Soliton dynamics for classical scalar fields

Abstract: I will present some of my recent work with Jacek Jendrej. We study classical scalar fields in dimension 1+1 with a symmetric double-well self-interaction potential. Examples of such equations are the phi-4 model and the sine-Gordon equation. These nonlinear wave equations admit non-trivial static solutions called kinks and antikinks, which are amongst the simplest examples of topological solitons. We define an n-kink cluster to be a solution approaching, for large positive times, a superposition of n alternating kinks and antikinks whose velocities converge to zero and mutual distances grow to infinity. Our main result is a determination of the leading order asymptotic behavior of any n-kink cluster. We use this information to construct the n-dimensional invariant manifold of n-kink clusters, which plays the dynamical role of the stable/unstable manifold for an ideal "critical point at infinity” given by well separated multi-kink configurations. In this context we also explain the role of kink clusters as universal profiles for the formation/annihilation of multikink configurations.

analysis of PDEs

Audience: researchers in the topic

Comments: Session Chair: Atanas Stefanov, University of Alabama, Birminghan (stefanov@uab.edu)

---

Seminar In the Analysis and Methods of PDE (SIAM PDE)

Series comments: The Editorial Board of SIMA and the officers of the SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) are launching the Seminar In the Analysis and Methods of PDE (SIAM PDE) to be held on the first Thursday of the month at 11:30am ET, except in August and January. The COVID-19 pandemic and consequent social distancing call for online venues of research dissemination. This webinar will serve as a way to recognize achievements in our area, and it will be expected to help promoting the standing of both SIMA and APDE. Further, these webinars should be beneficial to mathematicians who are in relatively isolated places.

This webinar is intended to continue well beyond this social distancing period. There are two type of SIAM PDE webinars: (i) Colloquium-type webinars by outstanding speakers presenting a survey of results on a topic where critical progress was recently achieved (and not necessarily published in SIMA); (ii) Webinars by outstanding authors based on some of the best articles accepted by SIMA. To launch the webinar series the initial six seminars will be of the Colloquium-type.

The talks are open to the public and, due to security reasons, all attendees have to register by following the link siam.zoom.us/webinar/register/WN_wShvDipBQw6WafNN6QO5wg

The link will contain detailed information about the talks. The registration is quick (asks only for your name and email) and, once registered, you will receive an email with the link to the webinars, which is unique to you, so please do not share that email. The registration is valid for all this webinar series, and you will receive a reminder about future talks.

Further, you will receive a follow-up email up to 7 days after each webinar, with a link to the on-demand SIAM PDE YouTube playlist recording.

Updated information and schedule will be available at sinews.siam.org/Details-Page/seminar-in-the-analysis-and-methods-of-pde

The SIAM PDE Webinar was initiated on June 2020:

By the SIMA Editors:

Robert Lipton, SIMA Editor-in-Chief (Louisiana State University, USA)

Athanasios Tzavaras (KAUST, Saudi Arabia)

By the SIAG/APDE Officers:

---