and there are unstable point spectra. In regions III and IV there is unstable essential spectra, η 0,− < 0 and the data is inadmissible.…”

“…We emphasize this when using weighted norms via the notation (2.23) and similarly for the H 1 p,j . The norms without j subscript, · L r p and · H 1 p , denote the usual, nonwindowed, p-weighted norm with weight centered on the mid-point of the N -pulse, 1 2 (q N + q 1 ).…”

“…Proof.The H 1 bound on G j follows from (3.35) and(3.22) since ξ j L 2 1,j = O(1). It remains to establish the weighted part of the estimate.…”