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Session S22 - Deterministic and probabilistic aspects of nonlinear evolution equations

Thursday, July 22, 16:35 ~ 17:05 UTC-3

Instability of soliton-type profiles on metric graphs

Jaime Angulo Pava

Department of Mathematics, IME-USP, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakfa978468cbc5dc5e7e1a66b4115f9198').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyfa978468cbc5dc5e7e1a66b4115f9198 = '&#97;ng&#117;l&#111;' + '&#64;'; addyfa978468cbc5dc5e7e1a66b4115f9198 = addyfa978468cbc5dc5e7e1a66b4115f9198 + '&#105;m&#101;' + '&#46;' + '&#117;sp' + '&#46;' + 'br'; var addy_textfa978468cbc5dc5e7e1a66b4115f9198 = '&#97;ng&#117;l&#111;' + '&#64;' + '&#105;m&#101;' + '&#46;' + '&#117;sp' + '&#46;' + 'br';document.getElementById('cloakfa978468cbc5dc5e7e1a66b4115f9198').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyfa978468cbc5dc5e7e1a66b4115f9198 + '\'>'+addy_textfa978468cbc5dc5e7e1a66b4115f9198+'<\/a>';

In this talk we shed new light on the mathematical studies of nonlinear dispersive evolution equations on metric graphs. This trend has been mainly motivated by the demand of reliable mathematical models for diferent phenomena in branched systems which, in meso- or nano-scales, resemble a thin neighborhood of a graph, such as Josephson junction networks, electric circuits, blood pressure waves in large arteries, or nerve impulses in complex arrays of neurons, just to mention a few examples. Our dynamic problems here will be essentially related to the sine-Gordon model on a $\mathcal Y$-junction graph. We establish a general linear instability criterium for solitons profiles associated to nonlinear dispersive evolution equations on metric graphs. In particular, we see that some kink or kink/anti-kink soliton profiles for the sine-Gordon model are linearly (and nonlinearly) unstable.

The arguments presented in this talk have prospects for the study of the instability of soliton-profiles solutions of other nonlinear evolution equations on branched systems.

Joint work with Ramón G. Plaza (IIMAS, UNAM, Cd. de México, México).