Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations (Q2710411)

The author of this interesting paper considers a system of two nonlinearly-coupled Schrödinger equations: NEWLINE\[NEWLINE i\partial w/\partial \zeta + r \partial ^2w/\partial t^2-\theta w+w^{\ast }v=0,\quad i\partial v/\partial \zeta + s \partial ^2v/\partial t^2-\alpha v+1/2w^2=0. NEWLINE\]NEWLINE This system describes second-harmonic generation and parametric wave interaction in dispersive non-centrosymmetric materials. \(w(\zeta ,t)\) and \(v(\zeta ,t)\) are complex variables representing the envelope amplitudes of the fundamental and second-harmonic waves, respectively. The coordinate \(\zeta \) is the evolution variable of the partial differential equations. It measures distance along the optical fiber or longitudinal distance along a waveguide. \(t\) represents retarded time in fiber-optics case. The stability of multipulses for this conservative system is analyzed. The multibump solitary-waves are steady-state solutions generated from a homoclinic bifurcation arising near a semi-simple eigenvalue scenario. The author introduces an interesting approach reducing the problem under consideration to two selfadjoint linear operators. An extension of the method which uncovers the existence of multiple pulses is utilized. It turns out that all of the multihumped waves are unstable.