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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1)$\cong$ GL(2,$\mathbb R$) $\cong$ M\"{o}bius group point of view, which mightbe a keystone to exactly solve some special non-linear differential equations.If we construct the $N$-soliton solutions through the KdV type B\"{a}cklundtransformation, we can transform different KdV/mKdV/sinh-Gordon equations andthe B\"{a}cklund transformations of the standard form into the same commonHirota form and the same common B\"{a}cklund transformation except the equationwhich has the time-derivative term. The difference is only the time-dependenceand the main structure of the $N$-soliton solutions has same common form forKdV/mKdV/sinh-Gordon systems. Then the $N$-soliton solutions for thesinh-Gordon equation is obtained just by the replacement from KdV/mKdV$N$-soliton solutions. We also give general addition formulae coming from theKdV type B\"{a}cklund transformation which plays not only an important role toconstruct the trigonometric/hyperbolic $N$-soliton solutions but also anessential role to construct the elliptic $N$-soliton solutions. In contrast tothe KdV type B\"{a}cklund transformation, the well-known mKdV/sinh-Gordon typeB\"{a}cklund transformation gives the non-cyclic symmetric $N$-solitonsolutions. We give an explicit non-cyclic symmetric 3-soliton solution forKdV/mKdV/sinh-Gordon equations.

Common Hirota Form B\"{a}cklund Transformation for the Unified Soliton System