2014/12/28 by Diego Noja, Noja, Diego, Dmitry E. Pelinovsky +3 · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Electromagnetic Simulation and Numerical Methods #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1412.8232
openalex publication_date 2014/12/28 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We develop a detailed rigorous analysis of edge bifurcations of standing\nwaves in the nonlinear Schr "odinger (NLS) equation on a tadpole graph (a ring\nattached to a semi-infinite line subject to the Kirchhoff boundary conditions\nat the junction). It is shown in the recent work [7] by using explicit Jacobi\nelliptic functions that the cubic NLS equation on a tadpole graph admits a rich\nstructure of standing waves. Among these, there are different branches of\nlocalized waves bifurcating from the edge of the essential spectrum of an\nassociated Schr "odinger operator. We show by using a modified Lyapunov-Schmidt\nreduction method that the bifur- cation of localized standing waves occurs for\nevery positive power nonlinearity. We distinguish a primary branch of never\nvanishing standing waves bifurcating from the trivial solution and an infinite\nsequence of higher branches with oscillating behavior in the ring. The higher\nbranches bifurcate from the branches of degenerate standing waves with\nvanishing tail outside the ring. Moreover, we analyze stability of bifurcating\nstanding waves. Namely, we show that the primary branch is composed by\norbitally stable standing waves for subcritical power nonlinearities, while all\nnontrivial higher branches are linearly unstable near the bifurcation point.\nThe stability character of the degenerate branches remains inconclusive at the\nanalytical level, whereas heuristic arguments based on analysis of embedded\neigenvalues of negative Krein signatures support the conjecture of their linear\ninstability at least near the bifurcation point. Numerical results for the\ncubic NLS equation show that this conjecture is valid and that the degenerate\nbranches become spectrally stable far away from the bifurcation point.\n