# Applied Math Seminar

#### On Novel Solitary Patterns in a Class of Klein-Gordon Equations

**Speaker:**
Philip Rosenau, Tel-Aviv University

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, November 8, 2024, 2:30 p.m.

**Synopsis:**

We study the emergence, stability and evolution of solitons and compactons in a class of Klein-Gordon equations

\(u_{tt} - u_{xx} + u = u^{1+n} - \kappa_{1+2n} u^{1+2n}, \quad n= 1, 2, \ldots\)

endowed with trivial and non-trivial stable equilibria, and demonstrate that similarly to the classical \(\kappa_{1+2n} = 0\) cases, solitons are linearly unstable, but their instability weakens as \(\kappa_{1+2n}\) increases, and vanishes at a critical \(\kappa_{1+2n}^{crit} = (1+n)/(2+n)^2\), where solitons disappear and kinks form.

As the growing amplitude of the unstable soliton approaches the non-trivial equilibrium, it morphs into ”meson”, a robust box shaped sharp pulse with a flat-top plateau which expands at a sonic speed. In the \(\kappa_{1+2n}^{crit}\) vicinity, where the instability is suppressed, and the internal modes hardly change, solitons persist for a very long time and rather than turn into meson, convert to breather.

Linear damping tempers the conversion and slows it. When \(-1/2 < n < 0\), compactons emerge and being unstable morph either to meson or to breather.

Joint work with Slava Krylov.