English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, convergent, multi-infinite, series solutions are derived forthe homoclinic orbits of a canonical fourth-order ODE system, in bothreversible and non-reversible cases. This ODE includes traveling-wavereductions of many important nonlinear PDEs or PDE systems, for which theseanalytical solutions would correspond to regular or localized pulses of thePDE. As such, the homoclinic solutions derived here are clearly topical, andthey are shown to match closely to earlier results obtained by homoclinicnumerical shooting. In addition, the results for the non-reversible case gobeyond those that have been typically considered in analyses conducted withinbifurcation-theoretic settings.  We also comment on generalizing the treatment here to parameter regimes wheresolutions homoclinic to exponentially small periodic orbits are known to exist,as well as another possible extension placing the solutions derived here withinthe framework of a comprehensive categorization of ALL possible traveling-wavesolutions, both smooth and non-smooth, for our governing ODE.

Convergent Analytic Solutions for Homoclinic Orbits in Reversible and Non-reversible Systems