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Periodic Travelling Waves and its Inter-relation with Solitons for the 2D abc-Boussinesq System

Commun. Math. Anal.
Volume 20, Number 1 (2017), 27 - 49

Periodic Travelling Waves and its Inter-relation with Solitons for the 2D abc-Boussinesq System

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Abstract

Via a variational approach involving Concentration-Compactness principle, we show the existence of $x$-periodic travelling wave solutions for a general 2D-Boussinesq system that arises in the study of the evolution of long water waves with small amplitude in the presence of surface tension. We also establish that $x$-periodic travelling waves have almost the same shape of solitons as the period tends to infinity, by showing that a special sequence of $x$-periodic travelling wave solutions parameterized by the period converges to a solitary wave in a appropriate sense.