{"paper":{"title":"Non-monotone travelling waves in a single species reaction-diffusion equation with delay","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sergei Trofimchuk, Teresa Faria","submitted_at":"2005-08-04T20:04:36Z","abstract_excerpt":"We prove the existence of a continuous family of positive and generally non-monotone travelling fronts in delayed reaction-diffusion equations $u_t(t,x) = \\Delta u(t,x)- u(t,x) + g(u(t-h,x)) (*)$, when $g \\in C^2(R_+,R_+)$ has exactly two fixed points: $x_1= 0$ and $x_2= a >0$. Recently, non-monotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay $h$ grows. For the case of $g$ with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass ty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508098","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}