Multi-hump pulses in systems with reflection and phase invariance
First, the authors consider systems of PDEs that have two continuous symmetries, namely, translation and phase invariance. The paradigm for such systems is the cubic-quintic complex Ginzburg-Landau equation. The intention of the authors is to investigate the existence and stability of standing and travelling multi-hump waves in PDEs with reflection and phase symmetries. The considered systems of PDEs respects two continuous symmetries, namely, translation in the spatial derivative variable and a gauge invariance. Therefore, the eigenvalue of the primary pulse at the origin has multiplicity two. \(N\)-pulse will therefore have \(2N\) critical eigenvalues near the origin. The authors also use Lyapunov-Schmidt reduction to relate the existence and stability properties of \(N\)-pulses to the solutions of a reduced set of equations. In fact, the authors extend an known technique from equations with just translation symmetry to systems with translation and phase symmetries. The reduced equations are solved for 2- and 3-pulses. In the last part of the paper, the authors apply the above results to the complex cubic-quintic Ginzburg-Landau equation near the limit \(e=0\) that corresponds to the nonlinear Schrödinger equation.
scientific article; zbMATH DE number 5608031
scientific article