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

We consider the Klein-Gordon and sine-Gordon type equations with a point-likepotential, which describes the wave phenomenon in disordered media with adefect. The singular potential term yields a critical phenomenon--that is, thesolution behavior around the critical parameter value bifurcates into twoextreme cases. Pinpointing the critical value with arbitrary accuracy is evenmore challenging. In this work, we adopt the generalized polynomial chaos (gPC)method to determine the critical values and the mean solutions around suchvalues. First, we consider the critical value associated with the strength ofthe singular potential for the Klein-Gordon equation. We expand the solution inthe random variable associated with the parameter. The obtained partialdifferential equations are solved using the Chebyshev collocation method. Dueto the existence of the singularity, the Gibbs phenomenon appears in thesolution, yielding a slow convergence of the numerically computed criticalvalue. To deal with the singularity, we adopt the consistent spectralcollocation method. The gPC method, along with the consistent Chebyshev method,determines the critical value and the mean solution highly efficiently. We thenconsider the sine-Gordon equation, for which the critical value is associatedwith the initial velocity of the kink soliton solution. The critical behaviorin this case is that the solution passes through (particle-pass), is trapped by(particle-capture), or is reflected by (particle-reflection) the singularpotential if the initial velocity of the soliton solution is greater than,equal to, or less than the critical value, respectively. We use the gPC meanvalue rather than reconstructing the solution to find the critical parameter.Numerical results show that the critical value can be determined efficientlyand accurately by using the proposed method.

Efficient determination of the critical parameters and the statistical  quantities for Klein-Gordon and sine-Gordon equations with a singular  potential using generalized polynomial chaos methods