| Abstract: |
| In this talk, we report our recent work on the long-time asymptotics of good Boussinesq equation and Sawada-Kotera equation with decaying initial data. Especially, the self-similar Painlev\`{e} regions in the two integrable systems are investigated in detail. For the good Boussinesq equation, the self-similar region is described by the Painlev\`{e} IV equation, while for the Sawada-Kotera equation, the self-similar region is described by the fourth-order analogues of Painlev\`{e} transcendent. The Miura transformations along with the modified Boussinesq equation and modified Sawada-Kotera equation are used in the asymptotic analysis. |
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