On the variational structure of breather solutions II: periodic mKdV equation




Alejo, Miguel
Munoz, Claudio
Palacios, Jose

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Texas State University, Department of Mathematics


We study the periodic modified KdV equation, where a periodic in space and time breather solution is known from the work of Kevrekidis et al. [19]. We show that these breathers satisfy a suitable elliptic equation, and we also discuss via numerics its spectral stability. We also identify a source of nonlinear instability for the case described in [19], and we conjecture that, even if spectral stability is satisfied, nonlinear stability/instability depends only on the sign of a suitable discriminant function, a condition that is trivially satisfied in the case of non-periodic (in space) mKdV breathers. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.



Modified KdV, Sine-Gordon equations, Periodic mKdV, Integrability, Breather, Stability


Alejo, M. A., Muñoz, C., & Palacios, J. M. (2017). On the variational structure of breather solutions II: periodic mKdV equation. <i>Electronic Journal of Differential Equations, 2017</i>(56), pp. 1-26.


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