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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In the present work we revisit the problem of the generalized Korteweg-deVries equation parametrically, as a function of the relevant nonlinearityexponent, to examine the emergence of blow-up solutions, as traveling waveformslose their stability past a critical point of the relevant parameter $p$, hereat $p=5$. We provide a {\it normal form} of the associated collapse dynamicsand illustrate how this captures the collapsing branch bifurcating from theunstable traveling branch. We also systematically characterize thelinearization spectrum of not only the traveling states, but importantly of theemergent collapsing waveforms in the so-called co-exploding frame where thesewaveforms are identified as stationary states. This spectrum, in addition totwo positive real eigenvalues which are shown to be associated with thesymmetries of translation and scaling invariance of the original(non-exploding) frame features complex patterns of negative eigenvalues that wealso fully characterize. We show that the phenomenology of the latter issignificantly affected by the boundary conditions and is far more complicatedthan in the corresponding symmetric Laplacian case of the nonlinearSchr{\"o}dinger problem that has recently been explored. In addition, weexplore the dynamics of the unstable solitary waves for $p>5$ in theco-exploding frame.

Self-similar blow-up solutions in the generalized Korteweg-de Vries equation: Spectral analysis, normal form and asymptotics